17 files changed, 6913 insertions, 25 deletions
diff --git a/src/Configuration.php b/src/Configuration.php
index ed9a342..6d1f03f 100644
--- a/src/Configuration.php
+++ b/src/Configuration.php
@@ -79,6 +79,11 @@ class Configuration
     protected $keepClasses = false;
 
     /**
+     * @var bool
+     */
+    protected $disableJSONLD = false;
+
+    /**
      * Configuration constructor.
      *
      * @param array $params
@@ -379,6 +384,26 @@ class Configuration
     /**
      * @return bool
      */
+    public function getDisableJSONLD()
+    {
+        return $this->disableJSONLD;
+    }
+
+    /**
+     * @param bool $disableJSONLD
+     *
+     * @return $this
+     */
+    public function setDisableJSONLD($disableJSONLD)
+    {
+        $this->disableJSONLD = $disableJSONLD;
+
+        return $this;
+    }
+
+    /**
+     * @return bool
+     */
     public function getSummonCthulhu()
     {
         return $this->summonCthulhu;
diff --git a/src/Readability.php b/src/Readability.php
index 23d7f33..9197891 100644
--- a/src/Readability.php
+++ b/src/Readability.php
@@ -94,6 +94,13 @@ class Readability
     private $logger;
 
     /**
+     * JSON-LD
+     *
+     * @var array
+     */
+    private $jsonld = [];
+
+    /**
      * Collection of attempted text extractions.
      *
      * @var array
@@ -319,6 +326,9 @@ class Readability
         // Unwrap image from noscript
         $this->unwrapNoscriptImages($dom);
 
+        // Extract JSON-LD metadata before removing scripts
+        $this->jsonld = $this->configuration->getDisableJSONLD() ? [] : $this->getJSONLD($dom);
+
         $this->removeScripts($dom);
 
         $this->prepDocument($dom);
@@ -329,6 +339,98 @@ class Readability
     }
 
     /**
+     * Try to extract metadata from JSON-LD object.
+     * For now, only Schema.org objects of type Article or its subtypes are supported.
+     *
+     * @param DOMDocument $dom
+     * @return Object with any metadata that could be extracted (possibly none)
+     */
+    private function getJSONLD(DOMDocument $dom)
+    {
+        $scripts = $this->_getAllNodesWithTag($dom, ['script']);
+
+        $jsonLdElement = $this->findNode($scripts, function($el) {
+            return $el->getAttribute('type') === 'application/ld+json';
+        });
+
+        if ($jsonLdElement) {
+            try {
+                // Strip CDATA markers if present
+                $content = preg_replace('/^\s*<!\[CDATA\[|\]\]>\s*$/', '', $jsonLdElement->textContent);
+                $parsed = json_decode($content, true);
+                $metadata = [];
+                if (
+                    !isset($parsed['@context']) ||
+                    !preg_match('/^https?\:\/\/schema\.org$/', $parsed['@context'])
+                ) {
+                    return $metadata;
+                }
+
+                if (!isset($parsed['@type']) && isset($parsed['@graph']) && is_array($parsed['@graph'])) {
+                    $_found = null;
+                    foreach ($parsed['@graph'] as $it) {
+                        if (isset($it['@type']) && is_string($it['@type']) && preg_match(NodeUtility::$regexps['jsonLdArticleTypes'], $it['@type'])) {
+                            $_found = $it;
+                        }
+                    }
+                    $parsed = $_found;
+                }
+
+                if (
+                    !$parsed ||
+                    !isset($parsed['@type']) ||
+                    !is_string($parsed['@type']) ||
+                    !preg_match(NodeUtility::$regexps['jsonLdArticleTypes'], $parsed['@type'])
+                ) {
+                    return $metadata;
+                }
+                if (isset($parsed['name']) && is_string($parsed['name'])) {
+                    $metadata['title'] = trim($parsed['name']);
+                } elseif (isset($parsed['headline']) && is_string($parsed['headline'])) {
+                    $metadata['title'] = trim($parsed['headline']);
+                }
+                if (isset($parsed['author'])) {
+                    if (isset($parsed['author']['name']) && is_string($parsed['author']['name'])) {
+                        $metadata['byline'] = trim($parsed['author']['name']);
+                    } elseif (
+                        is_array($parsed['author']) && 
+                        isset($parsed['author'][0]) && 
+                        is_array($parsed['author'][0]) && 
+                        isset($parsed['author'][0]['name']) && 
+                        is_string($parsed['author'][0]['name'])
+                    ) {
+                        $metadata['byline'] = array_filter($parsed['author'], function($author) {
+                            return is_array($author) && isset($author['name']) && is_string($author['name']);
+                        });
+                        $metadata['byline'] = array_map(function($author) {
+                            return trim($author['name']);
+                        }, $metadata['byline']);
+                        $metadata['byline'] = implode(', ', $metadata['byline']);
+                    }
+                }
+                if (isset($parsed['description']) && is_string($parsed['description'])) {
+                    $metadata['excerpt'] = trim($parsed['description']);
+                }
+                if (
+                    isset($parsed['publisher']) &&
+                    is_array($parsed['publisher']) &&
+                    isset($parsed['publisher']['name']) &&
+                    is_string($parsed['publisher']['name'])
+                ) {
+                    $metadata['siteName'] = trim($parsed['publisher']['name']);
+                }
+                return $metadata;
+            } catch (\Exception $err) {
+                // The try-catch blocks are from the JS version. Not sure if there's anything
+                // here in the PHP version that would trigger an error or exception, so perhaps we can 
+                // remove the try-catch blocks here (or at least translate errors to exceptions for this bit)
+                $this->logger->debug('[JSON-LD] Error parsing: '.$err->getMessage());
+            }
+        }
+        return [];
+    }
+
+    /**
      * Tries to guess relevant info from metadata of the html. Sets the results in the Readability properties.
      */
     private function getMetadata()
@@ -392,7 +494,11 @@ class Readability
             'twitter:title'
         ], array_keys($values)));
 
-        $this->setTitle(isset($values[$key]) ? trim($values[$key]) : null);
+        if (isset($this->jsonld['title'])) {
+            $this->setTitle($this->jsonld['title']);
+        } else {
+            $this->setTitle(isset($values[$key]) ? trim($values[$key]) : null);
+        }
 
         if (!$this->getTitle()) {
             $this->setTitle($this->getArticleTitle());
@@ -405,7 +511,11 @@ class Readability
             'author'
         ], array_keys($values)));
 
-        $this->setAuthor(isset($values[$key]) ? $values[$key] : null);
+        if (isset($this->jsonld['byline'])) {
+            $this->setAuthor($this->jsonld['byline']);
+        } else {
+            $this->setAuthor(isset($values[$key]) ? $values[$key] : null);
+        }
 
         // get description
         $key = current(array_intersect([
@@ -418,7 +528,11 @@ class Readability
             'twitter:description'
         ], array_keys($values)));
 
-        $this->setExcerpt(isset($values[$key]) ? $values[$key] : null);
+        if (isset($this->jsonld['excerpt'])) {
+            $this->setExcerpt($this->jsonld['excerpt']);
+        } else {
+            $this->setExcerpt(isset($values[$key]) ? $values[$key] : null);
+        }
 
         // get main image
         $key = current(array_intersect([
@@ -433,7 +547,11 @@ class Readability
             'og:site_name'
         ], array_keys($values)));
 
-        $this->setSiteName(isset($values[$key]) ? $values[$key] : null);
+        if (isset($this->jsonld['siteName'])) {
+            $this->setSiteName($this->jsonld['siteName']);
+        } else {
+            $this->setSiteName(isset($values[$key]) ? $values[$key] : null);
+        }
 
         // in many sites the meta value is escaped with HTML entities,
         // so here we need to unescape it
@@ -1993,6 +2111,24 @@ class Readability
     }
 
     /**
+     * Iterate over a NodeList, and return the first node that passes
+     * the supplied test function
+     *
+     * @param  NodeList nodeList The NodeList.
+     * @param  Function fn       The test function.
+     * @return DOMNode|null
+     */
+    private function findNode(array $nodeList, callable $fn)
+    {
+        foreach ($nodeList as $node) {
+            if ($fn($node)) {
+                return $node;
+            }
+        }
+        return null;
+    }
+
+    /**
      * @return null|string
      */
     public function __toString()
diff --git a/test/test-pages/aclu/expected-metadata.json b/test/test-pages/aclu/expected-metadata.json
index 9b20703..ebb86ef 100644
--- a/test/test-pages/aclu/expected-metadata.json
+++ b/test/test-pages/aclu/expected-metadata.json
@@ -1,8 +1,8 @@
 {
-    "Author": "By Daniel Kahn Gillmor, Senior Staff Technologist, ACLU Speech, Privacy, and Technology Project",
+    "Author": "Daniel Kahn Gillmor",
     "Direction": "ltr",
-    "Excerpt": "I don't use Facebook. I'm not technophobic — I'm a geek. I've been using email since the early 1990s, I have accounts on hundreds of services around the net, and I do software development and internet protocol design both for work and for fun. I believe that a globe-spanning communications network like the internet can be a positive social force, and I publish much of my own work on the open web.",
+    "Excerpt": "Facebook collects data about people who have never even opted in. But there are ways these non-users can protect themselves.",
     "Image": "https:\/\/www.aclu.org\/sites\/default\/files\/styles\/metatag_og_image_1200x630\/public\/field_share_image\/web18-facebook-socialshare-1200x628-v02.png?itok=p77cQjOm",
-    "Title": "Facebook Is Tracking Me Even Though I’m Not on Facebook",
+    "Title": "Facebook Is Tracking Me Even Though I\u2019m Not on Facebook",
     "SiteName": "American Civil Liberties Union"
-}
+}
+\ No newline at end of file
diff --git a/test/test-pages/bbc-1/expected-metadata.json b/test/test-pages/bbc-1/expected-metadata.json
index 6af682f..8d5937b 100644
--- a/test/test-pages/bbc-1/expected-metadata.json
+++ b/test/test-pages/bbc-1/expected-metadata.json
@@ -3,6 +3,6 @@
     "Direction": null,
     "Excerpt": "President Barack Obama tells the BBC his failure to pass \"common sense gun safety laws\" is the greatest frustration of his presidency.",
     "Image": "http:\/\/ichef.bbci.co.uk\/news\/1024\/cpsprodpb\/3D8B\/production\/_84455751_84455749.jpg",
-    "Title": "Obama admits US gun laws are his 'biggest frustration' - BBC News",
+    "Title": "Obama admits US gun laws are his 'biggest frustration'",
     "SiteName": "BBC News"
-}
+}
+\ No newline at end of file
diff --git a/test/test-pages/folha/expected-images.json b/test/test-pages/folha/expected-images.json
new file mode 100644
index 0000000..a449683
--- /dev/null
+++ b/test/test-pages/folha/expected-images.json
@@ -0,0 +1,3 @@
+[
+    "https:\/\/f.i.uol.com.br\/fotografia\/2018\/12\/21\/15454034955c1cfc67131dc_1545403495_3x2_rt.jpg"
+]
+\ No newline at end of file
diff --git a/test/test-pages/folha/expected-metadata.json b/test/test-pages/folha/expected-metadata.json
new file mode 100644
index 0000000..4d12641
--- /dev/null
+++ b/test/test-pages/folha/expected-metadata.json
@@ -0,0 +1,8 @@
+{
+    "Author": "Bruno (Henrique Zecchin) Rodrigues",
+    "Direction": null,
+    "Excerpt": "Na ocasi\u00e3o, t\u00e9cnico do Corinthians entregou r\u00e9plica do trof\u00e9u ao ex-presidente",
+    "Image": "https:\/\/f.i.uol.com.br\/fotografia\/2018\/12\/21\/15454034955c1cfc67131dc_1545403495_3x2_rt.jpg",
+    "Title": "Tite diz que errou ao levar ta\u00e7a da Libertadores a Lula em 2012",
+    "SiteName": "Folha de S.Paulo"
+}
+\ No newline at end of file
diff --git a/test/test-pages/folha/expected.html b/test/test-pages/folha/expected.html
new file mode 100644
index 0000000..13f3ac3
--- /dev/null
+++ b/test/test-pages/folha/expected.html
@@ -0,0 +1,24 @@
+<div data-share-text data-news-content-text data-disable-copy data-continue-reading data-continue-reading-hide-others=".js-continue-reading-hidden" itemprop="articleBody">
+                                                <p>
+                                                    Após rechaçar <a href="https://www1.folha.uol.com.br/esporte/2018/12/tite-se-recusa-a-encontrar-bolsonaro-antes-da-disputa-da-copa-america.shtml">um encontro da seleção brasileira com o presidente eleito Jair</a> <a href="https://www1.folha.uol.com.br/esporte/2018/12/tite-se-recusa-a-encontrar-bolsonaro-antes-da-disputa-da-copa-america.shtml">Bolsonaro</a>, o técnico Tite declarou que errou ao levar a taça da Copa Libertadores de 2012, conquistada pelo Corinthians, ao ex-presidente Luiz Inácio Lula da Silva.
+                                                </p>
+                                                <p>
+                                                    Ao lado de representantes do clube paulista, o atual comandante do Brasil ainda entregou uma réplica do troféu a Lula.
+                                                </p>
+                                                <p>
+                                                    "Em 2012 eu errei. Ele não era presidente, mas fui ao Instituto e mandei felicitações por um aniversário. Não me posicionei politicamente. Não tenho partido político, tenho sim a torcida para que o Brasil seja melhor em igualdade social. E que nossas prioridades sejam educação e punição. Que seja dada a possibilidade de estudo ao garoto de São Braz, que não tem chão batido para ir à escola, ou da periferia de Caixas ou do morro do Rio de Janeiro. Seja dada a ele a prioridade de estudo e não a outras situações", falou Tite ao programa "Grande Círculo", que ainda irá ao ar no SporTV.
+                                                </p>
+                                                <p>
+                                                    Na ocasião, Tite e outros representantes do Corinthians <a href="https://www1.folha.uol.com.br/poder/1124743-corinthians-leva-a-taca-da-libertadores-para-lula.shtml">foram ao Instituto Lula para mostrar a taça</a> original da Libertadores ao ex-presidente.
+                                                </p>
+                                                
+                                                <p>
+                                                    O assunto foi levantado&nbsp;porque recentemente Tite foi questionado se aceitaria um encontro da seleção brasileira com Bolsonaro em uma conquista de título ou <a href="https://www1.folha.uol.com.br/esporte/2018/12/selecao-brasileira-jogara-duas-vezes-em-sao-paulo-na-copa-america.shtml">antes da Copa América de 2019</a>, por exemplo. O treinador deixou claro que preferiria evitar esse tipo de formalidade.
+                                                </p>
+                                                <p>
+                                                    Apesar disso, Tite não questionou a ação de Palmeiras e CBF, que <a href="https://www1.folha.uol.com.br/esporte/2018/12/cbf-usa-festa-do-palmeiras-para-se-aproximar-de-governo-bolsonaro.shtml">convidaram Bolsonaro para a festa do título do Campeonato Brasileiro</a>. O presidente eleito até levantou a taça conquistada pelo clube alviverde.
+                                                </p>
+                                                <p>
+                                                    "Em 2012 eu fiz e errei. O protocolo e a situação gerada no jogo do Palmeiras são fatos de opinião pessoal. CBF e Palmeiras, enquanto instituições têm a opinião. Errei lá atrás, não faria com o presidente antes da Copa e nem agora porque entendo que misturar esporte e política não é legal. Fiz errado lá atrás? Sim. Faria de novo? Não", acrescentou o comandante.
+                                                </p>
+                                            </div>
+\ No newline at end of file
diff --git a/test/test-pages/folha/source.html b/test/test-pages/folha/source.html
new file mode 100644
index 0000000..03f749c
--- /dev/null
+++ b/test/test-pages/folha/source.html
@@ -0,0 +1,2518 @@
+<!DOCTYPE html>
+<html lang="pt-BR" data-version="prod@5391c020" xmlns="http://www.w3.org/1999/xhtml" xml:lang="pt-BR">
+    <head>
+        <meta charset="utf-8" />
+        <meta http-equiv="X-UA-Compatible" content="IE=edge" />
+        <meta property="desliga:paywall" content="false" />
+        <title>
+            Tite diz que errou ao levar taça da Libertadores a Lula em 2012 - 21/12/2018 - Esporte - Folha
+        </title>
+        <meta name="description" content="Após rechaçar um encontro da seleção brasileira com o presidente eleito Jair Bolsonaro, o técnico Tite declarou que errou ao levar a taça da Copa Libertadores de 2012, conquistada pelo Corinthians, ao ex-presidente Luiz Inácio Lula da Silva." />
+        <meta name="keywords" content="Futebol, corinthians, libertadores, folha" />
+        <meta http-equiv="Content-Security-Policy" content="upgrade-insecure-requests" />
+        <link rel="canonical" href="https://www1.folha.uol.com.br/esporte/2018/12/tite-diz-que-errou-ao-levar-taca-da-libertadores-a-lula-em-2012.shtml" />
+        <link rel="amphtml" href="https://www1.folha.uol.com.br/amp/esporte/2018/12/tite-diz-que-errou-ao-levar-taca-da-libertadores-a-lula-em-2012.shtml" /><!-- Open Graph Meta Tags -->
+        <meta property="og:description" content="Após rechaçar um encontro da seleção brasileira com o presidente eleito Jair Bolsonaro, o técnico Tite declarou que errou ao levar a taça da Copa Libertadores de 2012, conquistada pelo Corinthians, ao ex-presidente Luiz Inácio Lula da Silva." />
+        <meta property="og:image" content="https://f.i.uol.com.br/fotografia/2018/12/21/15454034955c1cfc67131dc_1545403495_3x2_rt.jpg" />
+        <meta property="og:image:height" content="1600" />
+        <meta property="og:image:type" content="image/jpeg" />
+        <meta property="og:image:width" content="2400" />
+        <meta property="og:locale" content="pt_BR" />
+        <meta property="og:site_name" content="Folha de S.Paulo" />
+        <meta property="og:title" content="Tite diz que errou ao levar taça da Libertadores a Lula em 2012" />
+        <meta property="og:type" content="article" />
+        <meta property="og:url" content="https://www1.folha.uol.com.br/esporte/2018/12/tite-diz-que-errou-ao-levar-taca-da-libertadores-a-lula-em-2012.shtml" />
+        <meta property="article:published_time" content="2018-12-21 12:55:00" />
+        <meta property="article:modified_time" content="2018-12-21 12:56:02" />
+        <meta property="article:section" content="Esporte" />
+        <meta property="article:tag" content="Folha de S.Paulo" />
+        <meta property="article:tag" content="Futebol" />
+        <meta property="article:tag" content="corinthians" />
+        <meta property="article:tag" content="libertadores" /><!-- /Open Graph Meta Tags  -->
+        <!-- Twitter Meta Tags -->
+        <meta name="twitter:card" content="summary_large_image" />
+        <meta name="twitter:image" content="https://f.i.uol.com.br/fotografia/2018/12/21/15454034955c1cfc67131dc_1545403495_3x2_rt.jpg" />
+        <meta name="twitter:creator" content="folha" />
+        <meta name="twitter:description" content="Após rechaçar um encontro da seleção brasileira com o presidente eleito Jair Bolsonaro, o técnico Tite declarou que errou ao levar a taça da Copa Libertadores de 2012, conquistada pelo Corinthians, ao ex-presidente Luiz Inácio Lula da Silva." />
+        <meta name="twitter:site" content="folha" />
+        <meta name="twitter:title" content="Tite diz que errou ao levar taça da Libertadores a Lula em 2012 - 21/12/2018 - Esporte - Folha" /><!-- /Twitter Meta Tags -->
+        <meta name="viewport" content="width=device-width, initial-scale=1" />
+        <meta name="mobile-web-app-capable" content="yes" />
+        <link rel="manifest" href="/manifest.json" />
+        <link rel="apple-touch-icon" href="//f.i.uol.com.br/hunting/folha/1/common/icons/apple-touch-icon.png" />
+        <link rel="apple-touch-icon" sizes="76x76" href="//f.i.uol.com.br/hunting/folha/1/common/icons/touch-icon-ipad.png" />
+        <link rel="apple-touch-icon" sizes="120x120" href="//f.i.uol.com.br/hunting/folha/1/common/icons/touch-icon-iphone-retina.png" />
+        <link rel="apple-touch-icon" sizes="152x152" href="//f.i.uol.com.br/hunting/folha/1/common/icons/touch-icon-ipad-retina.png" />
+        <link rel="apple-touch-startup-image" href="//f.i.uol.com.br/hunting/folha/1/common/splash/launch-640x1136.png" media="(device-width: 320px) and (device-height: 568px) and (-webkit-device-pixel-ratio: 2) and (orientation: portrait)" />
+        <link rel="apple-touch-startup-image" href="//f.i.uol.com.br/hunting/folha/1/common/splash/launch-750x1294.png" media="(device-width: 375px) and (device-height: 667px) and (-webkit-device-pixel-ratio: 2) and (orientation: portrait)" />
+        <link rel="apple-touch-startup-image" href="//f.i.uol.com.br/hunting/folha/1/common/splash/launch-1242x2148.png" media="(device-width: 414px) and (device-height: 736px) and (-webkit-device-pixel-ratio: 3) and (orientation: portrait)" />
+        <link rel="apple-touch-startup-image" href="//f.i.uol.com.br/hunting/folha/1/common/splash/launch-1125x2436.png" media="(device-width: 375px) and (device-height: 812px) and (-webkit-device-pixel-ratio: 3) and (orientation: portrait)" />
+        <link rel="apple-touch-startup-image" href="//f.i.uol.com.br/hunting/folha/1/common/splash/launch-1536x2048.png" media="(min-device-width: 768px) and (max-device-width: 1024px) and (-webkit-min-device-pixel-ratio: 2) and (orientation: portrait)" />
+        <link rel="apple-touch-startup-image" href="//f.i.uol.com.br/hunting/folha/1/common/splash/launch-1668x2224.png" media="(min-device-width: 834px) and (max-device-width: 834px) and (-webkit-min-device-pixel-ratio: 2) and (orientation: portrait)" />
+        <link rel="apple-touch-startup-image" href="//f.i.uol.com.br/hunting/folha/1/common/splash/launch-2048x2732.png" media="(min-device-width: 1024px) and (max-device-width: 1024px) and (-webkit-min-device-pixel-ratio: 2) and (orientation: portrait)" />
+        <link rel="icon" sizes="192x192" href="//f.i.uol.com.br/hunting/folha/1/common/icons/favicon-192.png" />
+        <link rel="shortcut icon" sizes="16x16 24x24 32x32 48x48 62x62" href="//f.i.uol.com.br/hunting/folha/1/common/icons/favicon.ico" />
+        <meta name="apple-mobile-web-app-capable" content="yes" />
+        <meta name="apple-mobile-web-app-status-bar-style" content="black" />
+        <meta name="apple-mobile-web-app-title" content="Folha de S.Paulo" />
+        <meta name="msapplication-TileImage" content="//f.i.uol.com.br/hunting/folha/1/common/icons/favicon-144.png" />
+        <meta name="msapplication-TileColor" content="#fff" />
+        <meta name="theme-color" content="#262626" />
+        <script>
+        <![CDATA[
+        //UOLTM
+        window.universal_variable = window.universal_variable || {};
+        window.universal_variable.dfp = {"custom_params":{"subchan":"","keyword":"corinthians"}};
+        ]]>
+        </script>
+        <script>
+        <![CDATA[
+        //UOLTM
+        window.uolads = window.uolads || [];
+        window.UOLPD = window.UOLPD || {};
+        (function() {
+        var uv = window.universal_variable, up = window.UOLPD;
+        uv.aud = { userType: "none", loginType: "none", userGroup: "none" };
+        up.dataLayer = { "t": String(Math.round(Math.random()*9)+1), "swgt": "sub_na" };
+        //
+        function uoltm() {
+        var scr = document.createElement( "script" );
+        scr.async = true;
+        scr.src = "https://tm.jsuol.com.br/uoltm.js?id=1p0oxf";
+        var el = document.getElementsByTagName( "script" )[0];
+        el.parentNode.insertBefore( scr, el );
+        };
+        if ( document.cookie.indexOf( "folha_ga_userType" ) >= 0 ) {
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+                            <div class="c-avatar" data-avatar="">
+                                <div class="c-avatar__media js-avatar-media"></div>
+                            </div><a class="u-link-clean link-icon" href="#user-info" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="user-info" data-nodropdown=""><svg xmlns="https://www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                            <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                        </div>
+                        <ul class="c-list-menu__sublist collapse" id="user-info">
+                            <li class="c-list-menu__item">
+                                <a href="//login.folha.com.br/perfil" class="u-link-clean">Editar perfil</a>
+                            </li>
+                            <li class="c-list-menu__item">
+                                <a href="//login.folha.com.br/conta" class="u-link-clean">Senha e conta</a>
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+                            <li class="c-list-menu__item">
+                                <a href="//login.folha.com.br/newsletter" class="u-link-clean">Newsletter</a>
+                            </li>
+                            <li class="c-list-menu__item">
+                                <a href="//login.folha.com.br/assinante" class="u-link-clean">Assinatura</a>
+                            </li>
+                            <li class="c-list-menu__item">
+                                <a class="u-link-clean js-link-logout" href="#">Sair</a>
+                            </li>
+                        </ul>
+                    </li>
+                </ul><!-- /User -->
+                <div class="c-list-menu__divider"></div><!-- Nav -->
+                <nav>
+                    <!-- Main sections -->
+                    <ul class="c-list-menu c-list-menu--insider">
+                        <li class="c-list-menu__item">
+                            <a href="//www1.folha.uol.com.br/fsp/" class="u-link-clean">Edição Impressa</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a href="//www1.folha.uol.com.br/ultimas-noticias/" class="u-link-clean">Últimas</a>
+                        </li>
