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Alternative characterizations subsection</span> </button> <ul id="toc-Alternative_characterizations-sublist" class="vector-toc-list"> <li id="toc-Equality_with_the_adjoint" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Equality_with_the_adjoint"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Equality with the adjoint</span> </div> </a> <ul id="toc-Equality_with_the_adjoint-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real-valuedness_of_quadratic_forms" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real-valuedness_of_quadratic_forms"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Real-valuedness of quadratic forms</span> </div> </a> <ul id="toc-Real-valuedness_of_quadratic_forms-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spectral_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spectral_properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Spectral properties</span> </div> </a> <ul id="toc-Spectral_properties-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples_and_solutions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples_and_solutions"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Examples and solutions</span> </div> </a> <ul id="toc-Examples_and_solutions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Properties" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Properties"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Properties</span> </div> </a> <button aria-controls="toc-Properties-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Properties subsection</span> </button> <ul id="toc-Properties-sublist" class="vector-toc-list"> <li id="toc-Main_diagonal_values_are_real" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Main_diagonal_values_are_real"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Main diagonal values are real</span> </div> </a> <ul id="toc-Main_diagonal_values_are_real-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Symmetric" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Symmetric"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Symmetric</span> </div> </a> <ul id="toc-Symmetric-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normal" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normal"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Normal</span> </div> </a> <ul id="toc-Normal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diagonalizable" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diagonalizable"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Diagonalizable</span> </div> </a> <ul id="toc-Diagonalizable-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sum_of_Hermitian_matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sum_of_Hermitian_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Sum of Hermitian matrices</span> </div> </a> <ul id="toc-Sum_of_Hermitian_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Inverse_is_Hermitian" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Inverse_is_Hermitian"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Inverse is Hermitian</span> </div> </a> <ul id="toc-Inverse_is_Hermitian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Associative_product_of_Hermitian_matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Associative_product_of_Hermitian_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Associative product of Hermitian matrices</span> </div> </a> <ul id="toc-Associative_product_of_Hermitian_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-ABA_Hermitian" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#ABA_Hermitian"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span><i>ABA</i> Hermitian</span> </div> </a> <ul id="toc-ABA_Hermitian-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-vHAv_is_real_for_complex_v" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#vHAv_is_real_for_complex_v"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.9</span> <span><span>v<sup>H</sup><i>A</i>v</span> is real for complex <span>v</span></span> </div> </a> <ul id="toc-vHAv_is_real_for_complex_v-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_Hermitian_forms_vector_space_over_ℝ" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_Hermitian_forms_vector_space_over_ℝ"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.10</span> <span>Complex Hermitian forms vector space over <span>ℝ</span></span> </div> </a> <ul id="toc-Complex_Hermitian_forms_vector_space_over_ℝ-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigendecomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigendecomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.11</span> <span>Eigendecomposition</span> </div> </a> <ul id="toc-Eigendecomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Singular_values" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Singular_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.12</span> <span>Singular values</span> </div> </a> <ul id="toc-Singular_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Real_determinant" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Real_determinant"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.13</span> <span>Real determinant</span> </div> </a> <ul id="toc-Real_determinant-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Decomposition_into_Hermitian_and_skew-Hermitian_matrices" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Decomposition_into_Hermitian_and_skew-Hermitian_matrices"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Decomposition into Hermitian and skew-Hermitian matrices</span> </div> </a> <ul id="toc-Decomposition_into_Hermitian_and_skew-Hermitian_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Rayleigh_quotient" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Rayleigh_quotient"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Rayleigh quotient</span> </div> </a> <ul id="toc-Rayleigh_quotient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Hermitian matrix</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 34 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-34" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">34 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9_%D9%87%D9%8A%D8%B1%D9%85%D9%8A%D8%AA%D9%8A%D8%A9" title="مصفوفة هيرميتية – Arabic" lang="ar" hreflang="ar" data-title="مصفوفة هيرميتية" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B9%E0%A6%BE%E0%A6%B0%E0%A6%AE%E0%A6%BF%E0%A6%B6%E0%A6%BF%E0%A6%AF%E0%A6%BC%E0%A6%BE%E0%A6%A8" title="হারমিশিয়ান – Bangla" lang="bn" hreflang="bn" data-title="হারমিশিয়ান" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%AD%D1%80%D0%BC%D1%96%D1%82%D0%B0%D0%B2%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0" title="Эрмітава матрыца – Belarusian" lang="be" hreflang="be" data-title="Эрмітава матрыца" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matriu_herm%C3%ADtica" title="Matriu hermítica – Catalan" lang="ca" hreflang="ca" data-title="Matriu hermítica" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Hermitovsk%C3%A1_matice" title="Hermitovská matice – Czech" lang="cs" hreflang="cs" data-title="Hermitovská matice" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Hermitisk_matrix" title="Hermitisk matrix – Danish" lang="da" hreflang="da" data-title="Hermitisk matrix" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Hermitesche_Matrix" title="Hermitesche Matrix – German" lang="de" hreflang="de" data-title="Hermitesche Matrix" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Hermiitiline_maatriks" title="Hermiitiline maatriks – Estonian" lang="et" hreflang="et" data-title="Hermiitiline maatriks" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%95%CF%81%CE%BC%CE%B9%CF%84%CE%B9%CE%B1%CE%BD%CF%8C%CF%82_%CF%80%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82" title="Ερμιτιανός πίνακας – Greek" lang="el" hreflang="el" data-title="Ερμιτιανός πίνακας" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matriz_hermitiana" title="Matriz hermitiana – Spanish" lang="es" hreflang="es" data-title="Matriz hermitiana" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Memadjunkta_matrico" title="Memadjunkta matrico – Esperanto" lang="eo" hreflang="eo" data-title="Memadjunkta matrico" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3_%D9%87%D8%B1%D9%85%DB%8C%D8%AA%DB%8C" title="ماتریس هرمیتی – Persian" lang="fa" hreflang="fa" data-title="ماتریس هرمیتی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Hermitien" title="Hermitien – French" lang="fr" hreflang="fr" data-title="Hermitien" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%97%90%EB%A5%B4%EB%AF%B8%ED%8A%B8_%ED%96%89%EB%A0%AC" title="에르미트 행렬 – Korean" lang="ko" hreflang="ko" data-title="에르미트 행렬" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Matriks_Hermite" title="Matriks Hermite – Indonesian" lang="id" hreflang="id" data-title="Matriks Hermite" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matrice_hermitiana" title="Matrice hermitiana – Italian" lang="it" hreflang="it" data-title="Matrice hermitiana" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Ermita_matrica" title="Ermita matrica – Latvian" lang="lv" hreflang="lv" data-title="Ermita matrica" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Ermito_matrica" title="Ermito matrica – Lithuanian" lang="lt" hreflang="lt" data-title="Ermito matrica" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Hermitikus_m%C3%A1trix" title="Hermitikus mátrix – Hungarian" lang="hu" hreflang="hu" data-title="Hermitikus mátrix" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Hermitische_matrix" title="Hermitische matrix – Dutch" lang="nl" hreflang="nl" data-title="Hermitische matrix" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><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" data-title="エルミート行列" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Macierz_hermitowska" title="Macierz hermitowska – Polish" lang="pl" hreflang="pl" data-title="Macierz hermitowska" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Matriz_hermitiana" title="Matriz hermitiana – Portuguese" lang="pt" hreflang="pt" data-title="Matriz hermitiana" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><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" data-title="Эрмитова матрица" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Matrica_hermitiane" title="Matrica hermitiane – Albanian" lang="sq" hreflang="sq" data-title="Matrica hermitiane" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Hermitska_matrika" title="Hermitska matrika – Slovenian" lang="sl" hreflang="sl" data-title="Hermitska matrika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><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" data-title="Ермитска матрица" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Hermiittinen_matriisi" title="Hermiittinen matriisi – Finnish" lang="fi" hreflang="fi" data-title="Hermiittinen matriisi" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Hermitesk_matris" title="Hermitesk matris – Swedish" lang="sv" hreflang="sv" data-title="Hermitesk matris" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><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" data-title="ஹெர்மைட் அணி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><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" data-title="เมทริกซ์แอร์มิต" data-language-autonym="ไทย" data-language-local-name="Thai" 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class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Matrix equal to its conjugate-transpose </div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><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 class="mw-empty-elt"> </p><p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, 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>: <span class="mwe-math-element"><span 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 A{\text{ is 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> is Hermitian</mtext> </mrow> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <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{ is 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/590b4e83dcdd340900d19dc5684a5fdb0591665e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.913ex; height:3.009ex;" alt="{\displaystyle A{\text{ is Hermitian}}\quad \iff \quad a_{ij}={\overline {a_{ji}}}}"></span> </p><p>or in matrix form: <span class="mwe-math-element"><span 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 A{\text{ is 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> is Hermitian</mtext> </mrow> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <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> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\text{ is 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/a18200c9d49e413a07535a0e4cbca45ffcd877ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:34.984ex; height:3.509ex;" alt="{\displaystyle A{\text{ is Hermitian}}\quad \iff \quad A={\overline {A^{\mathsf {T}}}}.}"></span> </p><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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 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> <mo>,</mo> </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/3a3464c251708923c86896824882d00d79f02829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:3.009ex;" alt="{\displaystyle A^{\mathsf {H}},}"></span> then the Hermitian property can be written concisely as </p><p><span class="mwe-math-element"><span 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 A{\text{ is 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> is Hermitian</mtext> </mrow> <mspace width="1em" /> <mspace width="thickmathspace" /> <mo stretchy="false">⟺</mo> <mspace width="thickmathspace" /> <mspace width="1em" /> <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{ is Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f97baecca036548a8a5f22063f0c230d51bc5c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:34.267ex; height:2.676ex;" alt="{\displaystyle A{\text{ is Hermitian}}\quad \iff \quad A=A^{\mathsf {H}}}"></span> </p><p>Hermitian matrices are named after <a href="/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a>,<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> 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> <mo>,</mo> </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/ae07db5129cd41637e87255526bf6f5ef8890a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.485ex; height:3.009ex;" alt="{\displaystyle A^{\mathsf {H}}=A^{\dagger }=A^{\ast },}"></span> although 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="{\displaystyle 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> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Alternative_characterizations">Alternative characterizations</h2><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"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hermitian matrices can be characterized in a number of equivalent ways, some of which are listed below: </p> <div class="mw-heading mw-heading3"><h3 id="Equality_with_the_adjoint">Equality with the adjoint</h3><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"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is Hermitian if and only if it is equal to its <a href="/wiki/Hermitian_adjoint" title="Hermitian adjoint">conjugate transpose</a>, that is, it satisfies <span class="mwe-math-element"><span 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 \mathbf {w} ,A\mathbf {v} \rangle =\langle A\mathbf {w} ,\mathbf {v} \rangle ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>,</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo fence="false" stretchy="false">⟩</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {w} ,A\mathbf {v} \rangle =\langle A\mathbf {w} ,\mathbf {v} \rangle ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/223750da016fbe2c9d41611baf9cff897c832388" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.602ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {w} ,A\mathbf {v} \rangle =\langle A\mathbf {w} ,\mathbf {v} \rangle ,}"></span> 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 \mathbf {v} ,\mathbf {w} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ,\mathbf {w} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f8a1ac5fc88dc8c29b405adbd820180c98745a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.023ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} ,\mathbf {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 mw-invert skin-invert" 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><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> <div class="mw-heading mw-heading3"><h3 id="Real-valuedness_of_quadratic_forms">Real-valuedness of quadratic forms</h3><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: Real-valuedness of quadratic forms"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>An <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 n\times {}n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times {}n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd15b4dbbfb9c4acbaf4dee0f4e28aaa7c61d9aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times {}n}"></span> 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is Hermitian if and only if <span class="mwe-math-element"><span 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 \mathbf {v} ,A\mathbf {v} \rangle \in \mathbb {R} ,\quad {\text{for all }}\mathbf {v} \in \mathbb {C} ^{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <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" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all </mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle \mathbf {v} ,A\mathbf {v} \rangle \in \mathbb {R} ,\quad {\text{for all }}\mathbf {v} \in \mathbb {C} ^{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ac8de63f1dc3c3015258da83d648479af08c0e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.482ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {v} ,A\mathbf {v} \rangle \in \mathbb {R} ,\quad {\text{for all }}\mathbf {v} \in \mathbb {C} ^{n}.