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<div id="mw-content-text" lang="en" dir="ltr">
<p>
In mathematics, a <b>Hermitian matrix</b> (or <b>self-adjoint matrix</b>) is a <a href="http://fakehost/wiki/Complex_number" title="Complex number">complex</a> <a href="http://fakehost/wiki/Square_matrix" title="Square matrix">square matrix</a> that is equal to its own <a href="http://fakehost/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>—that is, the element in the <span>i</span>-th row and <span>j</span>-th column is equal to the <a href="http://fakehost/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> of the element in the <span>j</span>-th row and <span>i</span>-th column, for all indices <span>i</span> and <span>j</span>:
</p>
<p>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28a0aaa74b2267a48312e19321211cd9e3a39228" aria-hidden="true" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}"></span>
</p>
<p>
or in matrix form:
</p>
<dl>
<dd>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca00b61ff0e264e6c1e5adc9a00c0d2751feecf" aria-hidden="true" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}}"></span>.
</dd>
</dl>
<p>
Hermitian matrices can be understood as the complex extension of real <a href="http://fakehost/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrices</a>.
</p>
<p>
If the <a href="http://fakehost/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of a matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is denoted by <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9415702ab196cc26f5df37af2d90e07318e93df" aria-hidden="true" alt="{\displaystyle A^{\mathsf {H}}}"></span>, then the Hermitian property can be written concisely as
</p>
<p>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/291d260bf69b764e75818669ab27870d58771e1f" aria-hidden="true" alt="{\displaystyle A{\text{ Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}"></span>
</p>
<p>
Hermitian matrices are named after <a href="http://fakehost/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a>, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real <a href="http://fakehost/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a>. Other, equivalent notations in common use are <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa270391d183478251283d2c4b2c72ac4563352" aria-hidden="true" alt="{\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast }}"></span>, although note that in <a href="http://fakehost/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5541bfa07743be995242c892c344395e672d6fa2" aria-hidden="true" alt="A^{\ast }"></span> typically means the <a href="http://fakehost/wiki/Complex_conjugate" title="Complex conjugate">complex conjugate</a> only, and not the <a href="http://fakehost/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a>.
</p>
<h2>
<span id="Alternative_characterizations">Alternative characterizations</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=1" title="Edit section: Alternative characterizations">edit</a><span>]</span></span>
</h2>
<p>
Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below:
</p>
<h3>
<span id="Equality_with_the_adjoint">Equality with the adjoint</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=2" title="Edit section: Equality with the adjoint">edit</a><span>]</span></span>
</h3>
<p>
A square matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is Hermitian if and only if it is equal to its <a href="http://fakehost/wiki/Hermitian_adjoint" title="Hermitian adjoint">adjoint</a>, that is, it satisfies
</p>
<div>
<p><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459de45e76bace9d04a80d2e8efc2abbbc246047" aria-hidden="true" alt="{\displaystyle \langle w,Av\rangle =\langle Aw,v\rangle ,}">
</p></div><p>for any pair of vectors <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6425c6e94fa47976601cb44d7564b5d04dcfbfef" aria-hidden="true" alt="v,w"></span>, where <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a50080b735975d8001c9552ac2134b49ad534c0" aria-hidden="true" alt="{\displaystyle \langle \cdot ,\cdot \rangle }"></span> denotes <a href="http://fakehost/wiki/Dot_product" title="Dot product">the inner product</a> operation.
</p><p>
This is also the way that the more general concept of <a href="http://fakehost/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operator</a> is defined.