+                    </ul>
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+                    <ul class="c-list-menu c-list-menu--insider">
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/opiniao" class="u-link-clean">opinião</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-opiniao" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-opiniao"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-opiniao" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/folha-topicos/editoriais/">Editoriais</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://fotografia.folha.uol.com.br/galerias/nova/1618604763277631-charges-dezembro-2018">Charges</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/opiniao/tendenciasdebates/">Tendências/Debates</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/colunaseblogs" class="u-link-clean">colunas e blogs</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-colunas-e-blogs" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-colunas-e-blogs"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-colunas-e-blogs" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/colunaseblogs/#so-colunas">Colunas</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/colunaseblogs/#so-blogs">Blogs</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/maispopulares" class="u-link-clean">mais populares</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-mais-populares" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-mais-populares"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-mais-populares" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/maispopulares/#mais-lidas">Mais lidas</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/maispopulares/#mais-enviadas">Mais enviadas</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/maispopulares/#mais-comentadas">Mais comentadas</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://acervo.folha.uol.com.br/" class="u-link-clean">acervo folha</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-acervo-folha" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-acervo-folha"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-acervo-folha" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2017/ha-50-anos">Há 50 Anos</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2017/saiu-no-np">Saiu no NP</a>
+                                </li>
+                            </ul>
+                        </li>
+                    </ul>
+                    <div class="c-list-menu__divider"></div>
+                    <ul class="c-list-menu c-list-menu--insider">
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/poder/eleicoes/2018/" class="u-link-clean">eleições 2018</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-eleicoes-2018" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-eleicoes-2018"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-eleicoes-2018" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://aovivo.folha.uol.com.br/2018/08/16/5454-aovivo.shtml">em tempo real</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/poder/eleicoes/candidatos/">candidatos</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/poder/eleicoes/2018/apuracao/2turno/brasil/">apuração</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://temas.folha.uol.com.br/entrevistas-eleicoes-2018/introducao/o-que-eles-pensam.shtml">Entrevistas com candidatos</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/sabatinas/sabatinas.shtml">Sabatinas</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/folha-informacoes/">Folha Informações</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/debates/">Debates</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/pesquisas-eleitorais/">Pesquisas eleitorais</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://painel.blogfolha.uol.com.br/">Painel</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/poder/" class="u-link-clean">poder</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-poder" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-poder"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-poder" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/governo-bolsonaro/">Governo Bolsonaro</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2017/entrevista-da-segunda/">Entrevista da 2ª</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2014/petrolao/">lava jato</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/mercado" class="u-link-clean">mercado</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-mercado" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-mercado"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-mercado" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mercado/cifraseletras/">cifras &amp; letras</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mercado/consumo-consciente/">consumo consciente</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mercado/folhainvest/">folhainvest</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mercado/imposto-de-renda/">ir 2018</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/colunas/mercadoaberto/">mercado aberto</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://fotografia.folha.uol.com.br/galerias/nova/1618607510546314-hora-do-cafe-dezembro-de-2018">hora do café</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mpme/">mpme</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mercado/brasil-que-da-certo/">o brasil que dá certo</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mercado/previdencia/">previdência</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/colunas/vaivem/">vaivém das commodities</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://publicidadelegal.folha.uol.com.br/">Publicidade Legal</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/cotidiano" class="u-link-clean">cotidiano</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-cotidiano" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-cotidiano"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-cotidiano" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/educacao">educação</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/cotidiano/ao-seu-tempo/">ao seu tempo</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/febre-amarela/">febre amarela</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/noticias/2016/folha-verao/">Folha verão</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/cotidiano/riodejaneiro/">rio de janeiro</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/loterias/">loterias</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://tempo.folha.uol.com.br/">tempo</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/mundo" class="u-link-clean">mundo</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-mundo" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-mundo"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-mundo" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2017/coreia-do-norte/">coreia do norte</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2017/governo-trump/">governo trump</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/venezuela/">venezuela</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://arte.folha.uol.com.br/mundo/2018/hipercidades/">hipercidades</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/esporte/" class="u-link-clean">esporte</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-esporte" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-esporte"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-esporte" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/esporte/">esporte</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/campeonato-brasileiro/serie-a/">campeonato brasileiro</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/ilustrada" class="u-link-clean">ilustrada</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-ilustrada" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-ilustrada"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-ilustrada" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/colunas/monicabergamo/">mônica bergamo</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://www1.folha.uol.com.br/ilustrada/cartum/cartunsdiarios">quadrinhos</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://www1.folha.uol.com.br/comida/">comida</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/colunas/mauricio-meireles/">painel das letras</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://www1.folha.uol.com.br/ilustrissima">ilustríssima</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://f5.folha.uol.com.br" class="u-link-clean">F5</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-f5" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-f5"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-f5" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://f5.folha.uol.com.br/bichos">Bichos</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://f5.folha.uol.com.br/celebridades">celebridades</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://f5.folha.uol.com.br/horoscopo/">horóscopo</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://f5.folha.uol.com.br/voceviu">você viu?</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://f5.folha.uol.com.br/televisao/">televisão</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://guia.folha.uol.com.br">guia folha</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/ciencia" class="u-link-clean">ciência</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-ciencia" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-ciencia"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-ciencia" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/ambiente">Ambiente</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://www1.folha.uol.com.br/equilibrioesaude">equilíbrio e saúde</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/saopaulo" class="u-link-clean">sãopaulo</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-saopaulo" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-saopaulo"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-saopaulo" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/o-melhor-de-sao-paulo/restaurantes-bares-e-cozinha/2018/">o melhor de sãopaulo</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/saopaulo/viaja-sp/melhores/2018/">viaja sãopaulo</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/descubra-sao-paulo/">descubra sãopaulo</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/estradas/">estradas</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/">especiais</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://seminariosfolha.folha.com.br/">Seminários Folha</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/empreendedorsocial/" class="u-link-clean">empreendedor social</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-empreendedor-social" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-empreendedor-social"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-empreendedor-social" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/empreendedorsocial/fiis/">FIIS</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/top-of-mind/">top of mind</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="" class="u-link-clean">imagem</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-imagem" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-imagem"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-imagem" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://fotografia.folha.uol.com.br/galerias">fotografia</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/tv">tv folha</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="" class="u-link-clean">parceiros</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-parceiros" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-parceiros"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-parceiros" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/bbc/">BBC News Brasil</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/dw/">Deutsche Welle</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mundo/financialtimes/">Financial Times</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/especial/2018/globalmedia/">Global Media</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mundo/latinoamerica21/">Latino América 21</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/nova-escola/">Nova Escola</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://piaui.folha.uol.com.br/">Piauí</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/rfi/">Radio France Internationale</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mundo/nyt/">The New York Times</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mundo/wsj/">The Wall Street Journal</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mundo/washingtonpost/">The Washington Post</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://telepadi.folha.uol.com.br/">Telepadi</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://www1.folha.uol.com.br/banco-de-dados">banco de dados</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://www1.folha.uol.com.br/dias-melhores">dias melhores</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://www1.folha.uol.com.br/paineldoleitor">painel do leitor</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/sobretudo" class="u-link-clean">sobre tudo</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-sobre-tudo" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-sobre-tudo"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-sobre-tudo" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/sobretudo/carreiras">carreiras</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://classificados1.folha.uol.com.br/">classificados</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/sobretudo/morar">morar</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/sobretudo/rodas">rodas</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/sobretudo/vida-pratica">vida prática</a>
+                                </li>
+                            </ul>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://www1.folha.uol.com.br/turismo">turismo</a>
+                        </li>
+                        <li class="c-list-menu__item">
+                            <a class="u-link-clean" href="https://www1.folha.uol.com.br/serafina/">Serafina</a>
+                        </li>
+                    </ul>
+                    <div class="c-list-menu__divider"></div>
+                    <ul class="c-list-menu c-list-menu--insider">
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="" class="u-link-clean">mais seções</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-mais-secoes" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-mais-secoes"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-mais-secoes" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/internacional/es/">folha en español</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/internacional/en/">folha in english</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mapas/">Folha Mapas</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://folhaleaks.folha.com.br/">folhaleaks</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/tudosobre/">tudo sobre</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/folha-topicos/">folha tópicos</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://transparencia.folha.uol.com.br">folha transparência</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/fsp">versão impressa</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/mapa-do-site/">Mapa do site</a>
+                                </li>
+                            </ul>
+                        </li>
+                    </ul>
+                    <div class="c-list-menu__divider"></div>
+                    <ul class="c-list-menu c-list-menu--insider">
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/institucional/" class="u-link-clean">Sobre a Folha</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-sobre-a-folha" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-sobre-a-folha"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-sobre-a-folha" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://www1.folha.uol.com.br/institucional">sobre o grupo folha</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/institucional/expediente.shtml">expediente</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://www1.folha.uol.com.br/paineldoleitor/politicadeprivacidade/">política de privacidade</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://temas.folha.uol.com.br/projeto-editorial-da-folha/projeto-editorial-2017/">projeto editorial</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/treinamento/">treinamento</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://www1.folha.uol.com.br/folha/trabalhe/vagas.html">trabalhe na folha</a>
+                                </li>
+                            </ul>
+                        </li>
+                    </ul>
+                    <div class="c-list-menu__divider"></div>
+                    <ul class="c-list-menu c-list-menu--insider">
+                        <li class="c-list-menu__item">
+                            <div class="c-list-menu__wrap-links">
+                                <a href="https://www1.folha.uol.com.br/falecomafolha" class="u-link-clean">fale com a folha</a> <a class="u-link-clean c-list-menu__toggle link-icon" href="#sidebarmenu-fale-com-a-folha" data-toggle="collapse" data-icon=".icon--chevron-down" aria-expanded="false" aria-controls="sidebarmenu-fale-com-a-folha"><svg xmlns="//www.w3.org/2000/svg" viewbox="0 0 24 24" width="24" height="24" class="icon icon--chevron-down" aria-hidden="true">
+                                <path d="M 6.2499999,9.800625 7.60125,8.449375 12,12.838542 16.39875,8.449375 17.75,9.800625 l -5.75,5.75 -5.75,-5.75 z"></path></svg></a>
+                            </div>
+                            <ul class="c-list-menu__sublist collapse" id="sidebarmenu-fale-com-a-folha" aria-expanded="false">
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://www.publicidade.folha.com.br">anuncie (publicidade folha)</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="http://atendimento.folha.com.br">atendimento ao assinante</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/erramos">erramos</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/ombudsman">ombudsman</a>
+                                </li>
+                                <li class="c-list-menu__item">
+                                    <a class="u-link-clean" href="https://www1.folha.uol.com.br/paineldoleitor">painel do leitor</a>
+                                </li>
+                            </ul>
+                        </li>
+                    </ul>
+                    <div class="c-list-menu__divider"></div><!--!-->
+                    <!-- /Main sections -->
+                    <!-- Nav -->
+                    <aside class="c-sidebar-menu__group">
+                        <!-- Contrast -->
+                        <div class="c-contrast" data-set-contrast="">
+                            <h6 class="c-contrast__title">
+                                Contraste
+                            </h6>
+                            <div class="c-contrast__content">
+                                <div class="c-form__radio c-form__radio--inline">
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+                                                <h1 class="c-content-head__title" itemprop="headline">
+                                                    Tite diz que errou ao levar taça da Libertadores a Lula em 2012
+                                                </h1>
+                                                <h2 class="c-content-head__subtitle" itemprop="alternativeHeadline">
+                                                    Na ocasião, técnico do Corinthians entregou réplica do troféu ao ex-presidente
+                                                </h2>
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+                                                <div class="widget-image">
+                                                    <figure>
+                                                        <img alt="Luiz Inácio Lula da Silva (ao centro) recebe de Tite e outros representantes do Corinthians réplica da taça da Libertadores" class="img-responsive" srcset="https://f.i.uol.com.br/fotografia/2018/12/21/15454034955c1cfc67131dc_1545403495_3x2_th.jpg 100w, https://f.i.uol.com.br/fotografia/2018/12/21/15454034955c1cfc67131dc_1545403495_3x2_sm.jpg 480w, https://f.i.uol.com.br/fotografia/2018/12/21/15454034955c1cfc67131dc_1545403495_3x2_md.jpg 768w, https://f.i.uol.com.br/fotografia/2018/12/21/15454034955c1cfc67131dc_1545403495_3x2_lg.jpg 1024w, https://f.i.uol.com.br/fotografia/2018/12/21/15454034955c1cfc67131dc_1545403495_3x2_xl.jpg 1200w, https://f.i.uol.com.br/fotografia/2018/12/21/15454034955c1cfc67131dc_1545403495_3x2_rt.jpg 2400w" sizes="(min-width: 1024px) 68vw, 100vw" src="https://f.i.uol.com.br/fotografia/2018/12/21/15454034955c1cfc67131dc_1545403495_3x2_md.jpg" />
+                                                        <figcaption class="widget-image__subtitle">
+                                                            Luiz Inácio Lula da Silva (ao centro) recebe de Tite e outros representantes do Corinthians réplica da taça da Libertadores - <span class="widget-image__credits">Heinrich Aikawa/Instituto Lula</span>
+                                                        </figcaption>
+                                                    </figure>
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+                                                <time class="c-more-options__published-date" itemprop="datePublished" datetime="2018-12-21 12:55:00">21.dez.2018 às 12h55</time>
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+                                                <strong class="c-signature__location">São Paulo</strong>
+                                            </div>
+                                            <div class="c-news__body" data-share-text="" data-news-content-text="" data-disable-copy="" data-continue-reading="" data-continue-reading-hide-others=".js-continue-reading-hidden" itemprop="articleBody">
+                                                <p>
+                                                    Após rechaçar <a href="https://www1.folha.uol.com.br/esporte/2018/12/tite-se-recusa-a-encontrar-bolsonaro-antes-da-disputa-da-copa-america.shtml">um encontro da seleção brasileira com o presidente eleito Jair</a> <a href="https://www1.folha.uol.com.br/esporte/2018/12/tite-se-recusa-a-encontrar-bolsonaro-antes-da-disputa-da-copa-america.shtml">Bolsonaro</a>, o técnico Tite declarou que errou ao levar a taça da Copa Libertadores de 2012, conquistada pelo Corinthians, ao ex-presidente Luiz Inácio Lula da Silva.
+                                                </p>
+                                                <p>
+                                                    Ao lado de representantes do clube paulista, o atual comandante do Brasil ainda entregou uma réplica do troféu a Lula.
+                                                </p>
+                                                <p>
+                                                    "Em 2012 eu errei. Ele não era presidente, mas fui ao Instituto e mandei felicitações por um aniversário. Não me posicionei politicamente. Não tenho partido político, tenho sim a torcida para que o Brasil seja melhor em igualdade social. E que nossas prioridades sejam educação e punição. Que seja dada a possibilidade de estudo ao garoto de São Braz, que não tem chão batido para ir à escola, ou da periferia de Caixas ou do morro do Rio de Janeiro. Seja dada a ele a prioridade de estudo e não a outras situações", falou Tite ao programa "Grande Círculo", que ainda irá ao ar no SporTV.
+                                                </p>
+                                                <p>
+                                                    Na ocasião, Tite e outros representantes do Corinthians <a href="https://www1.folha.uol.com.br/poder/1124743-corinthians-leva-a-taca-da-libertadores-para-lula.shtml">foram ao Instituto Lula para mostrar a taça</a> original da Libertadores ao ex-presidente.
+                                                </p>
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+                                                            </figcaption><a href="https://fotografia.folha.uol.com.br/galerias/nova/1618838542352047-os-times-de-coracao-dos-presidentes" class="gallery-widget-pre__link"><img src="https://f.i.uol.com.br/fotografia/2018/12/03/15438447625c05339a96ad0_1543844762_3x2_md.jpg" alt="Os times de coração dos presidentes" class="gallery-widget-pre__photo" /></a>
+                                                        </figure>
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+                                                    O assunto foi levantado&#160;porque recentemente Tite foi questionado se aceitaria um encontro da seleção brasileira com Bolsonaro em uma conquista de título ou <a href="https://www1.folha.uol.com.br/esporte/2018/12/selecao-brasileira-jogara-duas-vezes-em-sao-paulo-na-copa-america.shtml">antes da Copa América de 2019</a>, por exemplo. O treinador deixou claro que preferiria evitar esse tipo de formalidade.
+                                                </p>
+                                                <p>
+                                                    Apesar disso, Tite não questionou a ação de Palmeiras e CBF, que <a href="https://www1.folha.uol.com.br/esporte/2018/12/cbf-usa-festa-do-palmeiras-para-se-aproximar-de-governo-bolsonaro.shtml">convidaram Bolsonaro para a festa do título do Campeonato Brasileiro</a>. O presidente eleito até levantou a taça conquistada pelo clube alviverde.
+                                                </p>
+                                                <p>
+                                                    "Em 2012 eu fiz e errei. O protocolo e a situação gerada no jogo do Palmeiras são fatos de opinião pessoal. CBF e Palmeiras, enquanto instituições têm a opinião. Errei lá atrás, não faria com o presidente antes da Copa e nem agora porque entendo que misturar esporte e política não é legal. Fiz errado lá atrás? Sim. Faria de novo? Não", acrescentou o comandante.