}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Spectral_properties">Spectral properties</h3><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"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 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> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><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"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Hermitian matrices are fundamental to <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> because they describe operators with necessarily real eigenvalues. An eigenvalue <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/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> of an operator <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 {\hat {A}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>A</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {A}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f595a6c73d1183d6a1b2ac21fe47ac28c1483821" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.843ex;" alt="{\displaystyle {\hat {A}}}"></span> on some quantum state <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 |\psi \rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>ψ</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\psi \rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc27f1893b769a08cd6b296e115a29e61cab675e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.065ex; height:2.843ex;" alt="{\displaystyle |\psi \rangle }"></span> is one of the possible measurement outcomes of the operator, which requires the operators to have real eigenvalues. </p><p>In <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, Hermitian matrices are utilized in tasks like <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a> and signal representation.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> The eigenvalues and eigenvectors of Hermitian matrices play a crucial role in analyzing signals and extracting meaningful information. </p><p>Hermitian matrices are extensively studied in <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> and <a href="/wiki/Numerical_analysis" title="Numerical analysis">numerical analysis</a>. They have well-defined spectral properties, and many numerical algorithms, such as the <a href="/wiki/Lanczos_algorithm" title="Lanczos algorithm">Lanczos algorithm</a>, exploit these properties for efficient computations. Hermitian matrices also appear in techniques like <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> (SVD) and <a href="/wiki/Eigenvalue_decomposition" class="mw-redirect" title="Eigenvalue decomposition">eigenvalue decomposition</a>. </p><p>In <a href="/wiki/Statistics" title="Statistics">statistics</a> and <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, Hermitian matrices are used in <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrices</a>, where they represent the relationships between different variables. The positive definiteness of a Hermitian covariance matrix ensures the well-definedness of multivariate distributions.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>Hermitian matrices are applied in the design and analysis of <a href="/wiki/Communications_system" title="Communications system">communications system</a>, especially in the field of <a href="/wiki/Multiple-input_multiple-output" class="mw-redirect" title="Multiple-input multiple-output">multiple-input multiple-output</a> (MIMO) systems. Channel matrices in MIMO systems often exhibit Hermitian properties. </p><p>In <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>, Hermitian matrices are used to study the <a href="/wiki/Spectral_graph_theory" title="Spectral graph theory">spectra of graphs</a>. The Hermitian Laplacian matrix is a key tool in this context, as it is used to analyze the spectra of mixed graphs.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> The Hermitian-adjacency matrix of a mixed graph is another important concept, as it is a Hermitian matrix that plays a role in studying the energies of mixed graphs.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Examples_and_solutions">Examples and solutions</h2><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 and solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 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> <mo>,</mo> </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/3a3464c251708923c86896824882d00d79f02829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.786ex; height:3.009ex;" 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 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 mw-invert skin-invert" 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 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> <mo>.</mo> </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/a0ea102747993704ea8507ea27d4691a485727b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.505ex; height:3.009ex;" alt="{\displaystyle {\overline {A}}.}"></span> </p><p>See the following example: </p><p><span class="mwe-math-element"><span 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 {\begin{bmatrix}0&a-ib&c-id\\a+ib&1&m-in\\c+id&m+in&2\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>0</mn> </mtd> <mtd> <mi>a</mi> <mo>−</mo> <mi>i</mi> <mi>b</mi> </mtd> <mtd> <mi>c</mi> <mo>−</mo> <mi>i</mi> <mi>d</mi> </mtd> </mtr> <mtr> <mtd> <mi>a</mi> <mo>+</mo> <mi>i</mi> <mi>b</mi> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mi>m</mi> <mo>−</mo> <mi>i</mi> <mi>n</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo>+</mo> <mi>i</mi> <mi>d</mi> </mtd> <mtd> <mi>m</mi> <mo>+</mo> <mi>i</mi> <mi>n</mi> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&a-ib&c-id\\a+ib&1&m-in\\c+id&m+in&2\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8d4ddd393d8f3538c2a2fe61898ccea12970ab5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:28.523ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}0&a-ib&c-id\\a+ib&1&m-in\\c+id&m+in&2\end{bmatrix}}}"></span> </p><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 Hermitian matrices include the <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a>, the <a href="/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann matrices</a> and their generalizations. 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-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> which results in <a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">skew-Hermitian matrices</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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> equals the <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">product of a matrix</a> with 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> <mo>,</mo> </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/3fd0bc2032d2c77cb13875a52274c51265e75144" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.412ex; height:3.009ex;" 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> is positive definite. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><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"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Main_diagonal_values_are_real">Main diagonal values are real</h3><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: Main diagonal values are real"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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>. </p> <style data-mw-deduplicate="TemplateStyles:r1174254338">.mw-parser-output .math_proof{border:thin solid #aaa;margin:1em 2em;padding:0.5em 1em 0.4em}@media(max-width:500px){.mw-parser-output .math_proof{margin:1em 0;padding:0.5em 0.5em 0.4em}}</style><div class="math_proof" style=""><strong>Proof</strong> <p>By definition of the Hermitian matrix <span class="mwe-math-element"><span 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 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-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.249ex; height:3.676ex;" alt="{\displaystyle H_{ij}={\overline {H}}_{ji}}"></span> so for <span class="texhtml"><i>i</i> = <i>j</i></span> the above follows. </p> </div> <p>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. </p> <div class="mw-heading mw-heading3"><h3 id="Symmetric">Symmetric</h3><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: Symmetric"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A matrix that has only real entries is <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> it is a Hermitian matrix. A real and symmetric matrix is simply a special case of a Hermitian matrix. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <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 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 mw-invert skin-invert" 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="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}=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> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{ij}=H_{ji}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95dafb8df16452080ff28e30e04015dba7f8c083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.915ex; height:2.843ex;" alt="{\displaystyle H_{ij}=H_{ji}}"></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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.249ex; height:3.676ex;" alt="{\displaystyle H_{ij}={\overline {H}}_{ij}}"></span> (<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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb13feb15799a5df604f075b03c85cf59572d2bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.408ex; height:2.843ex;" alt="{\displaystyle H_{ij}}"></span> is real). </p> </div> <p>So, if a real anti-symmetric matrix is multiplied by a real multiple of the imaginary unit <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> <mo>,</mo> </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/d3d0f7dadba3056fa3c06a6bee5c0b4182471152" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.449ex; height:2.509ex;" alt="{\displaystyle i,}"></span> then it becomes Hermitian. </p> <div class="mw-heading mw-heading3"><h3 id="Normal">Normal</h3><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: Normal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Every Hermitian matrix is a <a href="/wiki/Normal_matrix" title="Normal matrix">normal matrix</a>. That is to say, <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 AA^{\mathsf {H}}=A^{\mathsf {H}}A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AA^{\mathsf {H}}=A^{\mathsf {H}}A.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89d6550ee93dbce053148fa9f31a82f42da8cd1f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.509ex; height:2.676ex;" alt="{\displaystyle AA^{\mathsf {H}}=A^{\mathsf {H}}A.}"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <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=A^{\mathsf {H}},}"> <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> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=A^{\mathsf {H}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c382392b7cfc00ed9b6236ae3ce40fb0e038e58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.627ex; height:3.009ex;" alt="{\displaystyle A=A^{\mathsf {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 AA^{\mathsf {H}}=AA=A^{\mathsf {H}}A.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <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>A</mi> <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> <mi>A</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AA^{\mathsf {H}}=AA=A^{\mathsf {H}}A.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab57bd9ed60c43df68821a97509b6a5aab1c929b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.094ex; height:2.676ex;" alt="{\displaystyle AA^{\mathsf {H}}=AA=A^{\mathsf {H}}A.}"></span> </p> </div> <div class="mw-heading mw-heading3"><h3 id="Diagonalizable">Diagonalizable</h3><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: Diagonalizable"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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"><b>C</b><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>. </p> <div class="mw-heading mw-heading3"><h3 id="Sum_of_Hermitian_matrices">Sum of Hermitian matrices</h3><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: Sum of Hermitian matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sum of any two Hermitian matrices is Hermitian. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p><span class="mwe-math-element"><span 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 (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-display mw-invert skin-invert" 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. </p> </div> <div class="mw-heading mw-heading3"><h3 id="Inverse_is_Hermitian">Inverse is Hermitian</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=13" title="Edit section: Inverse is Hermitian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Inverse_matrix" class="mw-redirect" title="Inverse matrix">inverse</a> of an invertible Hermitian matrix is Hermitian as well. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p>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> <mo>,</mo> </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/403a9911d140c65e720975b6f17a6c08123df7e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.736ex; height:3.009ex;" 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^{\mathsf {H}}=\left(A^{-1}A\right)^{\mathsf {H}}=A^{\mathsf {H}}\left(A^{-1}\right)^{\mathsf {H}}=A\left(A^{-1}\right)^{\mathsf {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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> </mrow> <mo>)</mo> </mrow> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <mo>=</mo> <mi>A</mi> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=I^{\mathsf {H}}=\left(A^{-1}A\right)^{\mathsf {H}}=A^{\mathsf {H}}\left(A^{-1}\right)^{\mathsf {H}}=A\left(A^{-1}\right)^{\mathsf {H}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b173ca91308b9df2fd36f836e94d436403e0f53e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:46.255ex; height:3.843ex;" alt="{\displaystyle I=I^{\mathsf {H}}=\left(A^{-1}A\right)^{\mathsf {H}}=A^{\mathsf {H}}\left(A^{-1}\right)^{\mathsf {H}}=A\left(A^{-1}\right)^{\mathsf {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}=\left(A^{-1}\right)^{\mathsf {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> <msup> <mrow> <mo>(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <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 A^{-1}=\left(A^{-1}\right)^{\mathsf {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f30bf59feb41584787fb73659f12c24b5331fe7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.776ex; height:3.843ex;" alt="{\displaystyle A^{-1}=\left(A^{-1}\right)^{\mathsf {H}}}"></span> as claimed. </p> </div> <div class="mw-heading mw-heading3"><h3 id="Associative_product_of_Hermitian_matrices">Associative product of Hermitian matrices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=14" title="Edit section: Associative product of Hermitian matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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>. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p><span class="mwe-math-element"><span 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 (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> <mtext> </mtext> <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/7116fe39ccaf471350ee20246c5b57b2233698d3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:52.786ex; 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 mw-invert skin-invert" 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> <mo>.</mo> </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/c11e0bfc4b5c228062a3f3e8f29c759d8f2a77e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.76ex; height:2.176ex;" alt="{\displaystyle AB=BA.}"></span> </p><p>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. </p> </div> <div class="mw-heading mw-heading3"><h3 id="ABA_Hermitian"><i>ABA</i> Hermitian</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=15" title="Edit section: ABA Hermitian"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>A</i> and <i>B</i> are Hermitian, then <i>ABA</i> is also Hermitian. </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <p><span class="mwe-math-element"><span 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 (ABA)^{\mathsf {H}}=(A(BA))^{\mathsf {H}}=(BA)^{\mathsf {H}}A^{\mathsf {H}}=A^{\mathsf {H}}B^{\mathsf {H}}A^{\mathsf {H}}=ABA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mi>A</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> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mi>A</mi> <mo stretchy="false">)</mo> <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> <mo stretchy="false">(</mo> <mi>B</mi> <mi>A</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> <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"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <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>A</mi> <mi>B</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (ABA)^{\mathsf {H}}=(A(BA))^{\mathsf {H}}=(BA)^{\mathsf {H}}A^{\mathsf {H}}=A^{\mathsf {H}}B^{\mathsf {H}}A^{\mathsf {H}}=ABA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d564916ae75ce912335c438d9448b2a7762ab4f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.653ex; height:3.176ex;" alt="{\displaystyle (ABA)^{\mathsf {H}}=(A(BA))^{\mathsf {H}}=(BA)^{\mathsf {H}}A^{\mathsf {H}}=A^{\mathsf {H}}B^{\mathsf {H}}A^{\mathsf {H}}=ABA}"></span> </p> </div> <div class="mw-heading mw-heading3"><h3 id="vHAv_is_real_for_complex_v"><span class="texhtml">v<sup>H</sup><i>A</i>v</span> is real for complex <span class="texhtml">v</span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=16" title="Edit section: vHAv is real for complex v"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For an arbitrary complex valued vector <span class="texhtml"><b>v</b></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 \mathbf {v} ^{\mathsf {H}}A\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ^{\mathsf {H}}A\mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9401ca469fa4149e80c5ed5ac5dea322daa725d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.961ex; height:2.676ex;" alt="{\displaystyle \mathbf {v} ^{\mathsf {H}}A\mathbf {v} }"></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 \mathbf {v} ^{\mathsf {H}}A\mathbf {v} =\left(\mathbf {v} ^{\mathsf {H}}A\mathbf {v} \right)^{\mathsf {H}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <msup> <mrow> <mo>(</mo> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ^{\mathsf {H}}A\mathbf {v} =\left(\mathbf {v} ^{\mathsf {H}}A\mathbf {v} \right)^{\mathsf {H}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f6aeefdda5f4128f2473fa45bde1b44cf6419ed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.193ex; height:3.843ex;" alt="{\displaystyle \mathbf {v} ^{\mathsf {H}}A\mathbf {v} =\left(\mathbf {v} ^{\mathsf {H}}A\mathbf {v} \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. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_Hermitian_forms_vector_space_over_ℝ"><span id="Complex_Hermitian_forms_vector_space_over_.E2.84.9D"></span>Complex Hermitian forms vector space over <span class="texhtml">ℝ</span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=17" title="Edit section: Complex Hermitian forms vector space over ℝ"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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"><b>ℂ</b></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"><b>ℝ</b></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"><b>ℝ</b></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 with respect to the Frobenius inner product) can be described as follows: <span class="mwe-math-element"><span 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 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" /> <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-display mw-invert skin-invert" 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> </p><p>together with the set of matrices of the form <span class="mwe-math-element"><span 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 {\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" /> <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-display mw-invert skin-invert" 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> </p><p>and the matrices <span class="mwe-math-element"><span 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 {\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" /> <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-display mw-invert skin-invert" 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> </p><p>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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span> denotes the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</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 i={\sqrt {-1}}~.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo>−</mo> <mn>1</mn> </msqrt> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i={\sqrt {-1}}~.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f056f41fd9edb481fee2a41b38138d2e0d0d12a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.035ex; height:3.009ex;" alt="{\displaystyle i={\sqrt {-1}}~.}"></span> </p><p>An example is that the four <a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli matrices</a> form a complete basis for the vector space of all complex 2-by-2 Hermitian matrices over <span class="texhtml"><b>ℝ</b></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Eigendecomposition">Eigendecomposition</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=18" title="Edit section: Eigendecomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>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 \mathbf {u} _{1},\dots ,\mathbf {u} _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25058c5d748ce5cef7fa04c15166242860d6a3fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.422ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1},\dots ,\mathbf {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</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 mw-invert skin-invert" 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 mw-invert skin-invert" 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 <span class="mwe-math-element"><span 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 A=\sum _{j}\lambda _{j}\mathbf {u} _{j}\mathbf {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</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <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}\mathbf {u} _{j}\mathbf {u} _{j}^{\mathsf {H}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32cea30a71d185fb6d5a2ec4f9171416aafa764d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.772ex; height:5.843ex;" alt="{\displaystyle A=\sum _{j}\lambda _{j}\mathbf {u} _{j}\mathbf {u} _{j}^{\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 \lambda _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.265ex; height:2.843ex;" alt="{\displaystyle \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"> <mi mathvariant="normal">Λ</mi> <mo>.</mo> </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/69f31ae3ac22be0a40c4c1321e4792374285282e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.26ex; height:2.176ex;" alt="{\displaystyle \Lambda .}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Singular_values">Singular values</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=19" title="Edit section: Singular values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The singular values 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 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> are the absolute values of its eigenvalues: </p><p>Since <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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span> has an eigendecomposition <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^{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>U</mi> <mi mathvariant="normal">Λ</mi> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=U\Lambda U^{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/560e477f6900b308cedbc4c9a96da62a3406e658" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.77ex; height:2.676ex;" alt="{\displaystyle A=U\Lambda U^{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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> is a <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a> (its columns are orthonormal vectors; <a class="mw-selflink-fragment" href="#Eigendecomposition">see above</a>), a <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> 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 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle 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 |{\text{sgn}}(\Lambda )U^{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mtext>sgn</mtext> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=U|\Lambda |{\text{sgn}}(\Lambda )U^{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/709dcd092b29e1a105436f5afed1005498072b00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.857ex; height:3.176ex;" alt="{\displaystyle A=U|\Lambda |{\text{sgn}}(\Lambda )U^{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 |\Lambda |}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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/ae7b580a54ad8d36cecac6ebbbe5537965846f76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.907ex; height:2.843ex;" alt="{\displaystyle |\Lambda |}"></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 {\text{sgn}}(\Lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>sgn</mtext> </mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{sgn}}(\Lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42fdf652f128e997b2987578434ba0e3e8e301eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.794ex; height:2.843ex;" alt="{\displaystyle {\text{sgn}}(\Lambda )}"></span> are diagonal matrices containing the absolute values <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"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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/e92d2419f2d2ad612edb802a7727b4f4eb07f3c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.649ex; height:2.843ex;" alt="{\displaystyle |\lambda |}"></span> and signs <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 {\text{sgn}}(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtext>sgn</mtext> </mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\text{sgn}}(\lambda )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45070018c413a42103e19eb9680bcdc27d1ca1b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.536ex; height:2.843ex;" alt="{\displaystyle {\text{sgn}}(\lambda )}"></span> 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 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>'s eigenvalues, respectively. <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 \operatorname {sgn}(\Lambda )U^{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo stretchy="false">)</mo> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn}(\Lambda )U^{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08d1838e7fbfc13d7f5e78bc6fa39b96736755c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.326ex; height:3.176ex;" alt="{\displaystyle \operatorname {sgn}(\Lambda )U^{H}}"></span> is unitary, since the columns 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 U^{H}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U^{H}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b89cff86c71c72179189290acb7dc61b6d55f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.533ex; height:2.676ex;" alt="{\displaystyle U^{H}}"></span> are only getting multiplied 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 \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bfeaa85da53ad1947d8000926cfea33827ef1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.971ex; height:2.176ex;" alt="{\displaystyle \pm 1}"></span>. <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"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi mathvariant="normal">Λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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/ae7b580a54ad8d36cecac6ebbbe5537965846f76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.907ex; height:2.843ex;" alt="{\displaystyle |\Lambda |}"></span> contains the singular values 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 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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"></span>, namely, the absolute values of its eigenvalues.