</p>
<h3>
<span id="Reality_of_quadratic_forms">Reality of quadratic forms</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=3" title="Edit section: Reality of quadratic forms">edit</a><span>]</span></span>
</h3>
<p>
A square matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is Hermitian if and only if it is such that
</p>
<div>
<p><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/997ea0350c18735926412de88420ac9ca989f50c" aria-hidden="true" alt="{\displaystyle \langle v,Av\rangle \in \mathbb {R} ,\quad v\in V.}">
</p></div>
<h3>
<span id="Spectral_properties">Spectral properties</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=4" title="Edit section: Spectral properties">edit</a><span>]</span></span>
</h3>
<p>
A square matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is Hermitian if and only if it is unitarily <a href="http://fakehost/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable</a> with real <a href="http://fakehost/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues</a>.
</p>
<h2>
<span id="Applications">Applications</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=5" title="Edit section: Applications">edit</a><span>]</span></span>
</h2>
<p>
Hermitian matrices are fundamental to the quantum theory of <a href="http://fakehost/wiki/Matrix_mechanics" title="Matrix mechanics">matrix mechanics</a> created by <a href="http://fakehost/wiki/Werner_Heisenberg" title="Werner Heisenberg">Werner Heisenberg</a>, <a href="http://fakehost/wiki/Max_Born" title="Max Born">Max Born</a>, and <a href="http://fakehost/wiki/Pascual_Jordan" title="Pascual Jordan">Pascual Jordan</a> in 1925.
</p>
<h2>
<span id="Examples">Examples</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=6" title="Edit section: Examples">edit</a><span>]</span></span>
</h2>
<p>
In this section, the conjugate transpose of matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is denoted as <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9415702ab196cc26f5df37af2d90e07318e93df" aria-hidden="true" alt="{\displaystyle A^{\mathsf {H}}}"></span>, the transpose of matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is denoted as <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54bf0331204e30cba9ec7f695dfea97e6745a7c2" aria-hidden="true" alt="{\displaystyle A^{\mathsf {T}}}"></span> and conjugate of matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is denoted as <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92efef0e89bdc77f6a848764195ef5b9d9bfcc6a" aria-hidden="true" alt="{\displaystyle {\overline {A}}}"></span>.
</p>
<p>
See the following example:
</p>
<dl>
<dd>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00ccf11c16396b6ddd4f2254f7132cd8bb2c57ee" aria-hidden="true" alt="{\displaystyle {\begin{bmatrix}2&2+i&4\\2-i&3&i\\4&-i&1\\\end{bmatrix}}}"></span>
</dd>
</dl>
<p>
The diagonal elements must be <a href="http://fakehost/wiki/Real_number" title="Real number">real</a>, as they must be their own complex conjugate.
</p>
<p>
Well-known families of <a href="http://fakehost/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>, <a href="http://fakehost/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann matrices</a> and their generalizations are Hermitian. In <a href="http://fakehost/wiki/Theoretical_physics" title="Theoretical physics">theoretical physics</a> such Hermitian matrices are often multiplied by <a href="http://fakehost/wiki/Imaginary_number" title="Imaginary number">imaginary</a> coefficients,<sup id="cite_ref-1"><a href="#cite_note-1">[1]</a></sup><sup id="cite_ref-2"><a href="#cite_note-2">[2]</a></sup> which results in <i>skew-Hermitian</i> matrices (see <a href="#facts">below</a>).
</p>
<p>
Here, we offer another useful Hermitian matrix using an abstract example. If a square matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> equals the <a href="http://fakehost/wiki/Matrix_multiplication" title="Matrix multiplication">multiplication of a matrix</a> and its conjugate transpose, that is, <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f0efab2d7c3a4b4b7caf14cc0705dadd13195a9" aria-hidden="true" alt="{\displaystyle A=BB^{\mathsf {H}}}"></span>, then <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is a Hermitian <a href="http://fakehost/wiki/Positive_semi-definite_matrix" title="Positive semi-definite matrix">positive semi-definite matrix</a>. Furthermore, if <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" aria-hidden="true" alt="B"></span> is row full-rank, then <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" aria-hidden="true" alt="A"></span> is positive definite.