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+                                                    <div class="c-rotate-generic__item js-rotate-generic__item">
+                                                        <div class="c-estudio-folha__content">
+                                                            <div class="c-estudio-folha__image-main">
+                                                                <a href="http://bit.ly/2QQE5he" class="c-estudio-folha__link"><img src="//f.i.uol.com.br/estudiofolha/images/1834816.jpeg" alt="" /></a>
+                                                            </div><a href="http://bit.ly/2QQE5he" class="c-estudio-folha__link">
+                                                            <p>
+                                                                <b>Novas tendências de consumo apontam escolha por praticidade e segurança alimentar</b>
+                                                            </p><img src="http://f.i.uol.com.br/estudiofolha/images/1832651.jpeg" class="c-estudio-folha__sponsored-logo" alt="" /></a>
+                                                        </div>
+                                                    </div>
+                                                    <div class="c-rotate-generic__item js-rotate-generic__item">
+                                                        <div class="c-estudio-folha__content">
+                                                            <div class="c-estudio-folha__image-main">
+                                                                <a href="http://bit.ly/2Phy4VR" class="c-estudio-folha__link"><img src="//f.i.uol.com.br/estudiofolha/images/1834518.jpeg" alt="" /></a>
+                                                            </div><a href="http://bit.ly/2Phy4VR" class="c-estudio-folha__link">
+                                                            <p>
+                                                                <b>Banco do Brasil lança app que auxilia produtores rurais</b>
+                                                            </p><img src="https://f.i.uol.com.br/estudiofolha/images/1833351.jpeg" class="c-estudio-folha__sponsored-logo" alt="" /></a>
+                                                        </div>
+                                                    </div>
+                                                    <div class="c-rotate-generic__item js-rotate-generic__item">
+                                                        <div class="c-estudio-folha__content">
+                                                            <div class="c-estudio-folha__image-main">
+                                                                <a href="http://bit.ly/2EpMFxJ" class="c-estudio-folha__link"><img src="//f.i.uol.com.br/estudiofolha/images/183514.jpeg" alt="" /></a>
+                                                            </div><a href="http://bit.ly/2EpMFxJ" class="c-estudio-folha__link">
+                                                            <p>
+                                                                <b>Aplicação pouco conhecida, COE pode garantir ganhos maiores com capital protegido</b>
+                                                            </p><img src="//f.i.uol.com.br/estudiofolha/images/1825615.jpeg" class="c-estudio-folha__sponsored-logo" alt="" /></a>
+                                                        </div>
+                                                    </div>
+                                                    <div class="c-rotate-generic__item js-rotate-generic__item">
+                                                        <div class="c-estudio-folha__content">
+                                                            <div class="c-estudio-folha__image-main">
+                                                                <a href="http://bit.ly/2ry3b6k" class="c-estudio-folha__link"><img src="//f.i.uol.com.br/estudiofolha/images/1834519.jpeg" alt="" /></a>
+                                                            </div><a href="http://bit.ly/2ry3b6k" class="c-estudio-folha__link">
+                                                            <p>
+                                                                <b>Projetos premiados trabalham a relação tributos e cidadania</b>
+                                                            </p><img src="http://f.i.uol.com.br/estudiofolha/images/1832722.jpeg" class="c-estudio-folha__sponsored-logo" alt="" /></a>
+                                                        </div>
+                                                    </div>
+                                                    <div class="c-rotate-generic__item js-rotate-generic__item">
+                                                        <div class="c-estudio-folha__content">
+                                                            <div class="c-estudio-folha__image-main">
+                                                                <a href="http://bit.ly/2Aj1pKp" class="c-estudio-folha__link"><img src="//f.i.uol.com.br/estudiofolha/images/1832742.jpeg" alt="" /></a>
+                                                            </div><a href="http://bit.ly/2Aj1pKp" class="c-estudio-folha__link">
+                                                            <p>
+                                                                <b>Estímulo à inovação gera interesse estrangeiro no setor aéreo nacional</b>
+                                                            </p><img src="//f.i.uol.com.br/estudiofolha/images/1829216.jpeg" class="c-estudio-folha__sponsored-logo" alt="" /></a>
+                                                        </div>
+                                                    </div>
+                                                    <div class="c-rotate-generic__item js-rotate-generic__item">
+                                                        <div class="c-estudio-folha__content">
+                                                            <div class="c-estudio-folha__image-main">
+                                                                <a href="http://bit.ly/2KJ4yb3" class="c-estudio-folha__link"><img src="//f.i.uol.com.br/estudiofolha/images/1834516.jpeg" alt="" /></a>
+                                                            </div><a href="http://bit.ly/2KJ4yb3" class="c-estudio-folha__link">
+                                                            <p>
+                                                                <b>Programa TAP Miles&amp;Go otimiza tempo de seus passageiros e agiliza viagem</b>
+                                                            </p><img src="https://f.i.uol.com.br/estudiofolha/images/1831142.jpeg" class="c-estudio-folha__sponsored-logo" alt="" /></a>
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+                    <div class="c-most-read">
+                        <div class="c-most-read__head">
+                            <h2 class="c-most-read__title c-section-title">
+                                Mais lidas <span class="c-most-read__section c-section-title__thin">em Esporte</span>
+                            </h2><a href="//www1.folha.uol.com.br/maispopulares/" class="c-button c-most-read__see-more" title="Ver todas matérias mais lidas">Ver todas</a>
+                        </div>
+                        <ol class="c-most-read__list">
+                            <li>
+                                <a href="https://www1.folha.uol.com.br/esporte/2018/12/fenomeno-do-basquete-dos-eua-refutou-futebol-americano-e-deve-chegar-a-nba.shtml">Fenômeno do basquete dos EUA refutou futebol americano e deve chegar à NBA</a>
+                            </li>
+                            <li>
+                                <a href="https://www1.folha.uol.com.br/esporte/2018/12/secretario-de-esportes-e-bem-relacionado-com-militares-e-cupula-do-governo.shtml">Militar irá comandar secretaria de esporte do governo Bolsonaro</a>
+                            </li>
+                            <li>
+                                <a href="https://www1.folha.uol.com.br/esporte/2018/12/globo-muda-horario-e-rodadas-de-quarta-no-futebol-brasileiro-serao-as-21h30.shtml">Globo muda horário, e rodada de quarta no futebol brasileiro será às 21h30</a>
+                            </li>
+                            <li>
+                                <a href="https://www1.folha.uol.com.br/esporte/2018/12/ana-marcela-ve-toquio-20-como-ultima-chance-e-diz-temer-intolerancia-sexual.shtml">Ana Marcela vê Tóquio-20 como última chance e diz temer intolerância sexual</a>
+                            </li>
+                            <li>
+                                <a href="https://www1.folha.uol.com.br/esporte/2018/12/quem-e-o-al-ain-time-que-desbancou-o-river-plate-no-mundial-de-clubes.shtml">Quem é o Al Ain, time que desbancou o River Plate no Mundial de Clubes</a>
+                            </li>
+                        </ol>
+                    </div>
+                </div>
+            </div>
+            <div class="block u-no-print">
+                <div class="container">
+                    <div class="flex flex--gutter flex--col flex--md-row">
+                        <div class="flex-cell">
+                            <h4 class="c-section-title c-section-title--brand">
+                                Principais do dia
+                            </h4>
+                            <div class="c-headline c-headline--newslist">
+                                <div class="c-headline__media-wrapper">
+                                    <a href="https://www1.folha.uol.com.br/cotidiano/2018/12/covas-quebra-promessa-de-doria-e-preve-dinheiro-de-privatizacao-na-previdencia.shtml"><img class="c-headline__image" src="https://f.i.uol.com.br/fotografia/2018/04/06/15230315615ac79e095b8f3_1523031561_3x2_sm.jpg" alt="Covas quebra promessa de Doria e prevê dinheiro de privatização na previdência" /></a>
+                                </div>
+                                <div class="c-headline__wrapper">
+                                    <div class="c-headline__head">
+                                        <h3 class="c-headline__kicker c-kicker">
+                                            <a href="http://www1.folha.uol.com.br/folha-topicos/joao-doria">João Doria</a>
+                                        </h3>
+                                        <div data-modal-drop="" data-qty-collumn="3.4" class="c-modal-drop">
+                                            <button data-trigger="" type="button" name="button" class="c-headline__action"><svg xmlns="https://www.w3.org/2000/svg" viewbox="0 0 16 16" width="24" height="24" class="icon icon--share" aria-hidden="true">
+                                            <path d="m 12.818311,11.294921 c 1.280064,0 2.333667,1.054406 2.333667,2.333668 0,1.279261 -1.054406,2.371411 -2.333667,2.371411 -1.279262,0 -2.333668,-1.09215 -2.333668,-2.371411 0,-0.187915 0,-0.377435 0.03774,-0.526802 L 4.8407964,9.789199 A 2.4252158,2.4252158 0 0 1 0.772537,8.020076 2.4252158,2.4252158 0 0 1 4.8383872,6.250954 L 10.48384,2.9761092 A 2.8974102,2.8974102 0 0 1 10.40915,2.4091547 C 10.40915,1.0921502 11.5013,0 12.818304,0 c 1.317008,0 2.409159,1.0921502 2.409159,2.4091547 0,1.3170047 -1.092151,2.4091553 -2.409155,2.4091553 -0.640032,0 -1.204577,-0.263401 -1.656695,-0.677776 L 5.5161598,7.453925 c 0.036941,0.187914 0.074684,0.377434 0.074684,0.564545 0,0.187111 -0.037744,0.377434 -0.075486,0.562137 l 5.7217422,3.31339 c 0.417587,-0.377434 0.979724,-0.602289 1.582012,-0.602289 z"></path></svg> <span class="u-visually-hidden">Compartilhar</span></button>
+                                            <div aria-hidden="true" data-content="" class="c-modal-drop__content c-modal-drop__content--no-padding">
+                                                <div class="c-modal-drop__controls u-hidden-md">
+                                                    <button data-close="" class="c-modal-drop__close"><svg xmlns="https://www.w3.org/2000/svg" viewbox="0 0 16 16" width="24" height="24" class="icon icon--close icon--tiny" aria-hidden="true">
+                                                    <path d="M16 1.61L14.39 0 8 6.39 1.61 0 0 1.61 6.39 8 0 14.39 1.61 16 8 9.61 14.39 16 16 14.39 9.61 8z"></path></svg></button>
+                                                </div>
+                                                <div class="c-tools-share--larger">
+                                                    <ul class="c-tools-share__list" data-sharebar-utm-campaign-prefix="comp" data-sharebar-buttons="facebook whatsapp twitter messenger googlePlus linkedIn pinterest rss email" data-triggered-byclick=".c-modal-drop [data-trigger]" data-sharebar-url="https://www1.folha.uol.com.br/cotidiano/2018/12/covas-quebra-promessa-de-doria-e-preve-dinheiro-de-privatizacao-na-previdencia.shtml" data-sharebar-text="Covas quebra promessa de Doria e prevê dinheiro de privatização na previdência" data-sharebar-channel="cotidiano"></ul>
+                                                </div>
+                                            </div>
+                                        </div>
+                                    </div>
+                                    <div class="c-headline__content">
+                                        <a href="https://www1.folha.uol.com.br/cotidiano/2018/12/covas-quebra-promessa-de-doria-e-preve-dinheiro-de-privatizacao-na-previdencia.shtml">
+                                        <h2 class="c-headline__title">
+                                            Covas quebra promessa de Doria e prevê dinheiro de privatização na previdência
+                                        </h2>
+                                        <p class="c-headline__standfirst">
+                                            Atual gestão tucana em São Paulo havia prometido usar venda de equipamentos na área social
+                                        </p><time class="c-headline__dateline" itemprop="datePublished" datetime="2018-12-21 10:55:00">21.dez.2018 às 10h55</time></a>
+                                    </div>
+                                </div>
+                            </div>
+                            <div class="c-headline c-headline--newslist">
+                                <div class="c-headline__media-wrapper">
+                                    <a href="https://www1.folha.uol.com.br/poder/2018/12/pela-primeira-vez-na-republica-ministerio-excluira-norte-e-nordeste.shtml"><img class="c-headline__image" src="https://f.i.uol.com.br/fotografia/2018/12/19/15452261445c1a47a062deb_1545226144_3x2_sm.jpg" alt="Pela primeira vez na República, ministério que toma posse excluirá Norte e Nordeste" /></a>
+                                </div>
+                                <div class="c-headline__wrapper">
+                                    <div class="c-headline__head">
+                                        <h3 class="c-headline__kicker c-kicker">
+                                            <a href="https://www1.folha.uol.com.br/especial/2018/governo-bolsonaro">Governo Bolsonaro</a>
+                                        </h3>
+                                        <div data-modal-drop="" data-qty-collumn="3.4" class="c-modal-drop">
+                                            <button data-trigger="" type="button" name="button" class="c-headline__action"><svg xmlns="https://www.w3.org/2000/svg" viewbox="0 0 16 16" width="24" height="24" class="icon icon--share" aria-hidden="true">
+                                            <path d="m 12.818311,11.294921 c 1.280064,0 2.333667,1.054406 2.333667,2.333668 0,1.279261 -1.054406,2.371411 -2.333667,2.371411 -1.279262,0 -2.333668,-1.09215 -2.333668,-2.371411 0,-0.187915 0,-0.377435 0.03774,-0.526802 L 4.8407964,9.789199 A 2.4252158,2.4252158 0 0 1 0.772537,8.020076 2.4252158,2.4252158 0 0 1 4.8383872,6.250954 L 10.48384,2.9761092 A 2.8974102,2.8974102 0 0 1 10.40915,2.4091547 C 10.40915,1.0921502 11.5013,0 12.818304,0 c 1.317008,0 2.409159,1.0921502 2.409159,2.4091547 0,1.3170047 -1.092151,2.4091553 -2.409155,2.4091553 -0.640032,0 -1.204577,-0.263401 -1.656695,-0.677776 L 5.5161598,7.453925 c 0.036941,0.187914 0.074684,0.377434 0.074684,0.564545 0,0.187111 -0.037744,0.377434 -0.075486,0.562137 l 5.7217422,3.31339 c 0.417587,-0.377434 0.979724,-0.602289 1.582012,-0.602289 z"></path></svg> <span class="u-visually-hidden">Compartilhar</span></button>
+                                            <div aria-hidden="true" data-content="" class="c-modal-drop__content c-modal-drop__content--no-padding">
+                                                <div class="c-modal-drop__controls u-hidden-md">
+                                                    <button data-close="" class="c-modal-drop__close"><svg xmlns="https://www.w3.org/2000/svg" viewbox="0 0 16 16" width="24" height="24" class="icon icon--close icon--tiny" aria-hidden="true">
+                                                    <path d="M16 1.61L14.39 0 8 6.39 1.61 0 0 1.61 6.39 8 0 14.39 1.61 16 8 9.61 14.39 16 16 14.39 9.61 8z"></path></svg></button>
+                                                </div>
+                                                <div class="c-tools-share--larger">
+                                                    <ul class="c-tools-share__list" data-sharebar-utm-campaign-prefix="comp" data-sharebar-buttons="facebook whatsapp twitter messenger googlePlus linkedIn pinterest rss email" data-triggered-byclick=".c-modal-drop [data-trigger]" data-sharebar-url="https://www1.folha.uol.com.br/poder/2018/12/pela-primeira-vez-na-republica-ministerio-excluira-norte-e-nordeste.shtml" data-sharebar-text="Pela primeira vez na República, ministério que toma posse excluirá Norte e Nordeste" data-sharebar-channel="poder"></ul>
+                                                </div>
+                                            </div>
+                                        </div>
+                                    </div>
+                                    <div class="c-headline__content">
+                                        <a href="https://www1.folha.uol.com.br/poder/2018/12/pela-primeira-vez-na-republica-ministerio-excluira-norte-e-nordeste.shtml">
+                                        <h2 class="c-headline__title">
+                                            Pela primeira vez na República, ministério que toma posse excluirá Norte e Nordeste
+                                        </h2>
+                                        <p class="c-headline__standfirst">
+                                            No primeiro escalão, há 11 integrantes do Sudeste, 8 do Sul, 2 do Centro-Oeste e 1 colombiano
+                                        </p><time class="c-headline__dateline" itemprop="datePublished" datetime="2018-12-21 12:00:00">21.dez.2018 às 12h00</time></a>
+                                    </div>
+                                </div>
+                            </div>
+                            <div class="c-headline c-headline--newslist">
+                                <div class="c-headline__media-wrapper">
+                                    <a href="https://www1.folha.uol.com.br/colunas/reinaldoazevedo/2018/12/torcam-para-que-bolsonaro-consiga-modernizar-guedes.shtml"><img class="c-headline__image" src="https://f.i.uol.com.br/fotografia/2018/11/29/15435040395c0000a79a59e_1543504039_3x2_sm.jpg" alt="Torçam para que Bolsonaro consiga modernizar Guedes" /></a>
+                                </div>
+                                <div class="c-headline__wrapper">
+                                    <div class="c-headline__head">
+                                        <div data-modal-drop="" data-qty-collumn="3.4" class="c-modal-drop">
+                                            <button data-trigger="" type="button" name="button" class="c-headline__action"><svg xmlns="https://www.w3.org/2000/svg" viewbox="0 0 16 16" width="24" height="24" class="icon icon--share" aria-hidden="true">
+                                            <path d="m 12.818311,11.294921 c 1.280064,0 2.333667,1.054406 2.333667,2.333668 0,1.279261 -1.054406,2.371411 -2.333667,2.371411 -1.279262,0 -2.333668,-1.09215 -2.333668,-2.371411 0,-0.187915 0,-0.377435 0.03774,-0.526802 L 4.8407964,9.789199 A 2.4252158,2.4252158 0 0 1 0.772537,8.020076 2.4252158,2.4252158 0 0 1 4.8383872,6.250954 L 10.48384,2.9761092 A 2.8974102,2.8974102 0 0 1 10.40915,2.4091547 C 10.40915,1.0921502 11.5013,0 12.818304,0 c 1.317008,0 2.409159,1.0921502 2.409159,2.4091547 0,1.3170047 -1.092151,2.4091553 -2.409155,2.4091553 -0.640032,0 -1.204577,-0.263401 -1.656695,-0.677776 L 5.5161598,7.453925 c 0.036941,0.187914 0.074684,0.377434 0.074684,0.564545 0,0.187111 -0.037744,0.377434 -0.075486,0.562137 l 5.7217422,3.31339 c 0.417587,-0.377434 0.979724,-0.602289 1.582012,-0.602289 z"></path></svg> <span class="u-visually-hidden">Compartilhar</span></button>
+                                            <div aria-hidden="true" data-content="" class="c-modal-drop__content c-modal-drop__content--no-padding">
+                                                <div class="c-modal-drop__controls u-hidden-md">
+                                                    <button data-close="" class="c-modal-drop__close"><svg xmlns="https://www.w3.org/2000/svg" viewbox="0 0 16 16" width="24" height="24" class="icon icon--close icon--tiny" aria-hidden="true">
+                                                    <path d="M16 1.61L14.39 0 8 6.39 1.61 0 0 1.61 6.39 8 0 14.39 1.61 16 8 9.61 14.39 16 16 14.39 9.61 8z"></path></svg></button>
+                                                </div>
+                                                <div class="c-tools-share--larger">
+                                                    <ul class="c-tools-share__list" data-sharebar-utm-campaign-prefix="comp" data-sharebar-buttons="facebook whatsapp twitter messenger googlePlus linkedIn pinterest rss email" data-triggered-byclick=".c-modal-drop [data-trigger]" data-sharebar-url="https://www1.folha.uol.com.br/colunas/reinaldoazevedo/2018/12/torcam-para-que-bolsonaro-consiga-modernizar-guedes.shtml" data-sharebar-text="Torçam para que Bolsonaro consiga modernizar Guedes" data-sharebar-channel="colunistas-reinaldo-azevedo"></ul>
+                                                </div>
+                                            </div>
+                                        </div>
+                                    </div>
+                                    <div class="c-headline__content">
+                                        <a href="https://www1.folha.uol.com.br/colunas/reinaldoazevedo/2018/12/torcam-para-que-bolsonaro-consiga-modernizar-guedes.shtml">
+                                        <h2 class="c-headline__title">
+                                            Torçam para que Bolsonaro consiga modernizar Guedes
+                                        </h2>
+                                        <p class="c-headline__standfirst">
+                                            Ou o presidente vira o conselheiro do ministro da Economia ou a nau afunda
+                                        </p><time class="c-headline__dateline" itemprop="datePublished" datetime="2018-12-21 02:00:00">21.dez.2018 às 2h00</time></a>
+                                    </div>
+                                </div>
+                            </div>
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+                                                            <a href="https://www1.folha.uol.com.br/o-melhor-de-sao-paulo/2017/restaurantes-bares-e-cozinha/">O Melhor de sãopaulo</a>
+                                                        </li>
+                                                        <li>
+                                                            <a href="https://www1.folha.uol.com.br/ultimas-noticias/">Últimas</a>
+                                                        </li>
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+                                                            <a href="https://www1.folha.uol.com.br/fsp/">Versão Impressa</a>
+                                                        </li>
+                                                        <li>
+                                                            <a href="https://www1.folha.uol.com.br/mapa-do-site/">Mapa do site</a>
+                                                        </li>
+                                                    </ul>
+                                                </nav>
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+                                                <nav class="l-footer__section">
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+                                                            <a href="https://www1.folha.uol.com.br/cotidiano/aeroportos/">Aeroportos</a>
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+                                                            <a href="https://classificados1.folha.uol.com.br/">Classificados</a>
+                                                        </li>
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+                                                            <a href="https://www1.folha.uol.com.br/especial/2018/folha-informacoes/">Folha Informações</a>
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+                                                            <a href="https://f5.folha.uol.com.br/horoscopo/">Horóscopo</a>
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+                                                            <a href="https://tempo.folha.uol.com.br/">Tempo</a>
+                                                        </li>
+                                                    </ul>
+                                                </nav>
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+                                                <nav class="l-footer__section">
+                                                    <h4 class="c-heading c-heading--tiny">
+                                                        outros canais
+                                                    </h4>
+                                                    <ul class="l-footer__nav">
+                                                        <li>
+                                                            <a href="https://email.folha.uol.com.br/">e-mailFOLHA</a>
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+                                                        </li>
+                                                        <li>
+                                                            <a href="https://www1.folha.uol.com.br/especial/2018/top-of-mind/">Top of Mind</a>
+                                                        </li>
+                                                    </ul>
+                                                </nav>
+                                            </div>
+                                        </div>
+                                    </div>
+                                </div>
+                            </div>
+                            <div class="col col--xs-1-1 col--sm-1-1 col--md-1-4">
+                                <div class="row">
+                                    <div class="col col--xs-1-1 col--sm-1-1 col--md-1-1">
+                                        <div class="c-audience">
+                                            <h4 class="c-heading c-heading--tiny">
+                                                Audiência da Folha
+                                            </h4>
+                                            <table class="c-audience__table">
+                                                <tbody>
+                                                    <tr>
+                                                        <th class="c-audience__cell">
+                                                            Páginas vistas
+                                                        </th>
+                                                        <td class="c-audience__cell">
+                                                            231.506.740 <span id="tooltip-pagesviews" class="u-visually-hidden" role="tooltip" aria-hidden="true"><span class="c-tooltip__content">Total de páginas visualizadas no site da Folha em novembro de 2018, segundo o Google Analytics</span></span> <a aria-describedby="tooltip-pagesviews" data-tooltip="" data-direction="bottom" data-target="#tooltip-pagesviews" class="c-audience__cell--anchor"><svg xmlns="//www.w3.org/2000/svg" width="24" height="24" viewbox="0 0 24 24" version="1.1" class="icon icon--information" aria-hidden="true">
+                                                            <path d="M13,9H11V7H13M13,17H11V11H13M12,2A10,10 0 0,0 2,12A10,10 0 0,0 12,22A10,10 0 0,0 22,12A10,10 0 0,0 12,2Z"></path></svg><span class="u-visually-hidden">Informação</span></a> <span class="c-audience__date">nov.2018</span>
+                                                        </td>
+                                                    </tr>
+                                                    <tr>
+                                                        <th class="c-audience__cell">
+                                                            Visitantes únicos
+                                                        </th>
+                                                        <td class="c-audience__cell">
+                                                            35.681.468 <span id="tooltip-unique_visitors" class="u-visually-hidden" role="tooltip" aria-hidden="true"><span class="c-tooltip__content">Total de visitantes diferentes do site da Folha em novembro de 2018, segundo o Google Analytics</span></span> <a aria-describedby="tooltip-unique_visitors" data-tooltip="" data-direction="bottom" data-target="#tooltip-unique_visitors" class="c-audience__cell--anchor"><svg xmlns="//www.w3.org/2000/svg" width="24" height="24" viewbox="0 0 24 24" version="1.1" class="icon icon--information" aria-hidden="true">
+                                                            <path d="M13,9H11V7H13M13,17H11V11H13M12,2A10,10 0 0,0 2,12A10,10 0 0,0 12,22A10,10 0 0,0 22,12A10,10 0 0,0 12,2Z"></path></svg><span class="u-visually-hidden">Informação</span></a> <span class="c-audience__date">nov.2018</span>
+                                                        </td>
+                                                    </tr>
+                                                </tbody>
+                                            </table>
+                                        </div>
+                                    </div>
+                                    <div class="col col--xs-1-1 col--sm-1-1 col--md-1-1">
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+                                                Escolha suas newsletters
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+                                                </div><span class="c-form__error-message c-newsletter__error-message" id="validation-error"></span>
+                                            </form>
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+                                        <ul class="c-follow-social-media">
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+                                                <path d="M 0.5217573,3.4217496 C 0.8574099,3.7574022 1.3243328,3.925225 1.883477,3.925225 l 0.019112,0 c 0.5965319,0 1.062624,-0.1678228 1.416555,-0.5034754 C 3.6739051,3.0669849 3.8417314,2.656557 3.8417314,2.1348025 3.8417314,1.6313201 3.6547964,1.2026132 3.319144,0.86696056 2.9834872,0.531308 2.517395,0.3634851 1.9391422,0.3634851 1.3608886,0.3634851 0.8947967,0.531308 0.5408658,0.86696056 0.1861045,1.2026132 -1e-7,1.6313201 -1e-7,2.1348025 c 0,0.5217545 0.1678263,0.9321824 0.5217572,1.2869471 l 0,0 z m -0.3165439,12.2147654 3.4313014,0 0,-10.2939042 -3.4313014,0 0,10.2939042 z m 8.7460799,0 0,-5.7443117 c 0,-0.3539317 0.037384,-0.633921 0.1121617,-0.839135 C 9.2121719,8.6991367 9.4364945,8.4008753 9.7347601,8.1582701 10.013917,7.9156719 10.386957,7.8035124 10.834771,7.8035124 c 0.596531,0 1.025236,0.2052139 1.305224,0.6156349 0.279987,0.410428 0.428705,0.9886857 0.428705,1.7156547 l 0,5.501713 3.4313,0 0,-5.8930294 C 16,8.2139334 15.64607,7.0582579 14.919099,6.2747932 14.191298,5.4913284 13.240005,5.1000126 12.046942,5.1000126 c -0.428705,0 -0.839131,0.055677 -1.193893,0.1678228 -0.353931,0.093054 -0.652197,0.2425982 -0.9130756,0.446986 C 9.6973732,5.9017564 9.5104375,6.0878582 9.3617206,6.2565141 9.2121719,6.4235108 9.063455,6.6104458 8.9321847,6.8339388 l 0.019112,0 0,-1.491328 -3.4313016,0 0.018279,0.5034754 0,3.0765435 c 0,1.7156473 0,3.9538893 -0.018279,6.7138853 l 3.4313016,0 0,0 z"></path></svg></a>
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+                            </div>
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+                        </fieldset>
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e&&e.__esModule?e:{default:e}}Object.defineProperty(t,"__esModule",{value:!0}),t.default=function(e){if(!c&&(c=!0,(0,i.default)(window,"paywall.after"))){var t=(0,o.default)(window,"paywall.after");"function"==typeof t&&t(e)}};var n=r(61),o=_interopRequireDefault(n),a=r(37),i=_interopRequireDefault(a),c=!1;e.exports=t.default},function(e,t){"use strict";function showPaywall(e,t,r,n,o,a){var i=arguments.length>6&&void 0!==arguments[6]?arguments[6]:{login:"#"},c=i.login,s=(arguments[7],arguments[8]),u=document.querySelector("[data-paywall-box]"),l=document.querySelector("[data-paywall]").getAttribute("data-paywall-product"),f=l?" c-paywall--"+l:"",p=document.querySelector('meta[name="folhaapp-webview"][content="ios"]'),h=navigator.userAgent.match(/iphone|ipad/i),d=p&&h?"c-paywall__teaser--is-hidden":"",y=null!==document.querySelector(".c-image-full"),_=null!==document.querySelector(".c-news__head"),v=encodeURIComponent(window.location.href),m=s||"//secure.folha.com.br/folha/assine/380101",g="//login.folha.com.br/uol/login",b=void 0,w='<div class="c-paywall'+f+'"><div class="c-paywall__row">'+('<h4 class="c-paywall__title">Você atingiu o limite de '+n+" reportagens por mês</h4>")+'<div class="c-paywall__half-col c-paywall__login-area"><div class="c-paywall__row"><div class="c-paywall__half-col">'+('<p class="c-paywall__text">Tenha acesso a '+o+" reportagens por mês<br>")+"Caso seja assinante, o acesso será ilimitado</p>"+("<a onclick=\"UOLPD.Audience.countClick({category:'porteira-embed-"+a+"'")+",action:'clique',label:'crie-sua-conta'})\" "+('href="'+c+"?return_url="+v+'%3Floggedpaywall&urltype=register" class="c-paywall__button">')+'Crie sua conta</a></div><div class="c-paywall__half-col"><p class="c-paywall__text"><span class="c-paywall__text--custom">Para você que já tem conta</span></p>'+("<a onclick=\"UOLPD.Audience.countClick({category:'porteira-embed-"+a+"'")+",action:'clique',label:'faca-login'})\""+('href="'+c+"?return_url="+v+'%3Floggedpaywall&logintype=cadastrado&urltype=login" ')+'class="c-paywall__button">Faça login</a></div></div><div class="c-paywall__social"><p class="c-paywall__text">ou acesse com:</p><ul class="c-paywall__login-list"><li>'+("<a onclick=\"UOLPD.Audience.countClick({category:'porteira-embed-"+a+"',action:'clique',")+"label:'acessar-com-facebook'})\" "+('href="'+c+"?return_url="+v+'%3Floggedpaywall&amp;logintype=cadastrado&amp;urltype=facebook" ')+'class="c-paywall__social-login c-paywall__social-login--facebook rs_skip" role="button" aria-label="Facebook"><svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 16 16" width="24" height="24" class="icon 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fill="#34A853"></path></g></svg><svg xmlns="http://www.w3.org/2000/svg" width="46" height="14" viewBox="0 0 46 14" version="1.1" class="icon-google-text" aria-hidden="true"><path fill="#757575" fill-rule="evenodd" d="M8.64 9.708c-.36.47-.858.825-1.496 1.066-.638.242-1.363.363-2.174.363-.834 0-1.573-.19-2.215-.568a3.78 3.78 0 0 1-1.487-1.623C.92 8.242.738 7.42.725 6.48v-.779c0-1.504.36-2.678 1.08-3.524C2.525 1.333 3.53.91 4.819.91c1.108 0 1.987.274 2.639.82.652.547 1.044 1.336 1.176 2.366H6.938c-.19-1.199-.886-1.798-2.084-1.798-.775 0-1.364.279-1.768.837-.403.559-.6111.378-.625 2.458v.765c0 1.076.227 1.911.68 2.506.454.594 1.084.892 1.89.892.884 0 1.513-.2 1.887-.602V7.206h-2.05V5.894H8.64v3.814zm1.648-2.475c0-.724.144-1.377.43-1.958a3.145 3.145 0 0 1 1.21-1.34c.52-.312 1.117-.468 1.792-.468.998 0 1.808.321 2.43.964.622.642.958 1.494 1.008 2.556l.007.39c0 .73-.14 1.38-.42 1.955a3.108 3.108 0 0 1-1.203 1.333c-.522.314-1.125.472-1.809.472-1.043 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onclick=\"UOLPD.Audience.countClick({category:'porteira-embed-"+a+"',action:'clique',")+("label:'acessar-com-uol'})\" href=\""+g+"?done="+v+'%3Floggedpaywall" ')+'class="c-paywall__login-uol rs_skip" role="button"></a></li></ul></div></div><div class="c-paywall__half-col">'+('<div class="c-paywall__teaser '+d+'">')+'<figure class="c-paywall__devices"><img src="//f.i.uol.com.br/heimdall/assine-folha-todos-dispositivos-transparente-321x149.png" alt="Folha de São Paulo em todos devices"></figure><p class="c-paywall__text">Tenha acesso ilimitado, <span>descontos em produtos Folha e muito mais</span></p><p class="c-paywall__price">Folha digital por apenas R$1,90* <small>*no primeiro mês</small></p>'+("<a onclick=\"UOLPD.Audience.countClick({category:'porteira-embed-"+a+"'")+",action:'clique',label:'assine-a-folha'})\""+('href="'+m+'" class="c-paywall__button c-paywall__button--primary">Assine a Folha</a>')+"</div></div></div></div>",x=""+('<div class="c-paywall'+f+'">')+('<h4 class="c-paywall__title">Você atingiu o limite de '+o+" reportagens por mês</h4>")+('<div class="c-paywall__teaser '+d+'">')+'<figure class="c-paywall__devices"><img src="//f.i.uol.com.br/heimdall/assine-folha-todos-dispositivos-transparente-321x149.png" alt="Folha de São Paulo em todos devices"></figure><p class="c-paywall__text c-paywall__text-offers">Tenha acesso ilimitado, descontos em produtos Folha e muito mais</p><p class="c-paywall__price">Folha digital por apenas R$1,90* <small>*no primeiro mês</small></p>'+("<a onclick=\"UOLPD.Audience.countClick({category:'porteira-embed-"+a+"'")+",action:'clique',label:'assine-a-folha'})\""+('href="'+m+'" class="c-paywall__button c-paywall__button--primary">Assine a Folha</a>')+'</div><p class="c-paywall__text">Já é assinante?</p>'+("<a onclick=\"UOLPD.Audience.countClick({category:'porteira-embed-"+a+"'")+",action:'clique',label:'faca-login'})\""+('href="'+c+"?return_url="+v+'%3Floggedpaywall&logintype=cadastrado&urltype=login" 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Date})},Cookie})},function(e,t,r){r(65),r(64),r(62),e.exports=r(63)}]);
+        ]]>
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+        <div id="banner-1x1-area"></div>
+        <script>
+        <![CDATA[
+        //chartbeat
+        var _sf_async_config = {};
+        _sf_async_config.sections = "esporte";
+        ]]>
+        </script> 
+        <script>
+        <![CDATA[
+
+        //UOLTM - ads
+        document.querySelectorAll(".c-advertising__banner-area").forEach(function(ad){window.uolads.push({id:ad.id})});
+        window.uolads.push({id:"banner-1x1-area"});
+
+        //IVC
+        (function(p,l,o,w,i,n,g){if(!p[i]){p.GlobalIvcNamespace=p.GlobalIvcNamespace||[];
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+        window.ivc('newTracker', 'cf', 'ivccf.ivcbrasil.org.br', { idWeb: '125' });
+        window.ivc('trackPageView');
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+        //chartbeat
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+        _sf_async_config.uid = 50059;
+        _sf_async_config.domain = 'folha.com.br';
+        _sf_async_config.useCanonical = true;
+        _sf_async_config.authors = 'Folha';
+        (function(){
+        function loadChartbeat() {
+        window._sf_endpt=(new Date()).getTime();
+        var e = document.createElement('script');
+        e.setAttribute('language', 'javascript');
+        e.setAttribute('type', 'text/javascript');
+        e.setAttribute('src', 'https://static.chartbeat.com/js/chartbeat.js');
+        document.body.appendChild(e);
+        }
+        var oldonload = window.onload;
+        window.onload = (typeof window.onload != 'function') ?
