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]]> - </style> - </head> - <body class="mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Hermitian_matrix rootpage-Hermitian_matrix skin-vector action-view"> - <div id="mw-page-base" class="noprint"></div> - <div id="mw-head-base" class="noprint"></div> - <div id="content" class="mw-body" role="main"> - <a id="top"></a> - <div id="siteNotice" class="mw-body-content"> - <!-- CentralNotice --> - </div> - <div class="mw-indicators mw-body-content"></div> - <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> - <div id="contentSub"></div> - <div id="jump-to-nav"></div><a class="mw-jump-link" href="#mw-head">Jump to navigation</a> <a class="mw-jump-link" href="#p-search">Jump to search</a> - <div id="mw-content-text" lang="en" dir="ltr" class="mw-content-ltr" xml:lang="en"> - <div class="mw-parser-output"> - <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}}}}"> - <semantics> - <mrow class="MJX-TeXAtom-ORD"> - <mstyle displaystyle="true" scriptlevel="0"> - <mi> - A - </mi> - <mrow class="MJX-TeXAtom-ORD"> - <mtext> - Hermitian - </mtext> - </mrow> - <mspace width="1em"></mspace> - <mspace width="thickmathspace"></mspace> - <mo stretchy="false"> - ⟺<!-- ⟺ --> - </mo> - <mspace width="thickmathspace"></mspace> - <mspace width="1em"></mspace> - <msub> - <mi> - a - </mi> - <mrow class="MJX-TeXAtom-ORD"> - <mi> - i - </mi> - <mi> - j - </mi> - </mrow> - </msub> - <mo> - = - </mo> - <mrow class="MJX-TeXAtom-ORD"> - <mover> - <msub> - <mi> - a - </mi> - <mrow class="MJX-TeXAtom-ORD"> - <mi> - j - </mi> - <mi> - i - </mi> - </mrow> - </msub> - <mo accent="false"> - ¯<!-- ¯ --> - </mo> - </mover> - </mrow> - </mstyle> - </mrow> - <annotation encoding="application/x-tex"> - {\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> - 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> - 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&action=edit&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&action=edit&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&action=edit&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&action=edit&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&action=edit&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&action=edit&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&2+i&4\\2-i&3&i\\4&-i&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&2+i&4\\2-i&3&i\\4&-i&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&2+i&4\\2-i&3&i\\4&-i&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&action=edit&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&action=edit&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> - for - </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> - 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<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> - for - </mtext> - </mrow> - <mn> - 1 - </mn> - <mo> - ≤<!-- ≤ --> - </mo> - <mi> - j - </mi> - <mo> - < - </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> - 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<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<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<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> - for - </mtext> - </mrow> - <mn> - 1 - </mn> - <mo> - ≤<!-- ≤ --> - </mo> - <mi> - j - </mi> - <mo> - < - </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> - 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<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<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&action=edit&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> - <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> - <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&action=edit&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&action=edit&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&action=edit&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&lpg=PA652&dq=%22Lie%20algebra%22%20physics%20%22skew-Hermitian%22&pg=PA652#v=onepage&q&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. 652. <a href="/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <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> <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&action=edit&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> <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> - </ul><!-- -NewPP limit report -Parsed by mw1375 -Cached time: 20200224203345 -Cache expiry: 2592000 -Dynamic content: false -Complications: [vary‐revision‐sha1] -CPU time usage: 0.344 seconds -Real time usage: 0.623 seconds -Preprocessor visited node count: 1719/1000000 -Post‐expand include size: 18736/2097152 bytes -Template argument size: 1789/2097152 bytes -Highest expansion depth: 13/40 -Expensive parser function count: 3/500 -Unstrip recursion depth: 1/20 -Unstrip post‐expand size: 14154/5000000 bytes -Number of Wikibase entities loaded: 0/400 -Lua time usage: 0.107/10.000 seconds -Lua memory usage: 3.33 MB/50 MB ---> - <!-- -Transclusion expansion time report (%,ms,calls,template) -100.00% 423.690 1 -total - 26.69% 113.077 1 Template:Use_American_English - 21.93% 92.904 1 Template:Reflist - 19.39% 82.135 2 Template:Cite_book - 14.77% 62.599 1 Template:Short_description - 9.97% 42.248 1 Template:Expand_section - 7.81% 33.100 1 Template:Ambox - 7.76% 32.863 1 Template:Pagetype - 6.13% 25.975 1 Template:Citation_needed - 5.21% 22.058 1 Template:Fix ---> - <!-- Saved in parser cache with key enwiki:pcache:idhash:189682-0!canonical!math=5 and timestamp 20200224203346 and revision id 942460710 - --> - </div><noscript><img src="//en.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; 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<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> - <div class="after-portlet after-portlet-lang"> - <span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q652941#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span> - 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