</p>
<h2>
<span id="Properties">Properties</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=7" title="Edit section: Properties">edit</a><span>]</span></span>
</h2>
<table role="presentation" readabilityDataTable="0">
<tbody>
<tr>
<td>
<p><a href="http://fakehost/wiki/File:Wiki_letter_w_cropped.svg"><img alt="[icon]" src="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/44px-Wiki_letter_w_cropped.svg.png" decoding="async" width="44" height="31" srcset="http://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/66px-Wiki_letter_w_cropped.svg.png 1.5x, http://upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Wiki_letter_w_cropped.svg/88px-Wiki_letter_w_cropped.svg.png 2x" data-file-width="44" data-file-height="31"></a>
</p>
</td>
<td>
<p>
This section <b>needs expansion</b> with: Proof of the properties requested. <small>You can help by <a href="https://en.wikipedia.org/w/index.php?title=Hermitian_matrix&action=edit&section=1">adding to it</a>.</small> <small><i>(<span>February 2018</span>)</i></small>
</p>
</td>
</tr>
</tbody>
</table>
<ul>
<li>The entries on the <a href="http://fakehost/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> (top left to bottom right) of any Hermitian matrix are <a href="http://fakehost/wiki/Real_number" title="Real number">real</a>.
</li>
</ul>
<dl>
<dd>
<i>Proof:</i> By definition of the Hermitian matrix
<dl>
<dd>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa8265c5f6d4fc3b7cda6a06558c7d4d9aec855" aria-hidden="true" alt="{\displaystyle H_{ij}={\overline {H}}_{ji}}"></span>
</dd>
</dl>
</dd>
<dd>
so for <span><i>i</i> = <i>j</i></span> the above follows.
</dd>
<dd>
Only the <a href="http://fakehost/wiki/Main_diagonal" title="Main diagonal">main diagonal</a> entries are necessarily real; Hermitian matrices can have arbitrary complex-valued entries in their <a href="http://fakehost/wiki/Off-diagonal_element" title="Off-diagonal element">off-diagonal elements</a>, as long as diagonally-opposite entries are complex conjugates.
</dd>
</dl>
<ul>
<li>A matrix that has only real entries is Hermitian <a href="http://fakehost/wiki/If_and_only_if" title="If and only if">if and only if</a> it is <a href="http://fakehost/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>. A real and symmetric matrix is simply a special case of a Hermitian matrix.
</li>
</ul>
<dl>
<dd>
<i>Proof:</i> <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa8265c5f6d4fc3b7cda6a06558c7d4d9aec855" aria-hidden="true" alt="{\displaystyle H_{ij}={\overline {H}}_{ji}}"></span> by definition. Thus <span>H<sub><i>ij</i></sub> = H<sub><i>ji</i></sub></span> (matrix symmetry) if and only if <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f1862750b96d01100244370b3fca45f01923ce5" aria-hidden="true" alt="{\displaystyle H_{ij}={\overline {H}}_{ij}}"></span> (<span>H<sub><i>ij</i></sub></span> is real).
</dd>
</dl>
<ul>
<li>Every Hermitian matrix is a <a href="http://fakehost/wiki/Normal_matrix" title="Normal matrix">normal matrix</a>. That is to say, <span>AA<sup>H</sup> = A<sup>H</sup>A</span>.
</li>
</ul>
<dl>
<dd>
<i>Proof:</i> <span>A = A<sup>H</sup></span>, so <span>AA<sup>H</sup> = AA = A<sup>H</sup>A</span>.