+        loadChartbeat : function() { oldonload(); loadChartbeat(); };
+        })();
+
+        //UOLTM  - clique - menu
+        (function(){document.querySelector("header .container button[data-c-sidebar-menu-btn]").addEventListener("click",function(){
+        window.UOLPD.Audience.countClick({category:'menulateral',action:'clique',label:'menu'});},false);})();
+        ]]>
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+        <script src="//static.folha.uol.com.br/plugins/stats/1.0/stats.js"></script> <!--//-->
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+        <script src="https://static.folha.uol.com.br/folha/js/push/firebase-init.js?2018082001"></script>
+    </body>
+</html>
diff --git a/test/test-pages/lazy-image-1/expected-metadata.json b/test/test-pages/lazy-image-1/expected-metadata.json
index 1e1e19d..69d3d3b 100644
--- a/test/test-pages/lazy-image-1/expected-metadata.json
+++ b/test/test-pages/lazy-image-1/expected-metadata.json
@@ -1,8 +1,8 @@
 {
     "Author": "Vincent Vallet",
     "Direction": null,
-    "Excerpt": "Why CPU monitoring is important?",
+    "Excerpt": "How to run a CPU profiling with Node.js on your production in real-time and without interruption of service.",
     "Image": "https:\/\/miro.medium.com\/max\/1200\/1*EO-pr4RolgcAOj_Uk1rpDA.png",
     "Title": "Node.js and CPU profiling on production (in real-time without downtime)",
-    "SiteName": "Medium"
+    "SiteName": "Voodoo Engineering"
 }
 \ No newline at end of file
diff --git a/test/test-pages/lemonde-2/expected-metadata.json b/test/test-pages/lemonde-2/expected-metadata.json
index 916f4ab..eb7aee3 100644
--- a/test/test-pages/lemonde-2/expected-metadata.json
+++ b/test/test-pages/lemonde-2/expected-metadata.json
@@ -1,8 +1,8 @@
 {
     "Author": "Collectif",
     "Direction": null,
-    "Excerpt": "Le retour à la stabilité coûtera des milliards d'euros, mais l'Union européenne en vaut la peine, estime un collectif d'industriels.",
+    "Excerpt": "Le retour \u00e0 la stabilit\u00e9 co\u00fbtera des milliards d'euros, mais l'Union europ\u00e9enne en vaut la peine, estime un collectif d'industriels.",
     "Image": "http:\/\/s1.lemde.fr\/medias\/web\/1.2.705\/img\/placeholder\/default.png",
-    "Title": "La crise européenne est une crise de la dette, pas de l'euro",
-    "SiteName": "Le Monde.fr"
-}
+    "Title": "La crise europ\u00e9enne est une crise de la dette, pas de l'euro",
+    "SiteName": "Le Monde"
+}
+\ No newline at end of file
diff --git a/test/test-pages/videos-1/expected-metadata.json b/test/test-pages/videos-1/expected-metadata.json
index cf1f889..4d61d3b 100644
--- a/test/test-pages/videos-1/expected-metadata.json
+++ b/test/test-pages/videos-1/expected-metadata.json
@@ -1,8 +1,8 @@
 {
-    "Author": "By Alissa Wilkinson@alissamarie\n                                 Updated Jul 24, 2018, 2:15pm EDT",
+    "Author": "Alissa Wilkinson",
     "Direction": null,
-    "Excerpt": "How to watch the greatest movies of the year, from Lady Bird and Dunkirk to Get Out and The Big Sick.",
+    "Excerpt": "It was an extraordinary year for movies.",
     "Image": "https:\/\/cdn.vox-cdn.com\/thumbor\/7WyjCLC7n6i1IG2eJB06qi1o7kQ=\/0x148:2300x1352\/fit-in\/1200x630\/cdn.vox-cdn.com\/uploads\/chorus_asset\/file\/9871033\/Movies_end_of_year_2017.jpg",
-    "Title": "The 21 best movies of 2017",
+    "Title": "How to watch the 21 best films of 2017",
     "SiteName": "Vox"
 }
 \ No newline at end of file
diff --git a/test/test-pages/videos-2/expected-metadata.json b/test/test-pages/videos-2/expected-metadata.json
index 0230805..00c0019 100644
--- a/test/test-pages/videos-2/expected-metadata.json
+++ b/test/test-pages/videos-2/expected-metadata.json
@@ -1,8 +1,8 @@
 {
-    "Author": "Par Alexandre Hervaud et J\u00e9r\u00e9my Piette",
+    "Author": "Alexandre Hervaud, J\u00e9r\u00e9my Piette",
     "Direction": null,
-    "Excerpt": "S\u00e9ries, documentaires, programmes jeunesse\u2026 Retrouvez les recommandations de\u00a0Lib\u00e9ration\u00a0pour savoir quoi regarder sur vos \u00e9crans cette semaine. Pour d\u00e9passer...",
+    "Excerpt": "S\u00e9ries, documentaires, programmes jeunesse\u2026 Retrouvez les recommandations de\u00a0Lib\u00e9ration\u00a0pour savoir quoi regarder sur vos \u00e9crans cette semaine.\nPour d\u00e9passer...",
     "Image": "https:\/\/medias.liberation.fr\/photo\/1075029-screenshot-alphonse-vape-wave-6-days.jpg?modified_at=1511536242&amp;picto=fb&amp;ratio_x=191&amp;ratio_y=100&amp;width=600",
     "Title": "Screenshot\u00a0: \u00abVape Wave\u00bb, \u00ab6\u00a0Days\u00bb, \u00abAlphonse Pr\u00e9sident\u00bb\u2026",
-    "SiteName": "Lib\u00e9ration.fr"
+    "SiteName": "Lib\u00e9ration"
 }
 \ No newline at end of file
diff --git a/test/test-pages/wikipedia-2/expected-metadata.json b/test/test-pages/wikipedia-2/expected-metadata.json
index 2d1afaa..115048e 100644
--- a/test/test-pages/wikipedia-2/expected-metadata.json
+++ b/test/test-pages/wikipedia-2/expected-metadata.json
@@ -1,8 +1,8 @@
 {
-    "Author": "Authority control",
+    "Author": "Contributors to Wikimedia projects",
     "Direction": "ltr",
     "Excerpt": "Coordinates: 42\u00b0S 174\u00b0E\ufeff \/ \ufeff42\u00b0S 174\u00b0E",
     "Image": "https:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/3\/3e\/Flag_of_New_Zealand.svg\/1200px-Flag_of_New_Zealand.svg.png",
-    "Title": "New Zealand - Wikipedia",
-    "SiteName": null
+    "Title": "New Zealand",
+    "SiteName": "Wikimedia Foundation, Inc."
 }
 \ No newline at end of file
diff --git a/test/test-pages/wikipedia-3/expected-images.json b/test/test-pages/wikipedia-3/expected-images.json
new file mode 100644
index 0000000..caa1f9b
--- /dev/null
+++ b/test/test-pages/wikipedia-3/expected-images.json
@@ -0,0 +1,53 @@
+[
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/28a0aaa74b2267a48312e19321211cd9e3a39228",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6ca00b61ff0e264e6c1e5adc9a00c0d2751feecf",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/7daff47fa58cdfd29dc333def748ff5fa4c923e3",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d9415702ab196cc26f5df37af2d90e07318e93df",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/291d260bf69b764e75818669ab27870d58771e1f",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/8aa270391d183478251283d2c4b2c72ac4563352",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/5541bfa07743be995242c892c344395e672d6fa2",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/459de45e76bace9d04a80d2e8efc2abbbc246047",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6425c6e94fa47976601cb44d7564b5d04dcfbfef",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/9a50080b735975d8001c9552ac2134b49ad534c0",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/997ea0350c18735926412de88420ac9ca989f50c",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/54bf0331204e30cba9ec7f695dfea97e6745a7c2",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/00ccf11c16396b6ddd4f2254f7132cd8bb2c57ee",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3f0efab2d7c3a4b4b7caf14cc0705dadd13195a9",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/47136aad860d145f75f3eed3022df827cee94d7a",
+    "http:\/\/upload.wikimedia.org\/wikipedia\/commons\/thumb\/1\/1c\/Wiki_letter_w_cropped.svg\/44px-Wiki_letter_w_cropped.svg.png",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4fa8265c5f6d4fc3b7cda6a06558c7d4d9aec855",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0f1862750b96d01100244370b3fca45f01923ce5",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/251bf4ebbe3b0d119e0a7d19f8fd134c4f072971",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/021893240ff7fa3148b6649b0ba4d88cd207b5f0",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a28a8250ab35ad60228bb0376eb4b7024f027815",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a0179c3a7aebe194ccd9a19ba02b972500f88a7a",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b6cf8185ca7a0687bf959bb65b16db6cf1714ca2",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d303a1ebcac8547489b170be5d0dd7d8e04e548e",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/992c8ea49fdd26b491180036c5a4d879fec77442",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/46eedb181c0bdae46e8c1526161b03d0ea97b4b4",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/ddeac51c423f6dbefc5f63e483d9aee96e6fa342",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/db65cce3a8fa33e5b7b96badd756c8573aa866c0",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/add78d8608ad86e54951b8c8bd6c8d8416533d20",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/4ea1ea9ac61e6e1e84ac39130f78143c18865719",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3b7d749931e5f709bcbc0a446638d3b6b8ed0c6c",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/fa91daf9145f27bb95746fd2a37537342d587b77",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/1934e7eadd31fbf6f7d6bcf9c0e9bec934ce8976",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/a1240df64c3010e0be6eae865fdfcfe6f77bf5eb",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/43cc392bdcfbb134dd66d9b469847f6370e29d9d",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/3ef97bb04ce4ab682bcc84cf1059f8da235b483e",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f4ac665be4943ce769e33109e9f64abcf1e98050",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/0919d2e50fe1008af261f8301f243c002c328dbf",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/f8ed067bb4bc06662d6bdf6210d450779a529ce5",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/6ad9b0047f8437f7b012041d7b2fcd190a5a9ec2",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b431248ab2f121914608bbd1c2376715cecda9c8",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/d4ee4832d06e8560510d81237d0650c897d476e9",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/b1d54d3c850d35f99329591e3b57cef98d17237f",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/86a67b81c2de995bd608d5b2df50cd8cd7d92455",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/82c24522483ceaf1d54224b69af4244b60c3ac08",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/87f9e315fd7e2ba406057a97300593c4802b53e4",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/486623019ef451e0582b874018e0249a46e0f996",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/18fbf88c578fc9f75d4610ebd18ab55f4f2842ce",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/200db82bfdbc81cd227cb3470aa826d6f11a7653",
+    "https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/957584ae6a35f9edf293cb486d7436fb5b75e803"
+]
+\ No newline at end of file
diff --git a/test/test-pages/wikipedia-3/expected-metadata.json b/test/test-pages/wikipedia-3/expected-metadata.json
new file mode 100644
index 0000000..27df156
--- /dev/null
+++ b/test/test-pages/wikipedia-3/expected-metadata.json
@@ -0,0 +1,8 @@
+{
+    "Author": "Contributors to Wikimedia projects",
+    "Direction": "ltr",
+    "Excerpt": "In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose\u2014that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j:",
+    "Image": null,
+    "Title": "Hermitian matrix",
+    "SiteName": "Wikimedia Foundation, Inc."
+}
+\ No newline at end of file
diff --git a/test/test-pages/wikipedia-3/expected.html b/test/test-pages/wikipedia-3/expected.html
new file mode 100644
index 0000000..159b7e6
--- /dev/null
+++ b/test/test-pages/wikipedia-3/expected.html
@@ -0,0 +1,356 @@
+<div id="mw-content-text" lang="en" dir="ltr" xml:lang="en">
+                    <div>
+                        
+                        
+                        <p>
+                            In mathematics, a <b>Hermitian matrix</b> (or <b>self-adjoint matrix</b>) is a <a href="http://fakehost/wiki/Complex_number" title="Complex number">complex</a> <a href="http://fakehost/wiki/Square_matrix" title="Square matrix">square matrix</a> that is equal to its own <a href="http://fakehost/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>—that is, the element in the <span>i</span>-th row and <span>j</span>-th column is equal to the <a href="http://fakehost/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of the element in the <span>j</span>-th row and <span>i</span>-th column, for all indices <span>i</span> and <span>j</span>:
+                        </p>
+                        <p>
+                                <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28a0aaa74b2267a48312e19321211cd9e3a39228" aria-hidden="true" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}"></span>
+                            </p>
+                        <p>
+                            or in matrix form:
+                        </p>
+                        <dl>
+                            <dd>
+                                <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca00b61ff0e264e6c1e5adc9a00c0d2751feecf" aria-hidden="true" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}}"></span>.
+                            </dd>
+                        </dl>
+                        <p>
+                            Hermitian matrices can be understood as the complex extension of real <a href="http://fakehost/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrices</a>.
+                        </p>
+                        <p>
+                            If the <a href="http://fakehost/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of a matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is denoted by <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9415702ab196cc26f5df37af2d90e07318e93df" aria-hidden="true" alt="{\displaystyle A^{\mathsf {H}}}"></span>, then the Hermitian property can be written concisely as
+                        </p>
+                        <p>
+                                <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291d260bf69b764e75818669ab27870d58771e1f" aria-hidden="true" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}"></span>
+                            </p>
+                        <p>
+                            Hermitian matrices are named after <a href="http://fakehost/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a>, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real <a href="http://fakehost/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a>. Other, equivalent notations in common use are <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa270391d183478251283d2c4b2c72ac4563352" aria-hidden="true" alt="{\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}"></span>, although note that in <a href="http://fakehost/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5541bfa07743be995242c892c344395e672d6fa2" aria-hidden="true" alt="A^{\ast }"></span> typically means the <a href="http://fakehost/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> only, and not the <a href="http://fakehost/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>.
+                        </p>
+                        
+                        <h2>
+                            <span id="Alternative_characterizations">Alternative characterizations</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=1" title="Edit section: Alternative characterizations">edit</a><span>]</span></span>
+                        </h2>
+                        <p>
+                            Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
+                        </p>
+                        <h3>
+                            <span id="Equality_with_the_adjoint">Equality with the adjoint</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=2" title="Edit section: Equality with the adjoint">edit</a><span>]</span></span>
+                        </h3>
+                        <p>
+                            A square matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is Hermitian if and only if it is equal to its <a href="http://fakehost/wiki/Hermitian_adjoint" title="Hermitian adjoint">adjoint</a>, that is, it satisfies
+                        </p>
+                        <div>
+                            <p><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459de45e76bace9d04a80d2e8efc2abbbc246047" aria-hidden="true" alt="{\displaystyle \langle w,Av\rangle =\langle Aw,v\rangle ,}">
+                        </p></div><p>for any pair of vectors <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6425c6e94fa47976601cb44d7564b5d04dcfbfef" aria-hidden="true" alt="v,w"></span>, where <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" aria-hidden="true" alt="{\displaystyle \langle \cdot ,\cdot \rangle }"></span> denotes <a href="http://fakehost/wiki/Dot_product" title="Dot product">the inner product</a> operation.
+                        </p><p>
+                            This is also the way that the more general concept of <a href="http://fakehost/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operator</a> is defined.
+                        </p>
+                        <h3>
+                            <span id="Reality_of_quadratic_forms">Reality of quadratic forms</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=3" title="Edit section: Reality of quadratic forms">edit</a><span>]</span></span>
+                        </h3>
+                        <p>
+                            A square matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is Hermitian if and only if it is such that
+                        </p>
+                        <div>
+                            <p><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997ea0350c18735926412de88420ac9ca989f50c" aria-hidden="true" alt="{\displaystyle \langle v,Av\rangle \in \mathbb {R} ,\quad v\in V.}">
+                        </p></div>
+                        <h3>
+                            <span id="Spectral_properties">Spectral properties</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=4" title="Edit section: Spectral properties">edit</a><span>]</span></span>
+                        </h3>
+                        <p>
+                            A square matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is Hermitian if and only if it is unitarily <a href="http://fakehost/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable</a> with real <a href="http://fakehost/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a>.
+                        </p>
+                        <h2>
+                            <span id="Applications">Applications</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=5" title="Edit section: Applications">edit</a><span>]</span></span>
+                        </h2>
+                        <p>
+                            Hermitian matrices are fundamental to the quantum theory of <a href="http://fakehost/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> created by <a href="http://fakehost/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="http://fakehost/wiki/Max_Born" title="Max Born">Max Born</a>, and <a href="http://fakehost/wiki/Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a> in 1925.
+                        </p>
+                        <h2>
+                            <span id="Examples">Examples</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=6" title="Edit section: Examples">edit</a><span>]</span></span>
+                        </h2>
+                        <p>
+                            In this section, the conjugate transpose of matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is denoted as <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9415702ab196cc26f5df37af2d90e07318e93df" aria-hidden="true" alt="{\displaystyle A^{\mathsf {H}}}"></span>, the transpose of matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is denoted as <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54bf0331204e30cba9ec7f695dfea97e6745a7c2" aria-hidden="true" alt="{\displaystyle A^{\mathsf {T}}}"></span> and conjugate of matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is denoted as <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" aria-hidden="true" alt="{\displaystyle {\overline {A}}}"></span>.
+                        </p>
+                        <p>
+                            See the following example:
+                        </p>
+                        <dl>
+                            <dd>
+                                <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ccf11c16396b6ddd4f2254f7132cd8bb2c57ee" aria-hidden="true" alt="{\displaystyle {\begin{bmatrix}2&amp;2+i&amp;4\\2-i&amp;3&amp;i\\4&amp;-i&amp;1\\\end{bmatrix}}}"></span>
+                            </dd>
+                        </dl>
+                        <p>
+                            The diagonal elements must be <a href="http://fakehost/wiki/Real_number" title="Real number">real</a>, as they must be their own complex conjugate.
+                        </p>
+                        <p>
+                            Well-known families of <a href="http://fakehost/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>, <a href="http://fakehost/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann matrices</a> and their generalizations are Hermitian. In <a href="http://fakehost/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a> such Hermitian matrices are often multiplied by <a href="http://fakehost/wiki/Imaginary_number" title="Imaginary number">imaginary</a> coefficients,<sup id="cite_ref-1"><a href="#cite_note-1">[1]</a></sup><sup id="cite_ref-2"><a href="#cite_note-2">[2]</a></sup> which results in <i>skew-Hermitian</i> matrices (see <a href="#facts">below</a>).
+                        </p>
+                        <p>
+                            Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> equals the <a href="http://fakehost/wiki/Matrix_multiplication" title="Matrix multiplication">multiplication of a matrix</a> and its conjugate transpose, that is, <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0efab2d7c3a4b4b7caf14cc0705dadd13195a9" aria-hidden="true" alt="{\displaystyle A=BB^{\mathsf {H}}}"></span>, then <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is a Hermitian <a href="http://fakehost/wiki/Positive_semi-definite_matrix" title="Positive semi-definite matrix">positive semi-definite matrix</a>. Furthermore, if <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" aria-hidden="true" alt="B"></span> is row full-rank, then <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is positive definite.
+                        </p>
+                        <h2>
+                            <span id="Properties">Properties</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=7" title="Edit section: Properties">edit</a><span>]</span></span>
+                        </h2>
+                        <table role="presentation" readabilityDataTable="0">
+                            <tbody>
+                                <tr>
+                                    <td>
+                                        <p><a href="http://fakehost/wiki/File:Wiki_letter_w_cropped.svg"><img alt="[icon]" src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/44px-Wiki_letter_w_cropped.svg.png" decoding="async" width="44" height="31" srcset="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/66px-Wiki_letter_w_cropped.svg.png 1.5x, http://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/88px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31"></a>
+                                        </p>
+                                    </td>
+                                    <td>
+                                        <p>
+                                            This section <b>needs expansion</b> with: Proof of the properties requested. <small>You can help by <a href="https://en.wikipedia.org/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=1">adding to it</a>.</small> <small><i>(<span>February 2018</span>)</i></small>
+                                        </p>
+                                    </td>
+                                </tr>
+                            </tbody>
+                        </table>
+                        <ul>
+                            <li>The entries on the <a href="http://fakehost/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> (top left to bottom right) of any Hermitian matrix are <a href="http://fakehost/wiki/Real_number" title="Real number">real</a>.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> By definition of the Hermitian matrix
+                                <dl>
+                                    <dd>
+                                        <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa8265c5f6d4fc3b7cda6a06558c7d4d9aec855" aria-hidden="true" alt="{\displaystyle H_{ij}={\overline {H}}_{ji}}"></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                            <dd>
+                                so for <span><i>i</i> = <i>j</i></span> the above follows.