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Real_determinant">Real determinant</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=20" title="Edit section: Real determinant"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The determinant of a Hermitian matrix is real: </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1174254338"><div class="math_proof" style=""><strong>Proof</strong> <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 \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" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <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 mw-invert skin-invert" 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> 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" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <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> <mo>.</mo> </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/c50b9e388ab7113e1fd94f3a08ca9e943dcd0769" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.664ex; height:3.676ex;" alt="{\displaystyle A=A^{\mathsf {H}}\quad \Rightarrow \quad \det(A)={\overline {\det(A)}}.}"></span> </p> </div> <p>(Alternatively, the determinant is the product of the matrix's eigenvalues, and as mentioned before, the eigenvalues of a Hermitian matrix are real.) </p> <div class="mw-heading mw-heading2"><h2 id="Decomposition_into_Hermitian_and_skew-Hermitian_matrices">Decomposition into Hermitian and skew-Hermitian matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=21" title="Edit section: Decomposition into Hermitian and skew-Hermitian matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><span class="anchor" 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 mw-invert skin-invert" 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> <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 mw-invert skin-invert" 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> <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_9-0" class="reference"><a href="#cite_note-HornJohnson-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 227">: 227 </span></sup> <span class="mwe-math-element"><span 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 C=A+B\quad {\text{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\text{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" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>with</mtext> </mrow> <mspace width="1em" /> <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" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <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 {\text{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\text{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/dd0481a30075ad697ffa9a162a0668031758b232" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:63.495ex; height:5.176ex;" alt="{\displaystyle C=A+B\quad {\text{with}}\quad A={\frac {1}{2}}\left(C+C^{\mathsf {H}}\right)\quad {\text{and}}\quad B={\frac {1}{2}}\left(C-C^{\mathsf {H}}\right)}"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Rayleigh_quotient">Rayleigh quotient</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=22" title="Edit section: Rayleigh quotient"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><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 <span class="texhtml mvar" style="font-style:italic;">M</span> and nonzero vector <span class="texhtml"><b>x</b></span>, the Rayleigh quotient<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></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,\mathbf {x} ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(M,\mathbf {x} ),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32589bfbc470c528eb425a15346eaae2bdb045ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.107ex; height:2.843ex;" alt="{\displaystyle R(M,\mathbf {x} ),}"></span> is defined as:<sup id="cite_ref-HornJohnson_9-1" class="reference"><a href="#cite_note-HornJohnson-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: p. 234">: p. 234 </span></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span 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 R(M,\mathbf {x} ):={\frac {\mathbf {x} ^{\mathsf {H}}M\mathbf {x} }{\mathbf {x} ^{\mathsf {H}}\mathbf {x} }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>:=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">H</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(M,\mathbf {x} ):={\frac {\mathbf {x} ^{\mathsf {H}}M\mathbf {x} }{\mathbf {x} ^{\mathsf {H}}\mathbf {x} }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d06c7072650bea6e5dc3898e665553530d023356" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.349ex; height:6.009ex;" alt="{\displaystyle R(M,\mathbf {x} ):={\frac {\mathbf {x} ^{\mathsf {H}}M\mathbf {x} }{\mathbf {x} ^{\mathsf {H}}\mathbf {x} }}.}"></span> </p><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 \mathbf {x} ^{\mathsf {H}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </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 \mathbf {x} ^{\mathsf {H}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8da87ccc0bfe788346b8e8019a87244a5a3f5b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.807ex; height:2.676ex;" alt="{\displaystyle \mathbf {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 \mathbf {x} ^{\mathsf {T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif">T</mi> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ^{\mathsf {T}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da88ae32893609adb801ee3b8adc0f489c0db5e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.409ex; height:2.676ex;" alt="{\displaystyle \mathbf {x} ^{\mathsf {T}}.}"></span> <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,c\mathbf {x} )=R(M,\mathbf {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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>R</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(M,c\mathbf {x} )=R(M,\mathbf {x} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcd24cc10cb7f5dce90ecd114c702e71fe14fce8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.026ex; height:2.843ex;" alt="{\displaystyle R(M,c\mathbf {x} )=R(M,\mathbf {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> <mo>.</mo> </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/13b8d90daa52ffa8e5988459b6f10ef4d64ee5da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.654ex; height:1.676ex;" alt="{\displaystyle c.}"></span> Also, recall that a Hermitian (or real symmetric) matrix has real eigenvalues. </p><p>It can be shown<sup id="cite_ref-HornJohnson_9-2" class="reference"><a href="#cite_note-HornJohnson-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></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</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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.328ex; height:2.509ex;" alt="{\displaystyle \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 \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {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 \mathbf {v} _{\min }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{\min }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90d687b1c085a568b015944a1ad94e196b9c32cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.383ex; height:2.009ex;" alt="{\displaystyle \mathbf {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,\mathbf {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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>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,\mathbf {x} )\leq \lambda _{\max }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a83bee4928bf9e87c24d993937c944ff51c2e90" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.205ex; height:2.843ex;" alt="{\displaystyle R(M,\mathbf {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,\mathbf {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> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R(M,\mathbf {v} _{\max })=\lambda _{\max }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afa69f02fcb93f999b1cd2b36eb6917f19cfedfb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.143ex; height:2.843ex;" alt="{\displaystyle R(M,\mathbf {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</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 mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.646ex; height:2.509ex;" alt="{\displaystyle \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"><b>x</b></span> and <span class="texhtml"><i>M</i></span> varying through the algebra would be referred to as "vector state" of the algebra. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=23" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Complex_symmetric_matrix" class="mw-redirect" title="Complex symmetric matrix">Complex symmetric matrix</a> – Matrix equal to its transpose<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Haynsworth_inertia_additivity_formula" title="Haynsworth inertia additivity formula">Haynsworth inertia additivity formula</a> – Counts positive, negative, and zero eigenvalues of a block partitioned Hermitian matrix</li> <li><a href="/wiki/Hermitian_form" class="mw-redirect" title="Hermitian form">Hermitian form</a> – Generalization of a bilinear form<span style="display:none" class="category-annotation-with-redirected-description">Pages displaying short descriptions of redirect targets</span></li> <li><a href="/wiki/Normal_matrix" title="Normal matrix">Normal matrix</a> – Matrix that commutes with its conjugate transpose</li> <li><a href="/wiki/Schur%E2%80%93Horn_theorem" title="Schur–Horn theorem">Schur–Horn theorem</a> – Characterizes the diagonal of a Hermitian matrix with given eigenvalues</li> <li><a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">Self-adjoint operator</a> – Linear operator equal to its own adjoint</li> <li><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian matrix</a> – Matrix whose conjugate transpose is its negative (additive inverse) (anti-Hermitian matrix)</li> <li><a href="/wiki/Unitary_matrix" title="Unitary matrix">Unitary matrix</a> – Complex matrix whose conjugate transpose equals its inverse</li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a> – Algebraic structure in linear algebra</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=24" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><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"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFArchibald2010" class="citation cs2">Archibald, Tom (2010-12-31), Gowers, Timothy; Barrow-Green, June; Leader, Imre (eds.), <a rel="nofollow" class="external text" href="https://www.degruyter.com/document/doi/10.1515/9781400830398.773a/html">"VI.47 Charles Hermite"</a>, <i>The Princeton Companion to Mathematics</i>, Princeton University Press, p. 773, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9781400830398.773a">10.1515/9781400830398.773a</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4008-3039-8" title="Special:BookSources/978-1-4008-3039-8"><bdi>978-1-4008-3039-8</bdi></a><span class="reference-accessdate">, retrieved <span class="nowrap">2023-11-15</span></span></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Princeton+Companion+to+Mathematics&rft.atitle=VI.47+Charles+Hermite&rft.pages=773&rft.date=2010-12-31&rft_id=info%3Adoi%2F10.1515%2F9781400830398.773a&rft.isbn=978-1-4008-3039-8&rft.aulast=Archibald&rft.aufirst=Tom&rft_id=https%3A%2F%2Fwww.degruyter.com%2Fdocument%2Fdoi%2F10.1515%2F9781400830398.773a%2Fhtml&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHermitian+matrix" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRibeiro" class="citation web cs1">Ribeiro, Alejandro. <a rel="nofollow" class="external text" href="https://www.seas.upenn.edu/~ese2240/wiki/Lecture%20Notes/sip_PCA.pdf">"Signal and Information Processing"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Signal+and+Information+Processing&rft.aulast=Ribeiro&rft.aufirst=Alejandro&rft_id=https%3A%2F%2Fwww.seas.upenn.edu%2F~ese2240%2Fwiki%2FLecture%2520Notes%2Fsip_PCA.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHermitian+matrix" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://dspace.mit.edu/bitstream/handle/1721.1/121170/6-436j-fall-2008/contents/lecture-notes/MIT6_436JF08_lec15.pdf">"MULTIVARIATE NORMAL DISTRIBUTIONS"</a> <span 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title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Hermitian+Spectral+Theory+of+Mixed+Graphs&rft.aulast=Lau&rft.aufirst=Ivan&rft_id=https%3A%2F%2Fwww.sfu.ca%2F~iplau%2FEdinburgh_CS_Project.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHermitian+matrix" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLiuLi2015" class="citation journal cs1">Liu, Jianxi; Li, Xueliang (February 2015). <a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.laa.2014.10.028">"Hermitian-adjacency matrices and Hermitian energies of mixed graphs"</a>. <i>Linear Algebra and Its Applications</i>. <b>466</b>: 182–207. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.laa.2014.10.028">10.1016/j.laa.2014.10.028</a></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Linear+Algebra+and+Its+Applications&rft.atitle=Hermitian-adjacency+matrices+and+Hermitian+energies+of+mixed+graphs&rft.volume=466&rft.pages=182-207&rft.date=2015-02&rft_id=info%3Adoi%2F10.1016%2Fj.laa.2014.10.028&rft.aulast=Liu&rft.aufirst=Jianxi&rft.au=Li%2C+Xueliang&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.laa.2014.10.028&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHermitian+matrix" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFrankel2004" class="citation book cs1"><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&q=%22Lie%20algebra%22%20physics%20%22skew-Hermitian%22&pg=PA652"><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/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">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 title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Geometry+of+Physics%3A+an+introduction&rft.pages=652&rft.pub=Cambridge+University+Press&rft.date=2004&rft.isbn=0-521-53927-7&rft.aulast=Frankel&rft.aufirst=Theodore&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DDUnjs6nEn8wC%26q%3D%2522Lie%2520algebra%2522%2520physics%2520%2522skew-Hermitian%2522%26pg%3DPA652&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHermitian+matrix" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</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> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20220307172254/http://www.hep.caltech.edu/~fcp/physics/quantumMechanics/angularMomentum/angularMomentum.pdf">Archived</a> 2022-03-07 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</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-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrefethanBau,_III1997" class="citation book cs1">Trefethan, Lloyd N.; Bau, III, David (1997). <a rel="nofollow" class="external text" href="http://worldcat.org/oclc/1348374386"><i>Numerical linear algebra</i></a>. Philadelphia, PA, USA: <a href="/wiki/SIAM" class="mw-redirect" title="SIAM">SIAM</a>. p. 34. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-361-7" title="Special:BookSources/0-89871-361-7"><bdi>0-89871-361-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/1348374386">1348374386</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+linear+algebra&rft.place=Philadelphia%2C+PA%2C+USA&rft.pages=34&rft.pub=SIAM&rft.date=1997&rft_id=info%3Aoclcnum%2F1348374386&rft.isbn=0-89871-361-7&rft.aulast=Trefethan&rft.aufirst=Lloyd+N.&rft.au=Bau%2C+III%2C+David&rft_id=http%3A%2F%2Fworldcat.org%2Foclc%2F1348374386&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHermitian+matrix" class="Z3988"></span></span> </li> <li id="cite_note-HornJohnson-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-HornJohnson_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-HornJohnson_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-HornJohnson_9-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHornJohnson2013" class="citation book cs1">Horn, Roger A.; Johnson, Charles R. (2013). <i>Matrix Analysis, second edition</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780521839402" title="Special:BookSources/9780521839402"><bdi>9780521839402</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis%2C+second+edition&rft.pub=Cambridge+University+Press&rft.date=2013&rft.isbn=9780521839402&rft.aulast=Horn&rft.aufirst=Roger+A.&rft.au=Johnson%2C+Charles+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHermitian+matrix" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</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-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</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> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Hermitian_matrix&action=edit&section=25" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation cs2"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Hermitian_matrix">"Hermitian matrix"</a>, <i><a href="/wiki/Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>, <a href="/wiki/European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Hermitian+matrix&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DHermitian_matrix&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHermitian+matrix" class="Z3988"></span></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> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20170829203442/https://www.cyut.edu.tw/~ckhung/b/la/hermitian.en.php">Archived</a> 2017-08-29 at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>, by Chao-Kuei Hung from Chaoyang University, gives a more geometric explanation.