</dd>
</dl>
<ul>
<li>The finite-dimensional <a href="http://fakehost/wiki/Spectral_theorem" title="Spectral theorem">spectral theorem</a> says that any Hermitian matrix can be <a href="http://fakehost/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalized</a> by a <a href="http://fakehost/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a>, and that the resulting diagonal matrix has only real entries. This implies that all <a href="http://fakehost/wiki/Eigenvectors" title="Eigenvectors">eigenvalues</a> of a Hermitian matrix <span>A</span> with dimension <span>n</span> are real, and that <span>A</span> has <span>n</span> linearly independent <a href="http://fakehost/wiki/Eigenvector" title="Eigenvector">eigenvectors</a>. Moreover, a Hermitian matrix has <a href="http://fakehost/wiki/Orthogonal" title="Orthogonal">orthogonal</a> eigenvectors for distinct eigenvalues. Even if there are degenerate eigenvalues, it is always possible to find an <a href="http://fakehost/wiki/Orthogonal_basis" title="Orthogonal basis">orthogonal basis</a> of <span>ℂ<sup><i>n</i></sup></span> consisting of <span>n</span> eigenvectors of <span>A</span>.
</li>
</ul>
<ul>
<li>The sum of any two Hermitian matrices is Hermitian.
</li>
</ul>
<dl>
<dd>
<i>Proof:</i> <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251bf4ebbe3b0d119e0a7d19f8fd134c4f072971" aria-hidden="true" alt="{\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}={\overline {A}}_{ji}+{\overline {B}}_{ji}={\overline {(A+B)}}_{ji},}"></span> as claimed.
</dd>
</dl>
<ul>
<li>The <a href="http://fakehost/wiki/Inverse_matrix" title="Inverse matrix">inverse</a> of an invertible Hermitian matrix is Hermitian as well.
</li>
</ul>
<dl>
<dd>
<i>Proof:</i> If <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/021893240ff7fa3148b6649b0ba4d88cd207b5f0" aria-hidden="true" alt="{\displaystyle A^{-1}A=I}"></span>, then <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a28a8250ab35ad60228bb0376eb4b7024f027815" aria-hidden="true" alt="{\displaystyle I=I^{H}=(A^{-1}A)^{H}=A^{H}(A^{-1})^{H}=A(A^{-1})^{H}}"></span>, so <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0179c3a7aebe194ccd9a19ba02b972500f88a7a" aria-hidden="true" alt="{\displaystyle A^{-1}=(A^{-1})^{H}}"></span> as claimed.
</dd>
</dl>
<ul>
<li>The <a href="http://fakehost/wiki/Matrix_multiplication" title="Matrix multiplication">product</a> of two Hermitian matrices <span>A</span> and <span>B</span> is Hermitian if and only if <span><i>AB</i> = <i>BA</i></span>.
</li>
</ul>
<dl>
<dd>
<i>Proof:</i> Note that <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6cf8185ca7a0687bf959bb65b16db6cf1714ca2" aria-hidden="true" alt="{\displaystyle (AB)^{\mathsf {H}}={\overline {(AB)^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}A^{\mathsf {T}}}}={\overline {B^{\mathsf {T}}}}{\overline {A^{\mathsf {T}}}}=B^{\mathsf {H}}A^{\mathsf {H}}=BA.}"></span> Thus <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d303a1ebcac8547489b170be5d0dd7d8e04e548e" aria-hidden="true" alt="{\displaystyle (AB)^{\mathsf {H}}=AB}"></span> <a href="http://fakehost/wiki/If_and_only_if" title="If and only if">if and only if</a> <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992c8ea49fdd26b491180036c5a4d879fec77442" aria-hidden="true" alt="AB=BA"></span>.
</dd>
<dd>
Thus <span><i>A</i><sup><i>n</i></sup></span> is Hermitian if <span>A</span> is Hermitian and <span>n</span> is an integer.