+                            </dd>
+                            <dd>
+                                Only the <a href="http://fakehost/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their <a href="http://fakehost/wiki/Off-diagonal_element" title="Off-diagonal element">off-diagonal elements</a>, as long as diagonally-opposite entries are complex conjugates.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>A matrix that has only real entries is Hermitian <a href="http://fakehost/wiki/If_and_only_if" title="If and only if">if and only if</a> it is <a href="http://fakehost/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>. A real and symmetric matrix is simply a special case of a Hermitian matrix.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa8265c5f6d4fc3b7cda6a06558c7d4d9aec855" aria-hidden="true" alt="{\displaystyle H_{ij}={\overline {H}}_{ji}}"></span> by definition. Thus <span>H<sub><i>ij</i></sub> = H<sub><i>ji</i></sub></span> (matrix symmetry) if and only if <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1862750b96d01100244370b3fca45f01923ce5" aria-hidden="true" alt="{\displaystyle H_{ij}={\overline {H}}_{ij}}"></span> (<span>H<sub><i>ij</i></sub></span> is real).
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>Every Hermitian matrix is a <a href="http://fakehost/wiki/Normal_matrix" title="Normal matrix">normal matrix</a>. That is to say, <span>AA<sup>H</sup> = A<sup>H</sup>A</span>.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> <span>A = A<sup>H</sup></span>, so <span>AA<sup>H</sup> = AA = A<sup>H</sup>A</span>.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>The finite-dimensional <a href="http://fakehost/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a> says that any Hermitian matrix can be <a href="http://fakehost/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalized</a> by a <a href="http://fakehost/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a>, and that the resulting diagonal matrix has only real entries. This implies that all <a href="http://fakehost/wiki/Eigenvectors" title="Eigenvectors">eigenvalues</a> of a Hermitian matrix <span>A</span> with dimension <span>n</span> are real, and that <span>A</span> has <span>n</span> linearly independent <a href="http://fakehost/wiki/Eigenvector" title="Eigenvector">eigenvectors</a>. Moreover, a Hermitian matrix has <a href="http://fakehost/wiki/Orthogonal" title="Orthogonal">orthogonal</a> eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an <a href="http://fakehost/wiki/Orthogonal_basis" title="Orthogonal basis">orthogonal basis</a> of <span>ℂ<sup><i>n</i></sup></span> consisting of <span>n</span> eigenvectors of <span>A</span>.
+                            </li>
+                        </ul>
+                        <ul>
+                            <li>The sum of any two Hermitian matrices is Hermitian.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251bf4ebbe3b0d119e0a7d19f8fd134c4f072971" aria-hidden="true" alt="{\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},}"></span> as claimed.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>The <a href="http://fakehost/wiki/Inverse_matrix" title="Inverse matrix">inverse</a> of an invertible Hermitian matrix is Hermitian as well.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> If <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021893240ff7fa3148b6649b0ba4d88cd207b5f0" aria-hidden="true" alt="{\displaystyle A^{-1}A=I}"></span>, then <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a28a8250ab35ad60228bb0376eb4b7024f027815" aria-hidden="true" alt="{\displaystyle I=I^{H}=(A^{-1}A)^{H}=A^{H}(A^{-1})^{H}=A(A^{-1})^{H}}"></span>, so <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0179c3a7aebe194ccd9a19ba02b972500f88a7a" aria-hidden="true" alt="{\displaystyle A^{-1}=(A^{-1})^{H}}"></span> as claimed.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>The <a href="http://fakehost/wiki/Matrix_multiplication" title="Matrix multiplication">product</a> of two Hermitian matrices <span>A</span> and <span>B</span> is Hermitian if and only if <span><i>AB</i> = <i>BA</i></span>.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> Note that <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cf8185ca7a0687bf959bb65b16db6cf1714ca2" aria-hidden="true" alt="{\displaystyle (AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}{\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.}"></span> Thus <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d303a1ebcac8547489b170be5d0dd7d8e04e548e" aria-hidden="true" alt="{\displaystyle (AB)^{\mathsf {H}}=AB}"></span> <a href="http://fakehost/wiki/If_and_only_if" title="If and only if">if and only if</a> <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992c8ea49fdd26b491180036c5a4d879fec77442" aria-hidden="true" alt="AB=BA"></span>.
+                            </dd>
+                            <dd>
+                                Thus <span><i>A</i><sup><i>n</i></sup></span> is Hermitian if <span>A</span> is Hermitian and <span>n</span> is an integer.
+                            </dd>
+                        </dl>
+                        
+                        <ul>
+                            <li>The Hermitian complex <span>n</span>-by-<span>n</span> matrices do not form a <a href="http://fakehost/wiki/Vector_space" title="Vector space">vector space</a> over the <a href="http://fakehost/wiki/Complex_number" title="Complex number">complex numbers</a>, <span>ℂ</span>, since the identity matrix <span><i>I</i><sub><i>n</i></sub></span> is Hermitian, but <span><i>i</i> <i>I</i><sub><i>n</i></sub></span> is not. However the complex Hermitian matrices <i>do</i> form a vector space over the <a href="http://fakehost/wiki/Real_numbers" title="Real numbers">real numbers</a> <span>ℝ</span>. In the <span>2<i>n</i><sup>2</sup></span>-<a href="http://fakehost/wiki/Dimension_of_a_vector_space" title="Dimension of a vector space">dimensional</a> vector space of complex <span><i>n</i> × <i>n</i></span> matrices over <span>ℝ</span>, the complex Hermitian matrices form a subspace of dimension <span><i>n</i><sup>2</sup></span>. If <span><i>E</i><sub><i>jk</i></sub></span> denotes the <span>n</span>-by-<span>n</span> matrix with a <span>1</span> in the <span><i>j</i>,<i>k</i></span> position and zeros elsewhere, a basis (orthonormal w.r.t. the Frobenius inner product) can be described as follows:
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <dl>
+                                    <dd>
+                                        <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46eedb181c0bdae46e8c1526161b03d0ea97b4b4" aria-hidden="true" alt="{\displaystyle E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})}"></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                        </dl>
+                        <dl>
+                            <dd>
+                                together with the set of matrices of the form
+                            </dd>
+                        </dl>
+                        <dl>
+                            <dd>
+                                <dl>
+                                    <dd>
+                                        <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddeac51c423f6dbefc5f63e483d9aee96e6fa342" aria-hidden="true" alt="{\displaystyle {\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}"></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                        </dl>
+                        <dl>
+                            <dd>
+                                and the matrices
+                            </dd>
+                        </dl>
+                        <dl>
+                            <dd>
+                                <dl>
+                                    <dd>
+                                        <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db65cce3a8fa33e5b7b96badd756c8573aa866c0" aria-hidden="true" alt="{\displaystyle {\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}"></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                        </dl>
+                        <dl>
+                            <dd>
+                                where <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" aria-hidden="true" alt="i"></span> denotes the complex number <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" aria-hidden="true" alt="{\sqrt {-1}}"></span>, called the <i><a href="http://fakehost/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a></i>.
+                            </dd>
+                        </dl>
+                        
+                        <dl>
+                            <dd>
+                                <dl>
+                                    <dd>
+                                        <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7d749931e5f709bcbc0a446638d3b6b8ed0c6c" aria-hidden="true" alt="{\displaystyle A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\mathsf {H}},}"></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                            <dd>
+                                where <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa91daf9145f27bb95746fd2a37537342d587b77" aria-hidden="true" alt="\lambda _{j}"></span> are the eigenvalues on the diagonal of the diagonal matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1934e7eadd31fbf6f7d6bcf9c0e9bec934ce8976" aria-hidden="true" alt="\; \Lambda "></span>.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>The determinant of a Hermitian matrix is real:
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1240df64c3010e0be6eae865fdfcfe6f77bf5eb" aria-hidden="true" alt="{\displaystyle \det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}}"></span>
+                            </dd>
+                            <dd>
+                                Therefore if <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43cc392bdcfbb134dd66d9b469847f6370e29d9d" aria-hidden="true" alt="{\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}}"></span>.
+                            </dd>
+                            <dd>
+                                (Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
+                            </dd>
+                        </dl>
+                        <h2>
+                            <span id="Decomposition_into_Hermitian_and_skew-Hermitian">Decomposition into Hermitian and skew-Hermitian</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=8" title="Edit section: Decomposition into Hermitian and skew-Hermitian">edit</a><span>]</span></span>
+                        </h2>
+                        <p>
+                            <span id="facts"></span>Additional facts related to Hermitian matrices include:
+                        </p>
+                        <ul>
+                            <li>The sum of a square matrix and its conjugate transpose <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef97bb04ce4ab682bcc84cf1059f8da235b483e" aria-hidden="true" alt="{\displaystyle \left(A+A^{\mathsf {H}}\right)}"></span> is Hermitian.
+                            </li>
+                        </ul>
+                        <ul>
+                            <li>The difference of a square matrix and its conjugate transpose <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ac665be4943ce769e33109e9f64abcf1e98050" aria-hidden="true" alt="{\displaystyle \left(A-A^{\mathsf {H}}\right)}"></span> is <a href="http://fakehost/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">skew-Hermitian</a> (also called antihermitian). This implies that the <a href="http://fakehost/wiki/Commutator" title="Commutator">commutator</a> of two Hermitian matrices is skew-Hermitian.
+                            </li>
+                        </ul>
+                        <ul>
+                            <li>An arbitrary square matrix <span>C</span> can be written as the sum of a Hermitian matrix <span>A</span> and a skew-Hermitian matrix <span>B</span>. This is known as the Toeplitz decomposition of <span>C</span>.<sup id="cite_ref-HornJohnson_3-0"><a href="#cite_note-HornJohnson-3">[3]</a></sup><sup>:<span>p. 7</span></sup>
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <dl>
+                                    <dd>
+                                        <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0919d2e50fe1008af261f8301f243c002c328dbf" aria-hidden="true" alt="{\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}"></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                        </dl>
+                        <h2>
+                            <span id="Rayleigh_quotient">Rayleigh quotient</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=9" title="Edit section: Rayleigh quotient">edit</a><span>]</span></span>
+                        </h2>
+                        
+                        <p>
+                            In mathematics, for a given complex Hermitian matrix <i>M</i> and nonzero vector <i>x</i>, the Rayleigh quotient<sup id="cite_ref-4"><a href="#cite_note-4">[4]</a></sup> <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ed067bb4bc06662d6bdf6210d450779a529ce5" aria-hidden="true" alt="R(M, x)"></span>, is defined as:<sup id="cite_ref-HornJohnson_3-1"><a href="#cite_note-HornJohnson-3">[3]</a></sup><sup>:<span>p. 234</span></sup><sup id="cite_ref-5"><a href="#cite_note-5">[5]</a></sup>
+                        </p>
+                        <dl>
+                            <dd>
+                                <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad9b0047f8437f7b012041d7b2fcd190a5a9ec2" aria-hidden="true" alt="{\displaystyle R(M,x):={\frac {x^{\mathsf {H}}Mx}{x^{\mathsf {H}}x}}}"></span>.
+                            </dd>
+                        </dl>
+                        <p>
+                            For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b431248ab2f121914608bbd1c2376715cecda9c8" aria-hidden="true" alt="{\displaystyle x^{\mathsf {H}}}"></span> to the usual transpose <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ee4832d06e8560510d81237d0650c897d476e9" aria-hidden="true" alt="{\displaystyle x^{\mathsf {T}}}"></span>. Note that <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d54d3c850d35f99329591e3b57cef98d17237f" aria-hidden="true" alt="{\displaystyle R(M,cx)=R(M,x)}"></span> for any non-zero real scalar <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" aria-hidden="true" alt="c"></span>. Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.
+                        </p>
+                        <p>
+                            It can be shown<sup>[<i><a href="http://fakehost/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2019)">citation needed</span></a></i>]</sup> that, for a given matrix, the Rayleigh quotient reaches its minimum value <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c24522483ceaf1d54224b69af4244b60c3ac08" aria-hidden="true" alt="\lambda_\min"></span> (the smallest eigenvalue of M) when <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" aria-hidden="true" alt="x"></span> is <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/486623019ef451e0582b874018e0249a46e0f996" aria-hidden="true" alt="v_\min"></span> (the corresponding eigenvector). Similarly, <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18fbf88c578fc9f75d4610ebd18ab55f4f2842ce" aria-hidden="true" alt="R(M, x) \leq \lambda_\max"></span> and <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200db82bfdbc81cd227cb3470aa826d6f11a7653" aria-hidden="true" alt="R(M, v_\max) = \lambda_\max"></span>.
+                        </p>
+                        <p>
+                            The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.
+                        </p>
+                        <p>
+                            The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957584ae6a35f9edf293cb486d7436fb5b75e803" aria-hidden="true" alt="\lambda_\max"></span> is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to <span><i>M</i></span> associates the Rayleigh quotient <span><i>R</i>(<i>M</i>, <i>x</i>)</span> for a fixed <span><i>x</i></span> and <span><i>M</i></span> varying through the algebra would be referred to as "vector state" of the algebra.
+                        </p>
+                        <h2>
+                            <span id="See_also">See also</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=10" title="Edit section: See also">edit</a><span>]</span></span>
+                        </h2>
+                        <ul>
+                            <li>
+                                <a href="http://fakehost/wiki/Vector_space" title="Vector space">Vector space</a>
+                            </li>
+                            <li>
+                                <a href="http://fakehost/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian matrix</a> (anti-Hermitian matrix)
+                            </li>
+                            <li>
+                                <a href="http://fakehost/wiki/Haynsworth_inertia_additivity_formula" title="Haynsworth inertia additivity formula">Haynsworth inertia additivity formula</a>
+                            </li>
+                            <li>
+                                <a href="http://fakehost/wiki/Hermitian_form" title="Hermitian form">Hermitian form</a>
+                            </li>
+                            <li>
+                                <a href="http://fakehost/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint operator</a>
+                            </li>
+                            <li>
+                                <a href="http://fakehost/wiki/Unitary_matrix" title="Unitary matrix">Unitary matrix</a>
+                            </li>
+                        </ul>
+                        <h2>
+                            <span id="References">References</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=11" title="Edit section: References">edit</a><span>]</span></span>
+                        </h2>
+                        
+                        <h2>
+                            <span id="External_links">External links</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=12" title="Edit section: External links">edit</a><span>]</span></span>
+                        </h2>
+                        <ul>
+                            <li>
+                                <cite id="CITEREFHazewinkel2001"><a href="http://fakehost/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>, ed. (2001) [1994], <a rel="nofollow" href="https://www.encyclopediaofmath.org/index.php?title=p/h047070">"Hermitian matrix"</a>, <i><a href="http://fakehost/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, Springer Science+Business Media B.V. / Kluwer Academic Publishers, <a href="http://fakehost/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&nbsp;<a href="http://fakehost/wiki/Special:BookSources/978-1-55608-010-4" title="Special:BookSources/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a></cite>
+                                
+                            </li>
+                            <li>
+                                <a rel="nofollow" href="https://www.cyut.edu.tw/~ckhung/b/la/hermitian.en.php">Visualizing Hermitian Matrix as An Ellipse with Dr. Geo</a>, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.
+                            </li>
+                            <li>
+                                <cite><a rel="nofollow" href="http://www.mathpages.com/home/kmath306/kmath306.htm">"Hermitian Matrices"</a>. <i>MathPages.com</i>.</cite>
+                                
+                            </li>
+                        </ul>
+                        
+                        
+                    </div>
+                </div>
+\ No newline at end of file
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+            <h1 id="firstHeading" class="firstHeading" lang="en" xml:lang="en">
+                Hermitian matrix
+            </h1>
+            <div id="bodyContent" class="mw-body-content">
+                <div id="siteSub" class="noprint">
+                    From Wikipedia, the free encyclopedia
+                </div>
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+                        <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">
+                            Matrix equal to its conjugate-transpose
+                        </div>
+                        <div role="note" class="hatnote navigation-not-searchable">
+                            For matrices with symmetry over the <a href="/wiki/Real_number" title="Real number">real number</a> field, see <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrix</a>.
+                        </div>
+                        <p>
+                            In mathematics, a <b>Hermitian matrix</b> (or <b>self-adjoint matrix</b>) is a <a href="/wiki/Complex_number" title="Complex number">complex</a> <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> that is equal to its own <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>—that is, the element in the <span class="texhtml mvar" style="font-style:italic;">i</span>-th row and <span class="texhtml mvar" style="font-style:italic;">j</span>-th column is equal to the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of the element in the <span class="texhtml mvar" style="font-style:italic;">j</span>-th row and <span class="texhtml mvar" style="font-style:italic;">i</span>-th column, for all indices <span class="texhtml mvar" style="font-style:italic;">i</span> and <span class="texhtml mvar" style="font-style:italic;">j</span>:
+                        </p>
+                        <div class="equation-box" style="margin: ;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; background-color: #F5FFFA; text-align: center; display: table">
+                            <p>
+                                <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}">
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+                                        {\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28a0aaa74b2267a48312e19321211cd9e3a39228" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:32.77ex; height:3.009ex;" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}" /></span>
+                            </p>
+                        </div>
+                        <p>
+                            or in matrix form:
+                        </p>
+                        <dl>
+                            <dd>
+                                <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mtext>
+                                                    &nbsp;Hermitian
+                                                </mtext>
+                                            </mrow>
+                                            <mspace width="1em"></mspace>
+                                            <mspace width="thickmathspace"></mspace>
+                                            <mo stretchy="false">
+                                                ⟺<!-- ⟺ -->
+                                            </mo>
+                                            <mspace width="thickmathspace"></mspace>
+                                            <mspace width="1em"></mspace>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mover>
+                                                    <msup>
+                                                        <mi>
+                                                            A
+                                                        </mi>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mi mathvariant="sans-serif">
+                                                                    T
+                                                                </mi>
+                                                            </mrow>
+                                                        </mrow>
+                                                    </msup>
+                                                    <mo accent="false">
+                                                        ¯<!-- ¯ -->
+                                                    </mo>
+                                                </mover>
+                                            </mrow>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca00b61ff0e264e6c1e5adc9a00c0d2751feecf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:32.194ex; height:3.509ex;" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}}" /></span>.
+                            </dd>
+                        </dl>
+                        <p>
+                            Hermitian matrices can be understood as the complex extension of real <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrices</a>.
+                        </p>
+                        <p>
+                            If the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of a matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is denoted by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mathsf {H}}}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi mathvariant="sans-serif">
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </mrow>
+                                        </msup>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A^{\mathsf {H}}}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9415702ab196cc26f5df37af2d90e07318e93df" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.139ex; height:2.676ex;" alt="{\displaystyle A^{\mathsf {H}}}" /></span>, then the Hermitian property can be written concisely as
+                        </p>
+                        <div class="equation-box" style="margin: ;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; background-color: #F5FFFA; text-align: center; display: table">
+                            <p>
+                                <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mtext>
+                                                    &nbsp;Hermitian
+                                                </mtext>
+                                            </mrow>
+                                            <mspace width="1em"></mspace>
+                                            <mspace width="thickmathspace"></mspace>
+                                            <mo stretchy="false">
+                                                ⟺<!-- ⟺ -->
+                                            </mo>
+                                            <mspace width="thickmathspace"></mspace>
+                                            <mspace width="1em"></mspace>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msup>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291d260bf69b764e75818669ab27870d58771e1f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:32.123ex; height:2.676ex;" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}" /></span>
+                            </p>
+                        </div>
+                        <p>
+                            Hermitian matrices are named after <a href="/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a>, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a>. Other, equivalent notations in common use are <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi mathvariant="sans-serif">
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </mrow>
+                                        </msup>
+                                        <mo>
+                                            =
+                                        </mo>
+                                        <msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mo>
+                                                    †<!-- † -->
+                                                </mo>
+                                            </mrow>
+                                        </msup>
+                                        <mo>
+                                            =
+                                        </mo>
+                                        <msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mo>
+                                                    ∗<!-- ∗ -->
+                                                </mo>
+                                            </mrow>
+                                        </msup>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa270391d183478251283d2c4b2c72ac4563352" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:14.839ex; height:2.676ex;" alt="{\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}" /></span>, although note that in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\ast }}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mo>
+                                                    ∗<!-- ∗ -->
+                                                </mo>
+                                            </mrow>
+                                        </msup>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A^{\ast }}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5541bfa07743be995242c892c344395e672d6fa2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="A^{\ast }" /></span> typically means the <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> only, and not the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>.
+                        </p>
+                        <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading">
+                            <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" />
+                            <div class="toctitle" lang="en" dir="ltr" xml:lang="en">
+                                <h2 id="mw-toc-heading">
+                                    Contents
+                                </h2>
+                            </div>
+                            <ul>
+                                <li class="toclevel-1 tocsection-1">
+                                    <a href="#Alternative_characterizations"><span class="tocnumber">1</span> <span class="toctext">Alternative characterizations</span></a>
+                                    <ul>
+                                        <li class="toclevel-2 tocsection-2">
+                                            <a href="#Equality_with_the_adjoint"><span class="tocnumber">1.1</span> <span class="toctext">Equality with the adjoint</span></a>
+                                        </li>
+                                        <li class="toclevel-2 tocsection-3">
+                                            <a href="#Reality_of_quadratic_forms"><span class="tocnumber">1.2</span> <span class="toctext">Reality of quadratic forms</span></a>
+                                        </li>
+                                        <li class="toclevel-2 tocsection-4">
+                                            <a href="#Spectral_properties"><span class="tocnumber">1.3</span> <span class="toctext">Spectral properties</span></a>
+                                        </li>
+                                    </ul>
+                                </li>
+                                <li class="toclevel-1 tocsection-5">
+                                    <a href="#Applications"><span class="tocnumber">2</span> <span class="toctext">Applications</span></a>
+                                </li>
+                                <li class="toclevel-1 tocsection-6">
+                                    <a href="#Examples"><span class="tocnumber">3</span> <span class="toctext">Examples</span></a>
+                                </li>
+                                <li class="toclevel-1 tocsection-7">
+                                    <a href="#Properties"><span class="tocnumber">4</span> <span class="toctext">Properties</span></a>
+                                </li>
+                                <li class="toclevel-1 tocsection-8">
+                                    <a href="#Decomposition_into_Hermitian_and_skew-Hermitian"><span class="tocnumber">5</span> <span class="toctext">Decomposition into Hermitian and skew-Hermitian</span></a>
+                                </li>
+                                <li class="toclevel-1 tocsection-9">
+                                    <a href="#Rayleigh_quotient"><span class="tocnumber">6</span> <span class="toctext">Rayleigh quotient</span></a>
+                                </li>
+                                <li class="toclevel-1 tocsection-10">
+                                    <a href="#See_also"><span class="tocnumber">7</span> <span class="toctext">See also</span></a>
+                                </li>
+                                <li class="toclevel-1 tocsection-11">
+                                    <a href="#References"><span class="tocnumber">8</span> <span class="toctext">References</span></a>
+                                </li>
+                                <li class="toclevel-1 tocsection-12">
+                                    <a href="#External_links"><span class="tocnumber">9</span> <span class="toctext">External links</span></a>
+                                </li>
+                            </ul>
+                        </div>
+                        <h2>
+                            <span class="mw-headline" id="Alternative_characterizations">Alternative characterizations</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=1" title="Edit section: Alternative characterizations">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h2>
+                        <p>
+                            Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
+                        </p>
+                        <h3>
+                            <span class="mw-headline" id="Equality_with_the_adjoint">Equality with the adjoint</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=2" title="Edit section: Equality with the adjoint">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h3>
+                        <p>
+                            A square matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is Hermitian if and only if it is equal to its <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">adjoint</a>, that is, it satisfies
+                        </p>
+                        <div class="mwe-math-element">
+                            <div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;">
+                                <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle w,Av\rangle =\langle Aw,v\rangle ,}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mo fence="false" stretchy="false">
+                                                ⟨<!-- ⟨ -->
+                                            </mo>
+                                            <mi>
+                                                w
+                                            </mi>
+                                            <mo>
+                                                ,
+                                            </mo>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mi>
+                                                v
+                                            </mi>
+                                            <mo fence="false" stretchy="false">
+                                                ⟩<!-- ⟩ -->
+                                            </mo>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mo fence="false" stretchy="false">
+                                                ⟨<!-- ⟨ -->
+                                            </mo>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mi>
+                                                w
+                                            </mi>
+                                            <mo>
+                                                ,
+                                            </mo>
+                                            <mi>
+                                                v
+                                            </mi>
+                                            <mo fence="false" stretchy="false">
+                                                ⟩<!-- ⟩ -->
+                                            </mo>
+                                            <mo>
+                                                ,
+                                            </mo>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle \langle w,Av\rangle =\langle Aw,v\rangle ,}
+                                    </annotation>
+                                </semantics></math>
+                            </div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459de45e76bace9d04a80d2e8efc2abbbc246047" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:18.501ex; height:2.843ex;" alt="{\displaystyle \langle w,Av\rangle =\langle Aw,v\rangle ,}" />
+                        </div>for any pair of vectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v,w}">
+                        <semantics>
+                            <mrow class="MJX-TeXAtom-ORD">
+                                <mstyle displaystyle="true" scriptlevel="0">
+                                    <mi>
+                                        v
+                                    </mi>
+                                    <mo>
+                                        ,
+                                    </mo>
+                                    <mi>
+                                        w
+                                    </mi>
+                                </mstyle>
+                            </mrow>
+                            <annotation encoding="application/x-tex">
+                                {\displaystyle v,w}
+                            </annotation>
+                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6425c6e94fa47976601cb44d7564b5d04dcfbfef" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:3.826ex; height:2.009ex;" alt="v,w" /></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \cdot ,\cdot \rangle }">
+                        <semantics>
+                            <mrow class="MJX-TeXAtom-ORD">
+                                <mstyle displaystyle="true" scriptlevel="0">
+                                    <mo fence="false" stretchy="false">
+                                        ⟨<!-- ⟨ -->
+                                    </mo>
+                                    <mo>
+                                        ⋅<!-- ⋅ -->
+                                    </mo>
+                                    <mo>
+                                        ,
+                                    </mo>
+                                    <mo>
+                                        ⋅<!-- ⋅ -->
+                                    </mo>
+                                    <mo fence="false" stretchy="false">
+                                        ⟩<!-- ⟩ -->
+                                    </mo>
+                                </mstyle>
+                            </mrow>
+                            <annotation encoding="application/x-tex">
+                                {\displaystyle \langle \cdot ,\cdot \rangle }
+                            </annotation>
+                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.137ex; height:2.843ex;" alt="{\displaystyle \langle \cdot ,\cdot \rangle }" /></span> denotes <a href="/wiki/Dot_product" title="Dot product">the inner product</a> operation.
+                        <p>
+                            This is also the way that the more general concept of <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operator</a> is defined.
+                        </p>
+                        <h3>
+                            <span class="mw-headline" id="Reality_of_quadratic_forms">Reality of quadratic forms</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=3" title="Edit section: Reality of quadratic forms">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h3>
+                        <p>
+                            A square matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is Hermitian if and only if it is such that
+                        </p>
+                        <div class="mwe-math-element">
+                            <div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;">
+                                <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle v,Av\rangle \in \mathbb {R} ,\quad v\in V.}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mo fence="false" stretchy="false">
+                                                ⟨<!-- ⟨ -->
+                                            </mo>
+                                            <mi>
+                                                v
+                                            </mi>
+                                            <mo>
+                                                ,
+                                            </mo>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mi>
+                                                v
+                                            </mi>
+                                            <mo fence="false" stretchy="false">
+                                                ⟩<!-- ⟩ -->
+                                            </mo>
+                                            <mo>
+                                                ∈<!-- ∈ -->
+                                            </mo>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mi mathvariant="double-struck">
+                                                    R
+                                                </mi>
+                                            </mrow>
+                                            <mo>
+                                                ,
+                                            </mo>
+                                            <mspace width="1em"></mspace>
+                                            <mi>
+                                                v
+                                            </mi>
+                                            <mo>
+                                                ∈<!-- ∈ -->
+                                            </mo>
+                                            <mi>
+                                                V
+                                            </mi>
+                                            <mo>
+                                                .
+                                            </mo>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle \langle v,Av\rangle \in \mathbb {R} ,\quad v\in V.}
+                                    </annotation>
+                                </semantics></math>
+                            </div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997ea0350c18735926412de88420ac9ca989f50c" class="mwe-math-fallback-image-display" aria-hidden="true" style="vertical-align: -0.838ex; width:21.119ex; height:2.843ex;" alt="{\displaystyle \langle v,Av\rangle \in \mathbb {R} ,\quad v\in V.}" />
+                        </div>
+                        <h3>
+                            <span class="mw-headline" id="Spectral_properties">Spectral properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=4" title="Edit section: Spectral properties">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h3>
+                        <p>
+                            A square matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is Hermitian if and only if it is unitarily <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable</a> with real <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a>.
+                        </p>
+                        <h2>
+                            <span class="mw-headline" id="Applications">Applications</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=5" title="Edit section: Applications">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h2>
+                        <p>
+                            Hermitian matrices are fundamental to the quantum theory of <a href="/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> created by <a href="/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="/wiki/Max_Born" title="Max Born">Max Born</a>, and <a href="/wiki/Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a> in 1925.
+                        </p>
+                        <h2>
+                            <span class="mw-headline" id="Examples">Examples</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=6" title="Edit section: Examples">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h2>
+                        <p>
+                            In this section, the conjugate transpose of matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mathsf {H}}}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi mathvariant="sans-serif">
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </mrow>
+                                        </msup>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A^{\mathsf {H}}}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9415702ab196cc26f5df37af2d90e07318e93df" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.139ex; height:2.676ex;" alt="{\displaystyle A^{\mathsf {H}}}" /></span>, the transpose of matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\mathsf {T}}}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi mathvariant="sans-serif">
+                                                        T
+                                                    </mi>
+                                                </mrow>
+                                            </mrow>
+                                        </msup>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A^{\mathsf {T}}}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54bf0331204e30cba9ec7f695dfea97e6745a7c2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle A^{\mathsf {T}}}" /></span> and conjugate of matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is denoted as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {A}}}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mrow class="MJX-TeXAtom-ORD">
+                                            <mover>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mo accent="false">
+                                                    ¯<!-- ¯ -->
+                                                </mo>
+                                            </mover>
+                                        </mrow>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle {\overline {A}}}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.858ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}}" /></span>.
+                        </p>
+                        <p>
+                            See the following example:
+                        </p>
+                        <dl>
+                            <dd>
+                                <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}2&amp;2+i&amp;4\\2-i&amp;3&amp;i\\4&amp;-i&amp;1\\\end{bmatrix}}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mrow>
+                                                    <mo>
+                                                        [
+                                                    </mo>
+                                                    <mtable rowspacing="4pt" columnspacing="1em">
+                                                        <mtr>
+                                                            <mtd>
+                                                                <mn>
+                                                                    2
+                                                                </mn>
+                                                            </mtd>
+                                                            <mtd>
+                                                                <mn>
+                                                                    2
+                                                                </mn>
+                                                                <mo>
+                                                                    +
+                                                                </mo>
+                                                                <mi>
+                                                                    i
+                                                                </mi>
+                                                            </mtd>
+                                                            <mtd>
+                                                                <mn>
+                                                                    4
+                                                                </mn>
+                                                            </mtd>
+                                                        </mtr>
+                                                        <mtr>
+                                                            <mtd>
+                                                                <mn>
+                                                                    2
+                                                                </mn>
+                                                                <mo>
+                                                                    −<!-- − -->
+                                                                </mo>
+                                                                <mi>
+                                                                    i
+                                                                </mi>
+                                                            </mtd>
+                                                            <mtd>
+                                                                <mn>
+                                                                    3
+                                                                </mn>
+                                                            </mtd>
+                                                            <mtd>
+                                                                <mi>
+                                                                    i
+                                                                </mi>
+                                                            </mtd>
+                                                        </mtr>
+                                                        <mtr>
+                                                            <mtd>
+                                                                <mn>
+                                                                    4
+                                                                </mn>
+                                                            </mtd>
+                                                            <mtd>
+                                                                <mo>
+                                                                    −<!-- − -->
+                                                                </mo>
+                                                                <mi>
+                                                                    i
+                                                                </mi>
+                                                            </mtd>
+                                                            <mtd>
+                                                                <mn>
+                                                                    1
+                                                                </mn>
+                                                            </mtd>
+                                                        </mtr>
+                                                    </mtable>
+                                                    <mo>
+                                                        ]
+                                                    </mo>
+                                                </mrow>
+                                            </mrow>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle {\begin{bmatrix}2&amp;2+i&amp;4\\2-i&amp;3&amp;i\\4&amp;-i&amp;1\\\end{bmatrix}}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ccf11c16396b6ddd4f2254f7132cd8bb2c57ee" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -4.005ex; width:19.27ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}2&amp;2+i&amp;4\\2-i&amp;3&amp;i\\4&amp;-i&amp;1\\\end{bmatrix}}}" /></span>
+                            </dd>
+                        </dl>
+                        <p>
+                            The diagonal elements must be <a href="/wiki/Real_number" title="Real number">real</a>, as they must be their own complex conjugate.
+                        </p>
+                        <p>
+                            Well-known families of <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>, <a href="/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann matrices</a> and their generalizations are Hermitian. In <a href="/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a> such Hermitian matrices are often multiplied by <a href="/wiki/Imaginary_number" title="Imaginary number">imaginary</a> coefficients,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1">[1]</a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2">[2]</a></sup> which results in <i>skew-Hermitian</i> matrices (see <a href="#facts">below</a>).
+                        </p>
+                        <p>
+                            Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> equals the <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">multiplication of a matrix</a> and its conjugate transpose, that is, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=BB^{\mathsf {H}}}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                        <mo>
+                                            =
+                                        </mo>
+                                        <mi>
+                                            B
+                                        </mi>
+                                        <msup>
+                                            <mi>
+                                                B
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi mathvariant="sans-serif">
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </mrow>
+                                        </msup>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A=BB^{\mathsf {H}}}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0efab2d7c3a4b4b7caf14cc0705dadd13195a9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:9.765ex; height:2.676ex;" alt="{\displaystyle A=BB^{\mathsf {H}}}" /></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is a Hermitian <a href="/wiki/Positive_semi-definite_matrix" class="mw-redirect" title="Positive semi-definite matrix">positive semi-definite matrix</a>. Furthermore, if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            B
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle B}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="B" /></span> is row full-rank, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            A
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle A}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="A" /></span> is positive definite.
+                        </p>
+                        <h2>
+                            <span class="mw-headline" id="Properties">Properties</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=7" title="Edit section: Properties">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h2>
+                        <table class="box-Expand_section plainlinks metadata ambox ambox-content" role="presentation">
+                            <tbody>
+                                <tr>
+                                    <td class="mbox-image">
+                                        <div style="width:52px">
+                                            <a href="/wiki/File:Wiki_letter_w_cropped.svg" class="image"><img alt="[icon]" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/44px-Wiki_letter_w_cropped.svg.png" decoding="async" width="44" height="31" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/66px-Wiki_letter_w_cropped.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/88px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31" /></a>
+                                        </div>
+                                    </td>
+                                    <td class="mbox-text">
+                                        <div class="mbox-text-span">
+                                            This section <b>needs expansion</b> with: Proof of the properties requested. <small>You can help by <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=1">adding to it</a>.</small> <small class="date-container"><i>(<span class="date">February 2018</span>)</i></small>
+                                        </div>
+                                    </td>
+                                </tr>
+                            </tbody>
+                        </table>
+                        <ul>
+                            <li>The entries on the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> (top left to bottom right) of any Hermitian matrix are <a href="/wiki/Real_number" title="Real number">real</a>.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> By definition of the Hermitian matrix
+                                <dl>
+                                    <dd>
+                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{ij}={\overline {H}}_{ji}}">
+                                        <semantics>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mstyle displaystyle="true" scriptlevel="0">
+                                                    <msub>
+                                                        <mi>
+                                                            H
+                                                        </mi>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mi>
+                                                                i
+                                                            </mi>
+                                                            <mi>
+                                                                j
+                                                            </mi>
+                                                        </mrow>
+                                                    </msub>
+                                                    <mo>
+                                                        =
+                                                    </mo>
+                                                    <msub>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mover>
+                                                                <mi>
+                                                                    H
+                                                                </mi>
+                                                                <mo accent="false">
+                                                                    ¯<!-- ¯ -->
+                                                                </mo>
+                                                            </mover>
+                                                        </mrow>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mi>
+                                                                j
+                                                            </mi>
+                                                            <mi>
+                                                                i
+                                                            </mi>
+                                                        </mrow>
+                                                    </msub>
+                                                </mstyle>
+                                            </mrow>
+                                            <annotation encoding="application/x-tex">
+                                                {\displaystyle H_{ij}={\overline {H}}_{ji}}
+                                            </annotation>
+                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa8265c5f6d4fc3b7cda6a06558c7d4d9aec855" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:10.249ex; height:3.676ex;" alt="{\displaystyle H_{ij}={\overline {H}}_{ji}}" /></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                            <dd>
+                                so for <span class="texhtml"><i>i</i> = <i>j</i></span> the above follows.
+                            </dd>
+                            <dd>
+                                Only the <a href="/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their <a href="/wiki/Off-diagonal_element" class="mw-redirect" title="Off-diagonal element">off-diagonal elements</a>, as long as diagonally-opposite entries are complex conjugates.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>A matrix that has only real entries is Hermitian <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>. A real and symmetric matrix is simply a special case of a Hermitian matrix.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{ij}={\overline {H}}_{ji}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <msub>
+                                                <mi>
+                                                    H
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        i
+                                                    </mi>
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msub>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mover>
+                                                        <mi>
+                                                            H
+                                                        </mi>
+                                                        <mo accent="false">
+                                                            ¯<!-- ¯ -->
+                                                        </mo>
+                                                    </mover>
+                                                </mrow>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                    <mi>
+                                                        i
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle H_{ij}={\overline {H}}_{ji}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa8265c5f6d4fc3b7cda6a06558c7d4d9aec855" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:10.249ex; height:3.676ex;" alt="{\displaystyle H_{ij}={\overline {H}}_{ji}}" /></span> by definition. Thus <span class="texhtml">H<sub><i>ij</i></sub> = H<sub><i>ji</i></sub></span> (matrix symmetry) if and only if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{ij}={\overline {H}}_{ij}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <msub>
+                                                <mi>
+                                                    H
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        i
+                                                    </mi>
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msub>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mover>
+                                                        <mi>
+                                                            H
+                                                        </mi>
+                                                        <mo accent="false">
+                                                            ¯<!-- ¯ -->
+                                                        </mo>
+                                                    </mover>
+                                                </mrow>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        i
+                                                    </mi>
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle H_{ij}={\overline {H}}_{ij}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1862750b96d01100244370b3fca45f01923ce5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:10.249ex; height:3.676ex;" alt="{\displaystyle H_{ij}={\overline {H}}_{ij}}" /></span> (<span class="texhtml">H<sub><i>ij</i></sub></span> is real).
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>Every Hermitian matrix is a <a href="/wiki/Normal_matrix" title="Normal matrix">normal matrix</a>. That is to say, <span class="texhtml">AA<sup>H</sup> = A<sup>H</sup>A</span>.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> <span class="texhtml">A = A<sup>H</sup></span>, so <span class="texhtml">AA<sup>H</sup> = AA = A<sup>H</sup>A</span>.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>The finite-dimensional <a href="/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a> says that any Hermitian matrix can be <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalized</a> by a <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a>, and that the resulting diagonal matrix has only real entries. This implies that all <a href="/wiki/Eigenvectors" class="mw-redirect" title="Eigenvectors">eigenvalues</a> of a Hermitian matrix <span class="texhtml mvar" style="font-style:italic;">A</span> with dimension <span class="texhtml mvar" style="font-style:italic;">n</span> are real, and that <span class="texhtml mvar" style="font-style:italic;">A</span> has <span class="texhtml mvar" style="font-style:italic;">n</span> linearly independent <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a>. Moreover, a Hermitian matrix has <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an <a href="/wiki/Orthogonal_basis" title="Orthogonal basis">orthogonal basis</a> of <span class="texhtml">ℂ<sup><i>n</i></sup></span> consisting of <span class="texhtml mvar" style="font-style:italic;">n</span> eigenvectors of <span class="texhtml mvar" style="font-style:italic;">A</span>.
+                            </li>
+                        </ul>
+                        <ul>
+                            <li>The sum of any two Hermitian matrices is Hermitian.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mo stretchy="false">
+                                                (
+                                            </mo>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mo>
+                                                +
+                                            </mo>
+                                            <mi>
+                                                B
+                                            </mi>
+                                            <msub>
+                                                <mo stretchy="false">
+                                                    )
+                                                </mo>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        i
+                                                    </mi>
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msub>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        i
+                                                    </mi>
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                            <mo>
+                                                +
+                                            </mo>
+                                            <msub>
+                                                <mi>
+                                                    B
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        i
+                                                    </mi>
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msub>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mover>
+                                                        <mi>
+                                                            A
+                                                        </mi>
+                                                        <mo accent="false">
+                                                            ¯<!-- ¯ -->
+                                                        </mo>
+                                                    </mover>
+                                                </mrow>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                    <mi>
+                                                        i
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                            <mo>
+                                                +
+                                            </mo>
+                                            <msub>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mover>
+                                                        <mi>
+                                                            B
+                                                        </mi>
+                                                        <mo accent="false">
+                                                            ¯<!-- ¯ -->
+                                                        </mo>
+                                                    </mover>
+                                                </mrow>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                    <mi>
+                                                        i
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msub>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mover>
+                                                        <mrow>
+                                                            <mo stretchy="false">
+                                                                (
+                                                            </mo>
+                                                            <mi>
+                                                                A
+                                                            </mi>
+                                                            <mo>
+                                                                +
+                                                            </mo>
+                                                            <mi>
+                                                                B
+                                                            </mi>
+                                                            <mo stretchy="false">
+                                                                )
+                                                            </mo>
+                                                        </mrow>
+                                                        <mo accent="false">
+                                                            ¯<!-- ¯ -->
+                                                        </mo>
+                                                    </mover>
+                                                </mrow>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                    <mi>
+                                                        i
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                            <mo>
+                                                ,
+                                            </mo>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251bf4ebbe3b0d119e0a7d19f8fd134c4f072971" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.338ex; width:48.159ex; height:4.176ex;" alt="{\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},}" /></span> as claimed.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>The <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse</a> of an invertible Hermitian matrix is Hermitian as well.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{-1}A=I}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <msup>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mo>
+                                                        −<!-- − -->
+                                                    </mo>
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                </mrow>
+                                            </msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mi>
+                                                I
+                                            </mi>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle A^{-1}A=I}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021893240ff7fa3148b6649b0ba4d88cd207b5f0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:10.089ex; height:2.676ex;" alt="{\displaystyle A^{-1}A=I}" /></span>, then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I=I^{H}=(A^{-1}A)^{H}=A^{H}(A^{-1})^{H}=A(A^{-1})^{H}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mi>
+                                                I
+                                            </mi>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msup>
+                                                <mi>
+                                                    I
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </msup>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mo stretchy="false">
+                                                (
+                                            </mo>
+                                            <msup>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mo>
+                                                        −<!-- − -->
+                                                    </mo>
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                </mrow>
+                                            </msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <msup>
+                                                <mo stretchy="false">
+                                                    )
+                                                </mo>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </msup>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msup>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </msup>
+                                            <mo stretchy="false">
+                                                (
+                                            </mo>
+                                            <msup>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mo>
+                                                        −<!-- − -->
+                                                    </mo>
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                </mrow>
+                                            </msup>
+                                            <msup>
+                                                <mo stretchy="false">
+                                                    )
+                                                </mo>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </msup>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mo stretchy="false">
+                                                (
+                                            </mo>
+                                            <msup>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mo>
+                                                        −<!-- − -->
+                                                    </mo>
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                </mrow>
+                                            </msup>
+                                            <msup>
+                                                <mo stretchy="false">
+                                                    )
+                                                </mo>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </msup>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle I=I^{H}=(A^{-1}A)^{H}=A^{H}(A^{-1})^{H}=A(A^{-1})^{H}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a28a8250ab35ad60228bb0376eb4b7024f027815" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:46.124ex; height:3.176ex;" alt="{\displaystyle I=I^{H}=(A^{-1}A)^{H}=A^{H}(A^{-1})^{H}=A(A^{-1})^{H}}" /></span>, so <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{-1}=(A^{-1})^{H}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <msup>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mo>
+                                                        −<!-- − -->
+                                                    </mo>
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                </mrow>
+                                            </msup>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mo stretchy="false">
+                                                (
+                                            </mo>
+                                            <msup>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mo>
+                                                        −<!-- − -->
+                                                    </mo>
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                </mrow>
+                                            </msup>
+                                            <msup>
+                                                <mo stretchy="false">
+                                                    )
+                                                </mo>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </msup>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle A^{-1}=(A^{-1})^{H}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0179c3a7aebe194ccd9a19ba02b972500f88a7a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:14.751ex; height:3.176ex;" alt="{\displaystyle A^{-1}=(A^{-1})^{H}}" /></span> as claimed.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>The <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">product</a> of two Hermitian matrices <span class="texhtml mvar" style="font-style:italic;">A</span> and <span class="texhtml mvar" style="font-style:italic;">B</span> is Hermitian if and only if <span class="texhtml"><i>AB</i> = <i>BA</i></span>.
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}{\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mo stretchy="false">
+                                                (
+                                            </mo>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mi>
+                                                B
+                                            </mi>
+                                            <msup>
+                                                <mo stretchy="false">
+                                                    )
+                                                </mo>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mover>
+                                                    <mrow>
+                                                        <mo stretchy="false">
+                                                            (
+                                                        </mo>
+                                                        <mi>
+                                                            A
+                                                        </mi>
+                                                        <mi>
+                                                            B
+                                                        </mi>
+                                                        <msup>
+                                                            <mo stretchy="false">
+                                                                )
+                                                            </mo>
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mi mathvariant="sans-serif">
+                                                                        T
+                                                                    </mi>
+                                                                </mrow>
+                                                            </mrow>
+                                                        </msup>
+                                                    </mrow>
+                                                    <mo accent="false">
+                                                        ¯<!-- ¯ -->
+                                                    </mo>
+                                                </mover>
+                                            </mrow>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mover>
+                                                    <mrow>
+                                                        <msup>
+                                                            <mi>
+                                                                B
+                                                            </mi>
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mi mathvariant="sans-serif">
+                                                                        T
+                                                                    </mi>
+                                                                </mrow>
+                                                            </mrow>
+                                                        </msup>
+                                                        <msup>
+                                                            <mi>
+                                                                A
+                                                            </mi>
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mi mathvariant="sans-serif">
+                                                                        T
+                                                                    </mi>
+                                                                </mrow>
+                                                            </mrow>
+                                                        </msup>
+                                                    </mrow>
+                                                    <mo accent="false">
+                                                        ¯<!-- ¯ -->
+                                                    </mo>
+                                                </mover>
+                                            </mrow>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mover>
+                                                    <msup>
+                                                        <mi>
+                                                            B
+                                                        </mi>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mi mathvariant="sans-serif">
+                                                                    T
+                                                                </mi>
+                                                            </mrow>
+                                                        </mrow>
+                                                    </msup>
+                                                    <mo accent="false">
+                                                        ¯<!-- ¯ -->
+                                                    </mo>
+                                                </mover>
+                                            </mrow>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mover>
+                                                    <msup>
+                                                        <mi>
+                                                            A
+                                                        </mi>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mi mathvariant="sans-serif">
+                                                                    T
+                                                                </mi>
+                                                            </mrow>
+                                                        </mrow>
+                                                    </msup>
+                                                    <mo accent="false">
+                                                        ¯<!-- ¯ -->
+                                                    </mo>
+                                                </mover>
+                                            </mrow>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msup>
+                                                <mi>
+                                                    B
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                            <msup>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mi>
+                                                B
+                                            </mi>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mo>
+                                                .
+                                            </mo>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle (AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}{\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cf8185ca7a0687bf959bb65b16db6cf1714ca2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:52.205ex; height:4.009ex;" alt="{\displaystyle (AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}{\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.}" /></span> Thus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (AB)^{\mathsf {H}}=AB}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mo stretchy="false">
+                                                (
+                                            </mo>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mi>
+                                                B
+                                            </mi>
+                                            <msup>
+                                                <mo stretchy="false">
+                                                    )
+                                                </mo>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mi>
+                                                B
+                                            </mi>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle (AB)^{\mathsf {H}}=AB}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d303a1ebcac8547489b170be5d0dd7d8e04e548e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:13.318ex; height:3.176ex;" alt="{\displaystyle (AB)^{\mathsf {H}}=AB}" /></span> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB=BA}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mi>
+                                                B
+                                            </mi>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mi>
+                                                B
+                                            </mi>
+                                            <mi>
+                                                A
+                                            </mi>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle AB=BA}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992c8ea49fdd26b491180036c5a4d879fec77442" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:10.113ex; height:2.176ex;" alt="AB=BA" /></span>.