</dd>
</dl>
<ul>
<li>The Hermitian complex <span>n</span>-by-<span>n</span> matrices do not form a <a href="http://fakehost/wiki/Vector_space" title="Vector space">vector space</a> over the <a href="http://fakehost/wiki/Complex_number" title="Complex number">complex numbers</a>, <span>ℂ</span>, since the identity matrix <span><i>I</i><sub><i>n</i></sub></span> is Hermitian, but <span><i>i</i> <i>I</i><sub><i>n</i></sub></span> is not. However the complex Hermitian matrices <i>do</i> form a vector space over the <a href="http://fakehost/wiki/Real_numbers" title="Real numbers">real numbers</a> <span>ℝ</span>. In the <span>2<i>n</i><sup>2</sup></span>-<a href="http://fakehost/wiki/Dimension_of_a_vector_space" title="Dimension of a vector space">dimensional</a> vector space of complex <span><i>n</i> × <i>n</i></span> matrices over <span>ℝ</span>, the complex Hermitian matrices form a subspace of dimension <span><i>n</i><sup>2</sup></span>. If <span><i>E</i><sub><i>jk</i></sub></span> denotes the <span>n</span>-by-<span>n</span> matrix with a <span>1</span> in the <span><i>j</i>,<i>k</i></span> position and zeros elsewhere, a basis (orthonormal w.r.t. the Frobenius inner product) can be described as follows:
</li>
</ul>
<dl>
<dd>
<dl>
<dd>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46eedb181c0bdae46e8c1526161b03d0ea97b4b4" aria-hidden="true" alt="{\displaystyle E_{jj}{\text{ for }}1\leq j\leq n\quad (n{\text{ matrices}})}"></span>
</dd>
</dl>
</dd>
</dl>
<dl>
<dd>
together with the set of matrices of the form
</dd>
</dl>
<dl>
<dd>
<dl>
<dd>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ddeac51c423f6dbefc5f63e483d9aee96e6fa342" aria-hidden="true" alt="{\displaystyle {\frac {1}{\sqrt {2}}}\left(E_{jk}+E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}"></span>
</dd>
</dl>
</dd>
</dl>
<dl>
<dd>
and the matrices
</dd>
</dl>
<dl>
<dd>
<dl>
<dd>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db65cce3a8fa33e5b7b96badd756c8573aa866c0" aria-hidden="true" alt="{\displaystyle {\frac {i}{\sqrt {2}}}\left(E_{jk}-E_{kj}\right){\text{ for }}1\leq j<k\leq n\quad \left({\frac {n^{2}-n}{2}}{\text{ matrices}}\right)}"></span>
</dd>
</dl>
</dd>
</dl>
<dl>
<dd>
where <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" aria-hidden="true" alt="i"></span> denotes the complex number <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea1ea9ac61e6e1e84ac39130f78143c18865719" aria-hidden="true" alt="{\sqrt {-1}}"></span>, called the <i><a href="http://fakehost/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a></i>.
</dd>
</dl>
<dl>
<dd>
<dl>
<dd>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b7d749931e5f709bcbc0a446638d3b6b8ed0c6c" aria-hidden="true" alt="{\displaystyle A=\sum _{j}\lambda _{j}u_{j}u_{j}^{\mathsf {H}},}"></span>
</dd>
</dl>
</dd>
<dd>
where <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa91daf9145f27bb95746fd2a37537342d587b77" aria-hidden="true" alt="\lambda _{j}"></span> are the eigenvalues on the diagonal of the diagonal matrix <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1934e7eadd31fbf6f7d6bcf9c0e9bec934ce8976" aria-hidden="true" alt="\; \Lambda "></span>.
</dd>
</dl>
<ul>
<li>The determinant of a Hermitian matrix is real:
</li>
</ul>
<dl>
<dd>
<i>Proof:</i> <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1240df64c3010e0be6eae865fdfcfe6f77bf5eb" aria-hidden="true" alt="{\displaystyle \det(A)=\det \left(A^{\mathsf {T}}\right)\quad \Rightarrow \quad \det \left(A^{\mathsf {H}}\right)={\overline {\det(A)}}}"></span>
</dd>
<dd>
Therefore if <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43cc392bdcfbb134dd66d9b469847f6370e29d9d" aria-hidden="true" alt="{\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}}"></span>.
</dd>
<dd>
(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.)