+                            </dd>
+                            <dd>
+                                Thus <span class="texhtml"><i>A</i><sup><i>n</i></sup></span> is Hermitian if <span class="texhtml mvar" style="font-style:italic;">A</span> is Hermitian and <span class="texhtml mvar" style="font-style:italic;">n</span> is an integer.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>For an arbitrary complex valued vector <span class="texhtml mvar" style="font-style:italic;">v</span> the product <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{\mathsf {H}}Av}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <msup>
+                                                <mi>
+                                                    v
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mi>
+                                                v
+                                            </mi>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle v^{\mathsf {H}}Av}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c23d6757968a2267ee906cffc07cfe1bbc8aecc" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.394ex; height:2.676ex;" alt="{\displaystyle v^{\mathsf {H}}Av}" /></span> is real because of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v^{\mathsf {H}}Av=\left(v^{\mathsf {H}}Av\right)^{\mathsf {H}}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <msup>
+                                                <mi>
+                                                    v
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mi>
+                                                v
+                                            </mi>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msup>
+                                                <mrow>
+                                                    <mo>
+                                                        (
+                                                    </mo>
+                                                    <mrow>
+                                                        <msup>
+                                                            <mi>
+                                                                v
+                                                            </mi>
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mi mathvariant="sans-serif">
+                                                                        H
+                                                                    </mi>
+                                                                </mrow>
+                                                            </mrow>
+                                                        </msup>
+                                                        <mi>
+                                                            A
+                                                        </mi>
+                                                        <mi>
+                                                            v
+                                                        </mi>
+                                                    </mrow>
+                                                    <mo>
+                                                        )
+                                                    </mo>
+                                                </mrow>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle v^{\mathsf {H}}Av=\left(v^{\mathsf {H}}Av\right)^{\mathsf {H}}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7839ef3cdbc89c3ea1acd17a507d03a33ed79df" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:17.412ex; height:3.843ex;" alt="{\displaystyle v^{\mathsf {H}}Av=\left(v^{\mathsf {H}}Av\right)^{\mathsf {H}}}" /></span>. This is especially important in quantum physics where Hermitian matrices are operators that measure properties of a system e.g. total <a href="/wiki/Spin_(physics)" title="Spin (physics)">spin</a> which have to be real.
+                            </li>
+                        </ul>
+                        <ul>
+                            <li>The Hermitian complex <span class="texhtml mvar" style="font-style:italic;">n</span>-by-<span class="texhtml mvar" style="font-style:italic;">n</span> matrices do not form a <a href="/wiki/Vector_space" title="Vector space">vector space</a> over the <a href="/wiki/Complex_number" title="Complex number">complex numbers</a>, <span class="texhtml">ℂ</span>, since the identity matrix <span class="texhtml"><i>I</i><sub><i>n</i></sub></span> is Hermitian, but <span class="texhtml"><i>i</i> <i>I</i><sub><i>n</i></sub></span> is not. However the complex Hermitian matrices <i>do</i> form a vector space over the <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real numbers</a> <span class="texhtml">ℝ</span>. In the <span class="texhtml">2<i>n</i><sup>2</sup></span>-<a href="/wiki/Dimension_of_a_vector_space" class="mw-redirect" title="Dimension of a vector space">dimensional</a> vector space of complex <span class="texhtml"><i>n</i> × <i>n</i></span> matrices over <span class="texhtml">ℝ</span>, the complex Hermitian matrices form a subspace of dimension <span class="texhtml"><i>n</i><sup>2</sup></span>. If <span class="texhtml"><i>E</i><sub><i>jk</i></sub></span> denotes the <span class="texhtml mvar" style="font-style:italic;">n</span>-by-<span class="texhtml mvar" style="font-style:italic;">n</span> matrix with a <span class="texhtml">1</span> in the <span class="texhtml"><i>j</i>,<i>k</i></span> position and zeros elsewhere, a basis (orthonormal w.r.t. the Frobenius inner product) can be described as follows:
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <dl>
+                                    <dd>
+                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})}">
+                                        <semantics>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mstyle displaystyle="true" scriptlevel="0">
+                                                    <msub>
+                                                        <mi>
+                                                            E
+                                                        </mi>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mi>
+                                                                j
+                                                            </mi>
+                                                            <mi>
+                                                                j
+                                                            </mi>
+                                                        </mrow>
+                                                    </msub>
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mtext>
+                                                            &nbsp;for&nbsp;
+                                                        </mtext>
+                                                    </mrow>
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                    <mo>
+                                                        ≤<!-- ≤ -->
+                                                    </mo>
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                    <mo>
+                                                        ≤<!-- ≤ -->
+                                                    </mo>
+                                                    <mi>
+                                                        n
+                                                    </mi>
+                                                    <mspace width="1em"></mspace>
+                                                    <mo stretchy="false">
+                                                        (
+                                                    </mo>
+                                                    <mi>
+                                                        n
+                                                    </mi>
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mtext>
+                                                            &nbsp;matrices
+                                                        </mtext>
+                                                    </mrow>
+                                                    <mo stretchy="false">
+                                                        )
+                                                    </mo>
+                                                </mstyle>
+                                            </mrow>
+                                            <annotation encoding="application/x-tex">
+                                                {\displaystyle E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})}
+                                            </annotation>
+                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46eedb181c0bdae46e8c1526161b03d0ea97b4b4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:31.612ex; height:3.009ex;" alt="{\displaystyle E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})}" /></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                        </dl>
+                        <dl>
+                            <dd>
+                                together with the set of matrices of the form
+                            </dd>
+                        </dl>
+                        <dl>
+                            <dd>
+                                <dl>
+                                    <dd>
+                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}">
+                                        <semantics>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mstyle displaystyle="true" scriptlevel="0">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mfrac>
+                                                            <mn>
+                                                                1
+                                                            </mn>
+                                                            <msqrt>
+                                                                <mn>
+                                                                    2
+                                                                </mn>
+                                                            </msqrt>
+                                                        </mfrac>
+                                                    </mrow>
+                                                    <mrow>
+                                                        <mo>
+                                                            (
+                                                        </mo>
+                                                        <mrow>
+                                                            <msub>
+                                                                <mi>
+                                                                    E
+                                                                </mi>
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mi>
+                                                                        j
+                                                                    </mi>
+                                                                    <mi>
+                                                                        k
+                                                                    </mi>
+                                                                </mrow>
+                                                            </msub>
+                                                            <mo>
+                                                                +
+                                                            </mo>
+                                                            <msub>
+                                                                <mi>
+                                                                    E
+                                                                </mi>
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mi>
+                                                                        k
+                                                                    </mi>
+                                                                    <mi>
+                                                                        j
+                                                                    </mi>
+                                                                </mrow>
+                                                            </msub>
+                                                        </mrow>
+                                                        <mo>
+                                                            )
+                                                        </mo>
+                                                    </mrow>
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mtext>
+                                                            &nbsp;for&nbsp;
+                                                        </mtext>
+                                                    </mrow>
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                    <mo>
+                                                        ≤<!-- ≤ -->
+                                                    </mo>
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                    <mo>
+                                                        &lt;
+                                                    </mo>
+                                                    <mi>
+                                                        k
+                                                    </mi>
+                                                    <mo>
+                                                        ≤<!-- ≤ -->
+                                                    </mo>
+                                                    <mi>
+                                                        n
+                                                    </mi>
+                                                    <mspace width="1em"></mspace>
+                                                    <mrow>
+                                                        <mo>
+                                                            (
+                                                        </mo>
+                                                        <mrow>
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mfrac>
+                                                                    <mrow>
+                                                                        <msup>
+                                                                            <mi>
+                                                                                n
+                                                                            </mi>
+                                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                                <mn>
+                                                                                    2
+                                                                                </mn>
+                                                                            </mrow>
+                                                                        </msup>
+                                                                        <mo>
+                                                                            −<!-- − -->
+                                                                        </mo>
+                                                                        <mi>
+                                                                            n
+                                                                        </mi>
+                                                                    </mrow>
+                                                                    <mn>
+                                                                        2
+                                                                    </mn>
+                                                                </mfrac>
+                                                            </mrow>
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mtext>
+                                                                    &nbsp;matrices
+                                                                </mtext>
+                                                            </mrow>
+                                                        </mrow>
+                                                        <mo>
+                                                            )
+                                                        </mo>
+                                                    </mrow>
+                                                </mstyle>
+                                            </mrow>
+                                            <annotation encoding="application/x-tex">
+                                                {\displaystyle {\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}
+                                            </annotation>
+                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddeac51c423f6dbefc5f63e483d9aee96e6fa342" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:57.064ex; height:6.676ex;" alt="{\displaystyle {\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}" /></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                        </dl>
+                        <dl>
+                            <dd>
+                                and the matrices
+                            </dd>
+                        </dl>
+                        <dl>
+                            <dd>
+                                <dl>
+                                    <dd>
+                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}">
+                                        <semantics>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mstyle displaystyle="true" scriptlevel="0">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mfrac>
+                                                            <mi>
+                                                                i
+                                                            </mi>
+                                                            <msqrt>
+                                                                <mn>
+                                                                    2
+                                                                </mn>
+                                                            </msqrt>
+                                                        </mfrac>
+                                                    </mrow>
+                                                    <mrow>
+                                                        <mo>
+                                                            (
+                                                        </mo>
+                                                        <mrow>
+                                                            <msub>
+                                                                <mi>
+                                                                    E
+                                                                </mi>
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mi>
+                                                                        j
+                                                                    </mi>
+                                                                    <mi>
+                                                                        k
+                                                                    </mi>
+                                                                </mrow>
+                                                            </msub>
+                                                            <mo>
+                                                                −<!-- − -->
+                                                            </mo>
+                                                            <msub>
+                                                                <mi>
+                                                                    E
+                                                                </mi>
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mi>
+                                                                        k
+                                                                    </mi>
+                                                                    <mi>
+                                                                        j
+                                                                    </mi>
+                                                                </mrow>
+                                                            </msub>
+                                                        </mrow>
+                                                        <mo>
+                                                            )
+                                                        </mo>
+                                                    </mrow>
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mtext>
+                                                            &nbsp;for&nbsp;
+                                                        </mtext>
+                                                    </mrow>
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                    <mo>
+                                                        ≤<!-- ≤ -->
+                                                    </mo>
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                    <mo>
+                                                        &lt;
+                                                    </mo>
+                                                    <mi>
+                                                        k
+                                                    </mi>
+                                                    <mo>
+                                                        ≤<!-- ≤ -->
+                                                    </mo>
+                                                    <mi>
+                                                        n
+                                                    </mi>
+                                                    <mspace width="1em"></mspace>
+                                                    <mrow>
+                                                        <mo>
+                                                            (
+                                                        </mo>
+                                                        <mrow>
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mfrac>
+                                                                    <mrow>
+                                                                        <msup>
+                                                                            <mi>
+                                                                                n
+                                                                            </mi>
+                                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                                <mn>
+                                                                                    2
+                                                                                </mn>
+                                                                            </mrow>
+                                                                        </msup>
+                                                                        <mo>
+                                                                            −<!-- − -->
+                                                                        </mo>
+                                                                        <mi>
+                                                                            n
+                                                                        </mi>
+                                                                    </mrow>
+                                                                    <mn>
+                                                                        2
+                                                                    </mn>
+                                                                </mfrac>
+                                                            </mrow>
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mtext>
+                                                                    &nbsp;matrices
+                                                                </mtext>
+                                                            </mrow>
+                                                        </mrow>
+                                                        <mo>
+                                                            )
+                                                        </mo>
+                                                    </mrow>
+                                                </mstyle>
+                                            </mrow>
+                                            <annotation encoding="application/x-tex">
+                                                {\displaystyle {\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}
+                                            </annotation>
+                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db65cce3a8fa33e5b7b96badd756c8573aa866c0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.838ex; width:57.064ex; height:6.676ex;" alt="{\displaystyle {\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j&lt;k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}" /></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                        </dl>
+                        <dl>
+                            <dd>
+                                where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mi>
+                                                i
+                                            </mi>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle i}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="i" /></span> denotes the complex number <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\sqrt {-1}}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <msqrt>
+                                                    <mo>
+                                                        −<!-- − -->
+                                                    </mo>
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                </msqrt>
+                                            </mrow>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle {\sqrt {-1}}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:4.906ex; height:3.009ex;" alt="{\sqrt {-1}}" /></span>, called the <i><a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a></i>.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>If <span class="texhtml mvar" style="font-style:italic;">n</span> orthonormal eigenvectors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1},\dots ,u_{n}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <msub>
+                                                <mi>
+                                                    u
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mn>
+                                                        1
+                                                    </mn>
+                                                </mrow>
+                                            </msub>
+                                            <mo>
+                                                ,
+                                            </mo>
+                                            <mo>
+                                                …<!-- … -->
+                                            </mo>
+                                            <mo>
+                                                ,
+                                            </mo>
+                                            <msub>
+                                                <mi>
+                                                    u
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        n
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle u_{1},\dots ,u_{n}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba291b64c0d3afa90b4556a7a601116dfd74ef2e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:10.11ex; height:2.009ex;" alt="{\displaystyle u_{1},\dots ,u_{n}}" /></span> of a Hermitian matrix are chosen and written as the columns of the matrix <span class="texhtml mvar" style="font-style:italic;">U</span>, then one <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">eigendecomposition</a> of <span class="texhtml mvar" style="font-style:italic;">A</span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=U\Lambda U^{\mathsf {H}}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mi>
+                                                U
+                                            </mi>
+                                            <mi mathvariant="normal">
+                                                Λ<!-- Λ -->
+                                            </mi>
+                                            <msup>
+                                                <mi>
+                                                    U
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle A=U\Lambda U^{\mathsf {H}}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d89ef5bade4fb04b1aaf53c0d73e4763d4d154eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:11.474ex; height:2.676ex;" alt="{\displaystyle A=U\Lambda U^{\mathsf {H}}}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle UU^{\mathsf {H}}=I=U^{\mathsf {H}}U}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mi>
+                                                U
+                                            </mi>
+                                            <msup>
+                                                <mi>
+                                                    U
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mi>
+                                                I
+                                            </mi>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msup>
+                                                <mi>
+                                                    U
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                            <mi>
+                                                U
+                                            </mi>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle UU^{\mathsf {H}}=I=U^{\mathsf {H}}U}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b819b015c1984e6cb07153c675815d4657b90da8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:17.408ex; height:2.676ex;" alt="{\displaystyle UU^{\mathsf {H}}=I=U^{\mathsf {H}}U}" /></span> and therefore
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <dl>
+                                    <dd>
+                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\mathsf {H}},}">
+                                        <semantics>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mstyle displaystyle="true" scriptlevel="0">
+                                                    <mi>
+                                                        A
+                                                    </mi>
+                                                    <mo>
+                                                        =
+                                                    </mo>
+                                                    <munder>
+                                                        <mo>
+                                                            ∑<!-- ∑ -->
+                                                        </mo>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mi>
+                                                                j
+                                                            </mi>
+                                                        </mrow>
+                                                    </munder>
+                                                    <msub>
+                                                        <mi>
+                                                            λ<!-- λ -->
+                                                        </mi>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mi>
+                                                                j
+                                                            </mi>
+                                                        </mrow>
+                                                    </msub>
+                                                    <msub>
+                                                        <mi>
+                                                            u
+                                                        </mi>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mi>
+                                                                j
+                                                            </mi>
+                                                        </mrow>
+                                                    </msub>
+                                                    <msubsup>
+                                                        <mi>
+                                                            u
+                                                        </mi>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mi>
+                                                                j
+                                                            </mi>
+                                                        </mrow>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mi mathvariant="sans-serif">
+                                                                    H
+                                                                </mi>
+                                                            </mrow>
+                                                        </mrow>
+                                                    </msubsup>
+                                                    <mo>
+                                                        ,
+                                                    </mo>
+                                                </mstyle>
+                                            </mrow>
+                                            <annotation encoding="application/x-tex">
+                                                {\displaystyle A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\mathsf {H}},}
+                                            </annotation>
+                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7d749931e5f709bcbc0a446638d3b6b8ed0c6c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.338ex; width:16.46ex; height:5.843ex;" alt="{\displaystyle A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\mathsf {H}},}" /></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                            <dd>
+                                where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{j}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <msub>
+                                                <mi>
+                                                    λ<!-- λ -->
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi>
+                                                        j
+                                                    </mi>
+                                                </mrow>
+                                            </msub>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle \lambda _{j}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa91daf9145f27bb95746fd2a37537342d587b77" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:2.265ex; height:2.843ex;" alt="\lambda _{j}" /></span> are the eigenvalues on the diagonal of the diagonal matrix <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\Lambda }">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mspace width="thickmathspace"></mspace>
+                                            <mi mathvariant="normal">
+                                                Λ<!-- Λ -->
+                                            </mi>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle \;\Lambda }
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1934e7eadd31fbf6f7d6bcf9c0e9bec934ce8976" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.258ex; height:2.176ex;" alt="\; \Lambda " /></span>.
+                            </dd>
+                        </dl>
+                        <ul>
+                            <li>The determinant of a Hermitian matrix is real:
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <i>Proof:</i> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mo movablelimits="true" form="prefix">
+                                                det
+                                            </mo>
+                                            <mo stretchy="false">
+                                                (
+                                            </mo>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mo stretchy="false">
+                                                )
+                                            </mo>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mo movablelimits="true" form="prefix">
+                                                det
+                                            </mo>
+                                            <mrow>
+                                                <mo>
+                                                    (
+                                                </mo>
+                                                <msup>
+                                                    <mi>
+                                                        A
+                                                    </mi>
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mi mathvariant="sans-serif">
+                                                                T
+                                                            </mi>
+                                                        </mrow>
+                                                    </mrow>
+                                                </msup>
+                                                <mo>
+                                                    )
+                                                </mo>
+                                            </mrow>
+                                            <mspace width="1em"></mspace>
+                                            <mo stretchy="false">
+                                                ⇒<!-- ⇒ -->
+                                            </mo>
+                                            <mspace width="1em"></mspace>
+                                            <mo movablelimits="true" form="prefix">
+                                                det
+                                            </mo>
+                                            <mrow>
+                                                <mo>
+                                                    (
+                                                </mo>
+                                                <msup>
+                                                    <mi>
+                                                        A
+                                                    </mi>
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mi mathvariant="sans-serif">
+                                                                H
+                                                            </mi>
+                                                        </mrow>
+                                                    </mrow>
+                                                </msup>
+                                                <mo>
+                                                    )
+                                                </mo>
+                                            </mrow>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mover>
+                                                    <mrow>
+                                                        <mo movablelimits="true" form="prefix">
+                                                            det
+                                                        </mo>
+                                                        <mo stretchy="false">
+                                                            (
+                                                        </mo>
+                                                        <mi>
+                                                            A
+                                                        </mi>
+                                                        <mo stretchy="false">
+                                                            )
+                                                        </mo>
+                                                    </mrow>
+                                                    <mo accent="false">
+                                                        ¯<!-- ¯ -->
+                                                    </mo>
+                                                </mover>
+                                            </mrow>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle \det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1240df64c3010e0be6eae865fdfcfe6f77bf5eb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:45.862ex; height:3.843ex;" alt="{\displaystyle \det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}}" /></span>
+                            </dd>
+                            <dd>
+                                Therefore if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <msup>
+                                                <mi>
+                                                    A
+                                                </mi>
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mi mathvariant="sans-serif">
+                                                            H
+                                                        </mi>
+                                                    </mrow>
+                                                </mrow>
+                                            </msup>
+                                            <mspace width="1em"></mspace>
+                                            <mo stretchy="false">
+                                                ⇒<!-- ⇒ -->
+                                            </mo>
+                                            <mspace width="1em"></mspace>
+                                            <mo movablelimits="true" form="prefix">
+                                                det
+                                            </mo>
+                                            <mo stretchy="false">
+                                                (
+                                            </mo>
+                                            <mi>
+                                                A
+                                            </mi>
+                                            <mo stretchy="false">
+                                                )
+                                            </mo>
+                                            <mo>
+                                                =
+                                            </mo>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mover>
+                                                    <mrow>
+                                                        <mo movablelimits="true" form="prefix">
+                                                            det
+                                                        </mo>
+                                                        <mo stretchy="false">
+                                                            (
+                                                        </mo>
+                                                        <mi>
+                                                            A
+                                                        </mi>
+                                                        <mo stretchy="false">
+                                                            )
+                                                        </mo>
+                                                    </mrow>
+                                                    <mo accent="false">
+                                                        ¯<!-- ¯ -->
+                                                    </mo>
+                                                </mover>
+                                            </mrow>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43cc392bdcfbb134dd66d9b469847f6370e29d9d" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:33.017ex; height:3.676ex;" alt="{\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}}" /></span>.
+                            </dd>
+                            <dd>
+                                (Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
+                            </dd>
+                        </dl>
+                        <h2>
+                            <span class="mw-headline" id="Decomposition_into_Hermitian_and_skew-Hermitian">Decomposition into Hermitian and skew-Hermitian</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=8" title="Edit section: Decomposition into Hermitian and skew-Hermitian">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h2>
+                        <p>
+                            <span id="facts"></span>Additional facts related to Hermitian matrices include:
+                        </p>
+                        <ul>
+                            <li>The sum of a square matrix and its conjugate transpose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A+A^{\mathsf {H}}\right)}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mrow>
+                                                <mo>
+                                                    (
+                                                </mo>
+                                                <mrow>
+                                                    <mi>
+                                                        A
+                                                    </mi>
+                                                    <mo>
+                                                        +
+                                                    </mo>
+                                                    <msup>
+                                                        <mi>
+                                                            A
+                                                        </mi>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mi mathvariant="sans-serif">
+                                                                    H
+                                                                </mi>
+                                                            </mrow>
+                                                        </mrow>
+                                                    </msup>
+                                                </mrow>
+                                                <mo>
+                                                    )
+                                                </mo>
+                                            </mrow>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle \left(A+A^{\mathsf {H}}\right)}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef97bb04ce4ab682bcc84cf1059f8da235b483e" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:9.852ex; height:3.343ex;" alt="{\displaystyle \left(A+A^{\mathsf {H}}\right)}" /></span> is Hermitian.
+                            </li>
+                        </ul>
+                        <ul>
+                            <li>The difference of a square matrix and its conjugate transpose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A-A^{\mathsf {H}}\right)}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mrow>
+                                                <mo>
+                                                    (
+                                                </mo>
+                                                <mrow>
+                                                    <mi>
+                                                        A
+                                                    </mi>
+                                                    <mo>
+                                                        −<!-- − -->
+                                                    </mo>
+                                                    <msup>
+                                                        <mi>
+                                                            A
+                                                        </mi>
+                                                        <mrow class="MJX-TeXAtom-ORD">
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mi mathvariant="sans-serif">
+                                                                    H
+                                                                </mi>
+                                                            </mrow>
+                                                        </mrow>
+                                                    </msup>
+                                                </mrow>
+                                                <mo>
+                                                    )
+                                                </mo>
+                                            </mrow>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle \left(A-A^{\mathsf {H}}\right)}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ac665be4943ce769e33109e9f64abcf1e98050" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.005ex; width:9.852ex; height:3.343ex;" alt="{\displaystyle \left(A-A^{\mathsf {H}}\right)}" /></span> is <a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">skew-Hermitian</a> (also called antihermitian). This implies that the <a href="/wiki/Commutator" title="Commutator">commutator</a> of two Hermitian matrices is skew-Hermitian.