</dd>
</dl>
<h2>
<span id="Decomposition_into_Hermitian_and_skew-Hermitian">Decomposition into Hermitian and skew-Hermitian</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=8" title="Edit section: Decomposition into Hermitian and skew-Hermitian">edit</a><span>]</span></span>
</h2>
<p>
<span id="facts"></span>Additional facts related to Hermitian matrices include:
</p>
<ul>
<li>The sum of a square matrix and its conjugate transpose <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ef97bb04ce4ab682bcc84cf1059f8da235b483e" aria-hidden="true" alt="{\displaystyle \left(A+A^{\mathsf {H}}\right)}"></span> is Hermitian.
</li>
</ul>
<ul>
<li>The difference of a square matrix and its conjugate transpose <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4ac665be4943ce769e33109e9f64abcf1e98050" aria-hidden="true" alt="{\displaystyle \left(A-A^{\mathsf {H}}\right)}"></span> is <a href="http://fakehost/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">skew-Hermitian</a> (also called antihermitian). This implies that the <a href="http://fakehost/wiki/Commutator" title="Commutator">commutator</a> of two Hermitian matrices is skew-Hermitian.
</li>
</ul>
<ul>
<li>An arbitrary square matrix <span>C</span> can be written as the sum of a Hermitian matrix <span>A</span> and a skew-Hermitian matrix <span>B</span>. This is known as the Toeplitz decomposition of <span>C</span>.<sup id="cite_ref-HornJohnson_3-0"><a href="#cite_note-HornJohnson-3">[3]</a></sup><sup>:<span>p. 7</span></sup>
</li>
</ul>
<dl>
<dd>
<dl>
<dd>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0919d2e50fe1008af261f8301f243c002c328dbf" aria-hidden="true" alt="{\displaystyle C=A+B\quad {\mbox{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\mbox{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}"></span>
</dd>
</dl>
</dd>
</dl>
<h2>
<span id="Rayleigh_quotient">Rayleigh quotient</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=9" title="Edit section: Rayleigh quotient">edit</a><span>]</span></span>
</h2>
<p>
In mathematics, for a given complex Hermitian matrix <i>M</i> and nonzero vector <i>x</i>, the Rayleigh quotient<sup id="cite_ref-4"><a href="#cite_note-4">[4]</a></sup> <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8ed067bb4bc06662d6bdf6210d450779a529ce5" aria-hidden="true" alt="R(M, x)"></span>, is defined as:<sup id="cite_ref-HornJohnson_3-1"><a href="#cite_note-HornJohnson-3">[3]</a></sup><sup>:<span>p. 234</span></sup><sup id="cite_ref-5"><a href="#cite_note-5">[5]</a></sup>
</p>
<dl>
<dd>
<span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad9b0047f8437f7b012041d7b2fcd190a5a9ec2" aria-hidden="true" alt="{\displaystyle R(M,x):={\frac {x^{\mathsf {H}}Mx}{x^{\mathsf {H}}x}}}"></span>.
</dd>
</dl>
<p>
For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b431248ab2f121914608bbd1c2376715cecda9c8" aria-hidden="true" alt="{\displaystyle x^{\mathsf {H}}}"></span> to the usual transpose <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4ee4832d06e8560510d81237d0650c897d476e9" aria-hidden="true" alt="{\displaystyle x^{\mathsf {T}}}"></span>. Note that <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d54d3c850d35f99329591e3b57cef98d17237f" aria-hidden="true" alt="{\displaystyle R(M,cx)=R(M,x)}"></span> for any non-zero real scalar <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" aria-hidden="true" alt="c"></span>. Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues.
</p>
<p>
It can be shown<sup>[<i><a href="http://fakehost/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2019)">citation needed</span></a></i>]</sup> that, for a given matrix, the Rayleigh quotient reaches its minimum value <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82c24522483ceaf1d54224b69af4244b60c3ac08" aria-hidden="true" alt="\lambda_\min"></span> (the smallest eigenvalue of M) when <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" aria-hidden="true" alt="x"></span> is <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/486623019ef451e0582b874018e0249a46e0f996" aria-hidden="true" alt="v_\min"></span> (the corresponding eigenvector). Similarly, <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18fbf88c578fc9f75d4610ebd18ab55f4f2842ce" aria-hidden="true" alt="R(M, x) \leq \lambda_\max"></span> and <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200db82bfdbc81cd227cb3470aa826d6f11a7653" aria-hidden="true" alt="R(M, v_\max) = \lambda_\max"></span>.
</p>
<p>
The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues. It is also used in eigenvalue algorithms to obtain an eigenvalue approximation from an eigenvector approximation. Specifically, this is the basis for Rayleigh quotient iteration.
</p>
<p>
The range of the Rayleigh quotient (for matrix that is not necessarily Hermitian) is called a numerical range (or spectrum in functional analysis). When the matrix is Hermitian, the numerical range is equal to the spectral norm. Still in functional analysis, <span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/957584ae6a35f9edf293cb486d7436fb5b75e803" aria-hidden="true" alt="\lambda_\max"></span> is known as the spectral radius. In the context of C*-algebras or algebraic quantum mechanics, the function that to <span><i>M</i></span> associates the Rayleigh quotient <span><i>R</i>(<i>M</i>, <i>x</i>)</span> for a fixed <span><i>x</i></span> and <span><i>M</i></span> varying through the algebra would be referred to as "vector state" of the algebra.
</p>
<h2>
<span id="See_also">See also</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=10" title="Edit section: See also">edit</a><span>]</span></span>
</h2>
<ul>
<li>
<a href="http://fakehost/wiki/Vector_space" title="Vector space">Vector space</a>
</li>
<li>
<a href="http://fakehost/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian matrix</a> (anti-Hermitian matrix)
</li>
<li>
<a href="http://fakehost/wiki/Haynsworth_inertia_additivity_formula" title="Haynsworth inertia additivity formula">Haynsworth inertia additivity formula</a>
</li>
<li>
<a href="http://fakehost/wiki/Hermitian_form" title="Hermitian form">Hermitian form</a>
</li>
<li>
<a href="http://fakehost/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint operator</a>
</li>
<li>
<a href="http://fakehost/wiki/Unitary_matrix" title="Unitary matrix">Unitary matrix</a>
</li>
</ul>
<h2>
<span id="References">References</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=11" title="Edit section: References">edit</a><span>]</span></span>
</h2>
<h2>
<span id="External_links">External links</span><span><span>[</span><a href="http://fakehost/w/index.php?title=Hermitian_matrix&action=edit&section=12" title="Edit section: External links">edit</a><span>]</span></span>
</h2>
<ul>
<li>
<cite id="CITEREFHazewinkel2001"><a href="http://fakehost/wiki/Michiel_Hazewinkel" title="Michiel Hazewinkel">Hazewinkel, Michiel</a>, ed. (2001) [1994], <a rel="nofollow" href="https://www.encyclopediaofmath.org/index.php?title=p/h047070">"Hermitian matrix"</a>, <i><a href="http://fakehost/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, Springer Science+Business Media B.V. / Kluwer Academic Publishers, <a href="http://fakehost/wiki/International_Standard_Book_Number" title="International Standard Book Number">ISBN</a> <a href="http://fakehost/wiki/Special:BookSources/978-1-55608-010-4" title="Special:BookSources/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a></cite>
</li>
<li>
<a rel="nofollow" href="https://www.cyut.edu.tw/~ckhung/b/la/hermitian.en.php">Visualizing Hermitian Matrix as An Ellipse with Dr. Geo</a>, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.
</li>
<li>
<cite><a rel="nofollow" href="http://www.mathpages.com/home/kmath306/kmath306.htm">"Hermitian Matrices"</a>. <i>MathPages.com</i>.</cite>
</li>
</ul>
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