+                            </li>
+                        </ul>
+                        <ul>
+                            <li>An arbitrary square matrix <span class="texhtml mvar" style="font-style:italic;">C</span> can be written as the sum of a Hermitian matrix <span class="texhtml mvar" style="font-style:italic;">A</span> and a skew-Hermitian matrix <span class="texhtml mvar" style="font-style:italic;">B</span>. This is known as the Toeplitz decomposition of <span class="texhtml mvar" style="font-style:italic;">C</span>.<sup id="cite_ref-HornJohnson_3-0" class="reference"><a href="#cite_note-HornJohnson-3">[3]</a></sup><sup class="reference" style="white-space:nowrap;">:<span>p. 7</span></sup>
+                            </li>
+                        </ul>
+                        <dl>
+                            <dd>
+                                <dl>
+                                    <dd>
+                                        <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}">
+                                        <semantics>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mstyle displaystyle="true" scriptlevel="0">
+                                                    <mi>
+                                                        C
+                                                    </mi>
+                                                    <mo>
+                                                        =
+                                                    </mo>
+                                                    <mi>
+                                                        A
+                                                    </mi>
+                                                    <mo>
+                                                        +
+                                                    </mo>
+                                                    <mi>
+                                                        B
+                                                    </mi>
+                                                    <mspace width="1em"></mspace>
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mstyle displaystyle="false" scriptlevel="0">
+                                                            <mtext>
+                                                                with
+                                                            </mtext>
+                                                        </mstyle>
+                                                    </mrow>
+                                                    <mspace width="1em"></mspace>
+                                                    <mi>
+                                                        A
+                                                    </mi>
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+                                                    </mo>
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mfrac>
+                                                            <mn>
+                                                                1
+                                                            </mn>
+                                                            <mn>
+                                                                2
+                                                            </mn>
+                                                        </mfrac>
+                                                    </mrow>
+                                                    <mrow>
+                                                        <mo>
+                                                            (
+                                                        </mo>
+                                                        <mrow>
+                                                            <mi>
+                                                                C
+                                                            </mi>
+                                                            <mo>
+                                                                +
+                                                            </mo>
+                                                            <msup>
+                                                                <mi>
+                                                                    C
+                                                                </mi>
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                                        <mi mathvariant="sans-serif">
+                                                                            H
+                                                                        </mi>
+                                                                    </mrow>
+                                                                </mrow>
+                                                            </msup>
+                                                        </mrow>
+                                                        <mo>
+                                                            )
+                                                        </mo>
+                                                    </mrow>
+                                                    <mspace width="1em"></mspace>
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mstyle displaystyle="false" scriptlevel="0">
+                                                            <mtext>
+                                                                and
+                                                            </mtext>
+                                                        </mstyle>
+                                                    </mrow>
+                                                    <mspace width="1em"></mspace>
+                                                    <mi>
+                                                        B
+                                                    </mi>
+                                                    <mo>
+                                                        =
+                                                    </mo>
+                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                        <mfrac>
+                                                            <mn>
+                                                                1
+                                                            </mn>
+                                                            <mn>
+                                                                2
+                                                            </mn>
+                                                        </mfrac>
+                                                    </mrow>
+                                                    <mrow>
+                                                        <mo>
+                                                            (
+                                                        </mo>
+                                                        <mrow>
+                                                            <mi>
+                                                                C
+                                                            </mi>
+                                                            <mo>
+                                                                −<!-- − -->
+                                                            </mo>
+                                                            <msup>
+                                                                <mi>
+                                                                    C
+                                                                </mi>
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mrow class="MJX-TeXAtom-ORD">
+                                                                        <mi mathvariant="sans-serif">
+                                                                            H
+                                                                        </mi>
+                                                                    </mrow>
+                                                                </mrow>
+                                                            </msup>
+                                                        </mrow>
+                                                        <mo>
+                                                            )
+                                                        </mo>
+                                                    </mrow>
+                                                </mstyle>
+                                            </mrow>
+                                            <annotation encoding="application/x-tex">
+                                                {\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}
+                                            </annotation>
+                                        </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0919d2e50fe1008af261f8301f243c002c328dbf" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.838ex; width:63.495ex; height:5.176ex;" alt="{\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}" /></span>
+                                    </dd>
+                                </dl>
+                            </dd>
+                        </dl>
+                        <h2>
+                            <span class="mw-headline" id="Rayleigh_quotient">Rayleigh quotient</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=9" title="Edit section: Rayleigh quotient">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h2>
+                        <div role="note" class="hatnote navigation-not-searchable">
+                            Main article: <a href="/wiki/Rayleigh_quotient" title="Rayleigh quotient">Rayleigh quotient</a>
+                        </div>
+                        <p>
+                            In mathematics, for a given complex Hermitian matrix <i>M</i> and nonzero vector <i>x</i>, the Rayleigh quotient<sup id="cite_ref-4" class="reference"><a href="#cite_note-4">[4]</a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(M,x)}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            R
+                                        </mi>
+                                        <mo stretchy="false">
+                                            (
+                                        </mo>
+                                        <mi>
+                                            M
+                                        </mi>
+                                        <mo>
+                                            ,
+                                        </mo>
+                                        <mi>
+                                            x
+                                        </mi>
+                                        <mo stretchy="false">
+                                            )
+                                        </mo>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle R(M,x)}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ed067bb4bc06662d6bdf6210d450779a529ce5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.379ex; height:2.843ex;" alt="R(M, x)" /></span>, is defined as:<sup id="cite_ref-HornJohnson_3-1" class="reference"><a href="#cite_note-HornJohnson-3">[3]</a></sup><sup class="reference" style="white-space:nowrap;">:<span>p. 234</span></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5">[5]</a></sup>
+                        </p>
+                        <dl>
+                            <dd>
+                                <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(M,x):={\frac {x^{\mathsf {H}}Mx}{x^{\mathsf {H}}x}}}">
+                                <semantics>
+                                    <mrow class="MJX-TeXAtom-ORD">
+                                        <mstyle displaystyle="true" scriptlevel="0">
+                                            <mi>
+                                                R
+                                            </mi>
+                                            <mo stretchy="false">
+                                                (
+                                            </mo>
+                                            <mi>
+                                                M
+                                            </mi>
+                                            <mo>
+                                                ,
+                                            </mo>
+                                            <mi>
+                                                x
+                                            </mi>
+                                            <mo stretchy="false">
+                                                )
+                                            </mo>
+                                            <mo>
+                                                :=
+                                            </mo>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mfrac>
+                                                    <mrow>
+                                                        <msup>
+                                                            <mi>
+                                                                x
+                                                            </mi>
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mi mathvariant="sans-serif">
+                                                                        H
+                                                                    </mi>
+                                                                </mrow>
+                                                            </mrow>
+                                                        </msup>
+                                                        <mi>
+                                                            M
+                                                        </mi>
+                                                        <mi>
+                                                            x
+                                                        </mi>
+                                                    </mrow>
+                                                    <mrow>
+                                                        <msup>
+                                                            <mi>
+                                                                x
+                                                            </mi>
+                                                            <mrow class="MJX-TeXAtom-ORD">
+                                                                <mrow class="MJX-TeXAtom-ORD">
+                                                                    <mi mathvariant="sans-serif">
+                                                                        H
+                                                                    </mi>
+                                                                </mrow>
+                                                            </mrow>
+                                                        </msup>
+                                                        <mi>
+                                                            x
+                                                        </mi>
+                                                    </mrow>
+                                                </mfrac>
+                                            </mrow>
+                                        </mstyle>
+                                    </mrow>
+                                    <annotation encoding="application/x-tex">
+                                        {\displaystyle R(M,x):={\frac {x^{\mathsf {H}}Mx}{x^{\mathsf {H}}x}}}
+                                    </annotation>
+                                </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad9b0047f8437f7b012041d7b2fcd190a5a9ec2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.171ex; width:19.458ex; height:6.009ex;" alt="{\displaystyle R(M,x):={\frac {x^{\mathsf {H}}Mx}{x^{\mathsf {H}}x}}}" /></span>.
+                            </dd>
+                        </dl>
+                        <p>
+                            For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\mathsf {H}}}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <msup>
+                                            <mi>
+                                                x
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi mathvariant="sans-serif">
+                                                        H
+                                                    </mi>
+                                                </mrow>
+                                            </mrow>
+                                        </msup>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle x^{\mathsf {H}}}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b431248ab2f121914608bbd1c2376715cecda9c8" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.726ex; height:2.676ex;" alt="{\displaystyle x^{\mathsf {H}}}" /></span> to the usual transpose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x^{\mathsf {T}}}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <msup>
+                                            <mi>
+                                                x
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mrow class="MJX-TeXAtom-ORD">
+                                                    <mi mathvariant="sans-serif">
+                                                        T
+                                                    </mi>
+                                                </mrow>
+                                            </mrow>
+                                        </msup>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle x^{\mathsf {T}}}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ee4832d06e8560510d81237d0650c897d476e9" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:2.681ex; height:2.676ex;" alt="{\displaystyle x^{\mathsf {T}}}" /></span>. Note that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(M,cx)=R(M,x)}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            R
+                                        </mi>
+                                        <mo stretchy="false">
+                                            (
+                                        </mo>
+                                        <mi>
+                                            M
+                                        </mi>
+                                        <mo>
+                                            ,
+                                        </mo>
+                                        <mi>
+                                            c
+                                        </mi>
+                                        <mi>
+                                            x
+                                        </mi>
+                                        <mo stretchy="false">
+                                            )
+                                        </mo>
+                                        <mo>
+                                            =
+                                        </mo>
+                                        <mi>
+                                            R
+                                        </mi>
+                                        <mo stretchy="false">
+                                            (
+                                        </mo>
+                                        <mi>
+                                            M
+                                        </mi>
+                                        <mo>
+                                            ,
+                                        </mo>
+                                        <mi>
+                                            x
+                                        </mi>
+                                        <mo stretchy="false">
+                                            )
+                                        </mo>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle R(M,cx)=R(M,x)}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d54d3c850d35f99329591e3b57cef98d17237f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:20.864ex; height:2.843ex;" alt="{\displaystyle R(M,cx)=R(M,x)}" /></span> for any non-zero real scalar <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            c
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle c}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="c" /></span>. Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.
+                        </p>
+                        <p>
+                            It can be shown<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2019)">citation needed</span></a></i>]</sup> that, for a given matrix, the Rayleigh quotient reaches its minimum value <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\min }}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <msub>
+                                            <mi>
+                                                λ<!-- λ -->
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mo movablelimits="true" form="prefix">
+                                                    min
+                                                </mo>
+                                            </mrow>
+                                        </msub>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle \lambda _{\min }}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c24522483ceaf1d54224b69af4244b60c3ac08" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.328ex; height:2.509ex;" alt="\lambda_\min" /></span> (the smallest eigenvalue of M) when <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            x
+                                        </mi>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle x}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="x" /></span> is <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\min }}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <msub>
+                                            <mi>
+                                                v
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mo movablelimits="true" form="prefix">
+                                                    min
+                                                </mo>
+                                            </mrow>
+                                        </msub>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle v_{\min }}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/486623019ef451e0582b874018e0249a46e0f996" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.1ex; height:2.009ex;" alt="v_\min" /></span> (the corresponding eigenvector). Similarly, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(M,x)\leq \lambda _{\max }}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            R
+                                        </mi>
+                                        <mo stretchy="false">
+                                            (
+                                        </mo>
+                                        <mi>
+                                            M
+                                        </mi>
+                                        <mo>
+                                            ,
+                                        </mo>
+                                        <mi>
+                                            x
+                                        </mi>
+                                        <mo stretchy="false">
+                                            )
+                                        </mo>
+                                        <mo>
+                                            ≤<!-- ≤ -->
+                                        </mo>
+                                        <msub>
+                                            <mi>
+                                                λ<!-- λ -->
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mo movablelimits="true" form="prefix">
+                                                    max
+                                                </mo>
+                                            </mrow>
+                                        </msub>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle R(M,x)\leq \lambda _{\max }}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18fbf88c578fc9f75d4610ebd18ab55f4f2842ce" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:16.124ex; height:2.843ex;" alt="R(M, x) \leq \lambda_\max" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R(M,v_{\max })=\lambda _{\max }}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <mi>
+                                            R
+                                        </mi>
+                                        <mo stretchy="false">
+                                            (
+                                        </mo>
+                                        <mi>
+                                            M
+                                        </mi>
+                                        <mo>
+                                            ,
+                                        </mo>
+                                        <msub>
+                                            <mi>
+                                                v
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mo movablelimits="true" form="prefix">
+                                                    max
+                                                </mo>
+                                            </mrow>
+                                        </msub>
+                                        <mo stretchy="false">
+                                            )
+                                        </mo>
+                                        <mo>
+                                            =
+                                        </mo>
+                                        <msub>
+                                            <mi>
+                                                λ<!-- λ -->
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mo movablelimits="true" form="prefix">
+                                                    max
+                                                </mo>
+                                            </mrow>
+                                        </msub>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle R(M,v_{\max })=\lambda _{\max }}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200db82bfdbc81cd227cb3470aa826d6f11a7653" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:19.213ex; height:2.843ex;" alt="R(M, v_\max) = \lambda_\max" /></span>.
+                        </p>
+                        <p>
+                            The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.
+                        </p>
+                        <p>
+                            The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{\max }}">
+                            <semantics>
+                                <mrow class="MJX-TeXAtom-ORD">
+                                    <mstyle displaystyle="true" scriptlevel="0">
+                                        <msub>
+                                            <mi>
+                                                λ<!-- λ -->
+                                            </mi>
+                                            <mrow class="MJX-TeXAtom-ORD">
+                                                <mo movablelimits="true" form="prefix">
+                                                    max
+                                                </mo>
+                                            </mrow>
+                                        </msub>
+                                    </mstyle>
+                                </mrow>
+                                <annotation encoding="application/x-tex">
+                                    {\displaystyle \lambda _{\max }}
+                                </annotation>
+                            </semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957584ae6a35f9edf293cb486d7436fb5b75e803" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:4.646ex; height:2.509ex;" alt="\lambda_\max" /></span> is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to <span class="texhtml"><i>M</i></span> associates the Rayleigh quotient <span class="texhtml"><i>R</i>(<i>M</i>, <i>x</i>)</span> for a fixed <span class="texhtml"><i>x</i></span> and <span class="texhtml"><i>M</i></span> varying through the algebra would be referred to as "vector state" of the algebra.
+                        </p>
+                        <h2>
+                            <span class="mw-headline" id="See_also">See also</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=10" title="Edit section: See also">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h2>
+                        <ul>
+                            <li>
+                                <a href="/wiki/Vector_space" title="Vector space">Vector space</a>
+                            </li>
+                            <li>
+                                <a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian matrix</a> (anti-Hermitian matrix)
+                            </li>
+                            <li>
+                                <a href="/wiki/Haynsworth_inertia_additivity_formula" title="Haynsworth inertia additivity formula">Haynsworth inertia additivity formula</a>
+                            </li>
+                            <li>
+                                <a href="/wiki/Hermitian_form" class="mw-redirect" title="Hermitian form">Hermitian form</a>
+                            </li>
+                            <li>
+                                <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint operator</a>
+                            </li>
+                            <li>
+                                <a href="/wiki/Unitary_matrix" title="Unitary matrix">Unitary matrix</a>
+                            </li>
+                        </ul>
+                        <h2>
+                            <span class="mw-headline" id="References">References</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=11" title="Edit section: References">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h2>
+                        <div class="reflist" style="list-style-type: decimal;">
+                            <div class="mw-references-wrap">
+                                <ol class="references">
+                                    <li id="cite_note-1">
+                                        <span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/Theodore_Frankel" title="Theodore Frankel">Frankel, Theodore</a> (2004). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DUnjs6nEn8wC&amp;lpg=PA652&amp;dq=%22Lie%20algebra%22%20physics%20%22skew-Hermitian%22&amp;pg=PA652#v=onepage&amp;q&amp;f=false"><i>The Geometry of Physics: an introduction</i></a>. <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p.&#160;652. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/0-521-53927-7" title="Special:BookSources/0-521-53927-7"><bdi>0-521-53927-7</bdi></a>.</cite></span>
+                                    </li>
+                                    <li id="cite_note-2">
+                                        <span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a rel="nofollow" class="external text" href="http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf">Physics 125 Course Notes</a> at <a href="/wiki/California_Institute_of_Technology" title="California Institute of Technology">California Institute of Technology</a></span>
+                                    </li>
+                                    <li id="cite_note-HornJohnson-3">
+                                        <span class="mw-cite-backlink">^ <a href="#cite_ref-HornJohnson_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HornJohnson_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite class="citation book">Horn, Roger A.; Johnson, Charles R. (2013). <i>Matrix Analysis, second edition</i>. Cambridge University Press. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/9780521839402" title="Special:BookSources/9780521839402"><bdi>9780521839402</bdi></a>.</cite></span>
+                                        <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r935243608" />
+                                    </li>
+                                    <li id="cite_note-4">
+                                        <span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Also known as the <b>Rayleigh–Ritz ratio</b>; named after <a href="/wiki/Walther_Ritz" title="Walther Ritz">Walther Ritz</a> and <a href="/wiki/Lord_Rayleigh" class="mw-redirect" title="Lord Rayleigh">Lord Rayleigh</a>.</span>
+                                    </li>
+                                    <li id="cite_note-5">
+                                        <span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Parlet B. N. <i>The symmetric eigenvalue problem</i>, SIAM, Classics in Applied Mathematics,1998</span>
+                                    </li>
+                                </ol>
+                            </div>
+                        </div>
+                        <h2>
+                            <span class="mw-headline" id="External_links">External links</span><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&amp;action=edit&amp;section=12" title="Edit section: External links">edit</a><span class="mw-editsection-bracket">]</span></span>
+                        </h2>
+                        <ul>
+                            <li>
+                                <cite id="CITEREFHazewinkel2001" class="citation"><a href="/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>, ed. (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=p/h047070">"Hermitian matrix"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, Springer Science+Business Media B.V. / Kluwer Academic Publishers, <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a>&#160;<a href="/wiki/Special:BookSources/978-1-55608-010-4" title="Special:BookSources/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a></cite>
+                                <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r935243608" />
+                            </li>
+                            <li>
+                                <a rel="nofollow" class="external text" href="https://www.cyut.edu.tw/~ckhung/b/la/hermitian.en.php">Visualizing Hermitian Matrix as An Ellipse with Dr. Geo</a>, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.
+                            </li>
+                            <li>
+                                <cite class="citation web"><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath306/kmath306.htm">"Hermitian Matrices"</a>. <i>MathPages.com</i>.</cite>
+                                <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r935243608" />
+                            </li>
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+                    <h3 id="p-lang-label">
+                        Languages
+                    </h3>
+                    <div class="body">
+                        <ul>
+                            <li class="interlanguage-link interwiki-ar">
+                                <a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9_%D9%87%D9%8A%D8%B1%D9%85%D9%8A%D8%AA%D9%8A%D8%A9" title="مصفوفة هيرميتية – Arabic" lang="ar" hreflang="ar" class="interlanguage-link-target" xml:lang="ar">العربية</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-bn">
+                                <a href="https://bn.wikipedia.org/wiki/%E0%A6%B9%E0%A6%BE%E0%A6%B0%E0%A6%AE%E0%A6%BF%E0%A6%B6%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8" title="হারমিশিয়ান – Bangla" lang="bn" hreflang="bn" class="interlanguage-link-target" xml:lang="bn">বাংলা</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-be">
+                                <a href="https://be.wikipedia.org/wiki/%D0%AD%D1%80%D0%BC%D1%96%D1%82%D0%B0%D0%B2%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0" title="Эрмітава матрыца – Belarusian" lang="be" hreflang="be" class="interlanguage-link-target" xml:lang="be">Беларуская</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-ca">
+                                <a href="https://ca.wikipedia.org/wiki/Matriu_herm%C3%ADtica" title="Matriu hermítica – Catalan" lang="ca" hreflang="ca" class="interlanguage-link-target" xml:lang="ca">Català</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-cs">
+                                <a href="https://cs.wikipedia.org/wiki/Hermitovsk%C3%A1_matice" title="Hermitovská matice – Czech" lang="cs" hreflang="cs" class="interlanguage-link-target" xml:lang="cs">Čeština</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-de">
+                                <a href="https://de.wikipedia.org/wiki/Hermitesche_Matrix" title="Hermitesche Matrix – German" lang="de" hreflang="de" class="interlanguage-link-target" xml:lang="de">Deutsch</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-et">
+                                <a href="https://et.wikipedia.org/wiki/Hermiitiline_maatriks" title="Hermiitiline maatriks – Estonian" lang="et" hreflang="et" class="interlanguage-link-target" xml:lang="et">Eesti</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-es">
+                                <a href="https://es.wikipedia.org/wiki/Matriz_hermitiana" title="Matriz hermitiana – Spanish" lang="es" hreflang="es" class="interlanguage-link-target" xml:lang="es">Español</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-eo">
+                                <a href="https://eo.wikipedia.org/wiki/Memadjunkta_matrico" title="Memadjunkta matrico – Esperanto" lang="eo" hreflang="eo" class="interlanguage-link-target" xml:lang="eo">Esperanto</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-fa">
+                                <a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3_%D9%87%D8%B1%D9%85%DB%8C%D8%AA%DB%8C" title="ماتریس هرمیتی – Persian" lang="fa" hreflang="fa" class="interlanguage-link-target" xml:lang="fa">فارسی</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-fr">
+                                <a href="https://fr.wikipedia.org/wiki/Hermitien" title="Hermitien – French" lang="fr" hreflang="fr" class="interlanguage-link-target" xml:lang="fr">Français</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-ko">
+                                <a href="https://ko.wikipedia.org/wiki/%EC%97%90%EB%A5%B4%EB%AF%B8%ED%8A%B8_%ED%96%89%EB%A0%AC" title="에르미트 행렬 – Korean" lang="ko" hreflang="ko" class="interlanguage-link-target" xml:lang="ko">한국어</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-it">
+                                <a href="https://it.wikipedia.org/wiki/Matrice_hermitiana" title="Matrice hermitiana – Italian" lang="it" hreflang="it" class="interlanguage-link-target" xml:lang="it">Italiano</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-lv">
+                                <a href="https://lv.wikipedia.org/wiki/Ermita_matrica" title="Ermita matrica – Latvian" lang="lv" hreflang="lv" class="interlanguage-link-target" xml:lang="lv">Latviešu</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-lt">
+                                <a href="https://lt.wikipedia.org/wiki/Ermito_matrica" title="Ermito matrica – Lithuanian" lang="lt" hreflang="lt" class="interlanguage-link-target" xml:lang="lt">Lietuvių</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-hu">
+                                <a href="https://hu.wikipedia.org/wiki/Hermitikus_m%C3%A1trix" title="Hermitikus mátrix – Hungarian" lang="hu" hreflang="hu" class="interlanguage-link-target" xml:lang="hu">Magyar</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-nl">
+                                <a href="https://nl.wikipedia.org/wiki/Hermitische_matrix" title="Hermitische matrix – Dutch" lang="nl" hreflang="nl" class="interlanguage-link-target" xml:lang="nl">Nederlands</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-ja">
+                                <a href="https://ja.wikipedia.org/wiki/%E3%82%A8%E3%83%AB%E3%83%9F%E3%83%BC%E3%83%88%E8%A1%8C%E5%88%97" title="エルミート行列 – Japanese" lang="ja" hreflang="ja" class="interlanguage-link-target" xml:lang="ja">日本語</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-pl">
+                                <a href="https://pl.wikipedia.org/wiki/Macierz_hermitowska" title="Macierz hermitowska – Polish" lang="pl" hreflang="pl" class="interlanguage-link-target" xml:lang="pl">Polski</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-pt">
+                                <a href="https://pt.wikipedia.org/wiki/Matriz_hermitiana" title="Matriz hermitiana – Portuguese" lang="pt" hreflang="pt" class="interlanguage-link-target" xml:lang="pt">Português</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-ru">
+                                <a href="https://ru.wikipedia.org/wiki/%D0%AD%D1%80%D0%BC%D0%B8%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Эрмитова матрица – Russian" lang="ru" hreflang="ru" class="interlanguage-link-target" xml:lang="ru">Русский</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-sl">
+                                <a href="https://sl.wikipedia.org/wiki/Hermitska_matrika" title="Hermitska matrika – Slovenian" lang="sl" hreflang="sl" class="interlanguage-link-target" xml:lang="sl">Slovenščina</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-sr">
+                                <a href="https://sr.wikipedia.org/wiki/%D0%95%D1%80%D0%BC%D0%B8%D1%82%D1%81%D0%BA%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Ермитска матрица – Serbian" lang="sr" hreflang="sr" class="interlanguage-link-target" xml:lang="sr">Српски / srpski</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-fi">
+                                <a href="https://fi.wikipedia.org/wiki/Hermiittinen_matriisi" title="Hermiittinen matriisi – Finnish" lang="fi" hreflang="fi" class="interlanguage-link-target" xml:lang="fi">Suomi</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-sv">
+                                <a href="https://sv.wikipedia.org/wiki/Hermitesk_matris" title="Hermitesk matris – Swedish" lang="sv" hreflang="sv" class="interlanguage-link-target" xml:lang="sv">Svenska</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-ta">
+                                <a href="https://ta.wikipedia.org/wiki/%E0%AE%B9%E0%AF%86%E0%AE%B0%E0%AF%8D%E0%AE%AE%E0%AF%88%E0%AE%9F%E0%AF%8D_%E0%AE%85%E0%AE%A3%E0%AE%BF" title="ஹெர்மைட் அணி – Tamil" lang="ta" hreflang="ta" class="interlanguage-link-target" xml:lang="ta">தமிழ்</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-th">
+                                <a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A1%E0%B8%97%E0%B8%A3%E0%B8%B4%E0%B8%81%E0%B8%8B%E0%B9%8C%E0%B9%81%E0%B8%AD%E0%B8%A3%E0%B9%8C%E0%B8%A1%E0%B8%B4%E0%B8%95" title="เมทริกซ์แอร์มิต – Thai" lang="th" hreflang="th" class="interlanguage-link-target" xml:lang="th">ไทย</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-tr">
+                                <a href="https://tr.wikipedia.org/wiki/Hermisyen_matris" title="Hermisyen matris – Turkish" lang="tr" hreflang="tr" class="interlanguage-link-target" xml:lang="tr">Türkçe</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-uk">
+                                <a href="https://uk.wikipedia.org/wiki/%D0%95%D1%80%D0%BC%D1%96%D1%82%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F" title="Ермітова матриця – Ukrainian" lang="uk" hreflang="uk" class="interlanguage-link-target" xml:lang="uk">Українська</a>
+                            </li>
+                            <li class="interlanguage-link interwiki-zh">
+                                <a href="https://zh.wikipedia.org/wiki/%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E7%9F%A9%E9%98%B5" title="埃尔米特矩阵 – Chinese" lang="zh" hreflang="zh" class="interlanguage-link-target" xml:lang="zh">中文</a>
+                            </li>
+                        </ul>
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