</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.mathpages.com/home/kmath306/kmath306.htm">"Hermitian Matrices"</a>. <i>MathPages.com</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=MathPages.com&rft.atitle=Hermitian+Matrices&rft_id=http%3A%2F%2Fwww.mathpages.com%2Fhome%2Fkmath306%2Fkmath306.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AHermitian+matrix" class="Z3988"></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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.mw-parser-output .navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Matrix_classes" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar 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href="/wiki/Band_matrix" title="Band matrix">Band</a></li> <li><a href="/wiki/Bidiagonal_matrix" title="Bidiagonal matrix">Bidiagonal</a></li> <li><a href="/wiki/Bisymmetric_matrix" title="Bisymmetric matrix">Bisymmetric</a></li> <li><a href="/wiki/Block-diagonal_matrix" class="mw-redirect" title="Block-diagonal matrix">Block-diagonal</a></li> <li><a href="/wiki/Block_matrix" title="Block matrix">Block</a></li> <li><a href="/wiki/Block_tridiagonal_matrix" class="mw-redirect" title="Block tridiagonal matrix">Block tridiagonal</a></li> <li><a href="/wiki/Boolean_matrix" title="Boolean matrix">Boolean</a></li> <li><a href="/wiki/Cauchy_matrix" title="Cauchy matrix">Cauchy</a></li> <li><a href="/wiki/Centrosymmetric_matrix" title="Centrosymmetric matrix">Centrosymmetric</a></li> <li><a href="/wiki/Conference_matrix" title="Conference matrix">Conference</a></li> <li><a href="/wiki/Complex_Hadamard_matrix" title="Complex Hadamard matrix">Complex Hadamard</a></li> <li><a href="/wiki/Copositive_matrix" title="Copositive matrix">Copositive</a></li> <li><a href="/wiki/Diagonally_dominant_matrix" title="Diagonally dominant matrix">Diagonally dominant</a></li> <li><a href="/wiki/Diagonal_matrix" title="Diagonal matrix">Diagonal</a></li> <li><a href="/wiki/DFT_matrix" title="DFT matrix">Discrete Fourier Transform</a></li> <li><a href="/wiki/Elementary_matrix" title="Elementary matrix">Elementary</a></li> <li><a href="/wiki/Equivalent_matrix" class="mw-redirect" title="Equivalent matrix">Equivalent</a></li> <li><a href="/wiki/Frobenius_matrix" title="Frobenius matrix">Frobenius</a></li> <li><a href="/wiki/Generalized_permutation_matrix" title="Generalized permutation matrix">Generalized permutation</a></li> <li><a href="/wiki/Hadamard_matrix" title="Hadamard matrix">Hadamard</a></li> <li><a href="/wiki/Hankel_matrix" title="Hankel matrix">Hankel</a></li> <li><a class="mw-selflink selflink">Hermitian</a></li> <li><a href="/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg</a></li> <li><a href="/wiki/Hollow_matrix" title="Hollow matrix">Hollow</a></li> <li><a href="/wiki/Integer_matrix" title="Integer matrix">Integer</a></li> <li><a href="/wiki/Logical_matrix" title="Logical matrix">Logical</a></li> <li><a href="/wiki/Matrix_unit" title="Matrix unit">Matrix unit</a></li> <li><a href="/wiki/Metzler_matrix" title="Metzler matrix">Metzler</a></li> <li><a href="/wiki/Moore_matrix" title="Moore matrix">Moore</a></li> <li><a href="/wiki/Nonnegative_matrix" title="Nonnegative matrix">Nonnegative</a></li> <li><a href="/wiki/Pentadiagonal_matrix" class="mw-redirect" title="Pentadiagonal matrix">Pentadiagonal</a></li> <li><a href="/wiki/Permutation_matrix" title="Permutation matrix">Permutation</a></li> <li><a href="/wiki/Persymmetric_matrix" title="Persymmetric matrix">Persymmetric</a></li> <li><a href="/wiki/Polynomial_matrix" title="Polynomial matrix">Polynomial</a></li> <li><a href="/wiki/Quaternionic_matrix" title="Quaternionic matrix">Quaternionic</a></li> <li><a href="/wiki/Signature_matrix" title="Signature matrix">Signature</a></li> <li><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian</a></li> <li><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Skew-symmetric</a></li> <li><a href="/wiki/Skyline_matrix" title="Skyline matrix">Skyline</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse</a></li> <li><a href="/wiki/Sylvester_matrix" title="Sylvester matrix">Sylvester</a></li> <li><a href="/wiki/Symmetric_matrix" title="Symmetric matrix">Symmetric</a></li> <li><a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz</a></li> <li><a href="/wiki/Triangular_matrix" title="Triangular matrix">Triangular</a></li> <li><a href="/wiki/Tridiagonal_matrix" title="Tridiagonal matrix">Tridiagonal</a></li> <li><a href="/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde</a></li> <li><a href="/wiki/Walsh_matrix" title="Walsh matrix">Walsh</a></li> <li><a href="/wiki/Z-matrix_(mathematics)" title="Z-matrix (mathematics)">Z</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constant</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exchange_matrix" title="Exchange matrix">Exchange</a></li> <li><a href="/wiki/Hilbert_matrix" title="Hilbert matrix">Hilbert</a></li> <li><a href="/wiki/Identity_matrix" title="Identity matrix">Identity</a></li> <li><a href="/wiki/Lehmer_matrix" title="Lehmer matrix">Lehmer</a></li> <li><a href="/wiki/Matrix_of_ones" title="Matrix of ones">Of ones</a></li> <li><a href="/wiki/Pascal_matrix" title="Pascal matrix">Pascal</a></li> <li><a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli</a></li> <li><a href="/wiki/Redheffer_matrix" title="Redheffer matrix">Redheffer</a></li> <li><a href="/wiki/Shift_matrix" title="Shift matrix">Shift</a></li> <li><a href="/wiki/Zero_matrix" title="Zero matrix">Zero</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conditions on <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues or eigenvectors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Companion_matrix" title="Companion matrix">Companion</a></li> <li><a href="/wiki/Convergent_matrix" title="Convergent matrix">Convergent</a></li> <li><a href="/wiki/Defective_matrix" title="Defective matrix">Defective</a></li> <li><a href="/wiki/Definite_matrix" title="Definite matrix">Definite</a></li> <li><a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">Diagonalizable</a></li> <li><a href="/wiki/Hurwitz-stable_matrix" title="Hurwitz-stable matrix">Hurwitz-stable</a></li> <li><a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">Positive-definite</a></li> <li><a href="/wiki/Stieltjes_matrix" title="Stieltjes matrix">Stieltjes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satisfying conditions on <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">products</a> or <a href="/wiki/Inverse_of_a_matrix" class="mw-redirect" title="Inverse of a matrix">inverses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Matrix_congruence" title="Matrix congruence">Congruent</a></li> <li><a href="/wiki/Idempotent_matrix" title="Idempotent matrix">Idempotent</a> or <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Projection</a></li> <li><a href="/wiki/Invertible_matrix" title="Invertible matrix">Invertible</a></li> <li><a href="/wiki/Involutory_matrix" title="Involutory matrix">Involutory</a></li> <li><a href="/wiki/Nilpotent_matrix" title="Nilpotent matrix">Nilpotent</a></li> <li><a href="/wiki/Normal_matrix" title="Normal matrix">Normal</a></li> <li><a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">Orthogonal</a></li> <li><a href="/wiki/Unimodular_matrix" title="Unimodular matrix">Unimodular</a></li> <li><a href="/wiki/Unipotent" title="Unipotent">Unipotent</a></li> <li><a href="/wiki/Unitary_matrix" title="Unitary matrix">Unitary</a></li> <li><a href="/wiki/Totally_unimodular_matrix" class="mw-redirect" title="Totally unimodular matrix">Totally unimodular</a></li> <li><a href="/wiki/Weighing_matrix" title="Weighing matrix">Weighing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With specific applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjugate_matrix" title="Adjugate matrix">Adjugate</a></li> <li><a href="/wiki/Alternating_sign_matrix" title="Alternating sign matrix">Alternating sign</a></li> <li><a href="/wiki/Augmented_matrix" title="Augmented matrix">Augmented</a></li> <li><a href="/wiki/B%C3%A9zout_matrix" title="Bézout matrix">Bézout</a></li> <li><a href="/wiki/Carleman_matrix" title="Carleman matrix">Carleman</a></li> <li><a href="/wiki/Cartan_matrix" title="Cartan matrix">Cartan</a></li> <li><a href="/wiki/Circulant_matrix" title="Circulant matrix">Circulant</a></li> <li><a href="/wiki/Cofactor_matrix" class="mw-redirect" title="Cofactor matrix">Cofactor</a></li> <li><a href="/wiki/Commutation_matrix" title="Commutation matrix">Commutation</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion</a></li> <li><a href="/wiki/Coxeter_matrix" class="mw-redirect" title="Coxeter matrix">Coxeter</a></li> <li><a href="/wiki/Distance_matrix" title="Distance matrix">Distance</a></li> <li><a href="/wiki/Duplication_and_elimination_matrices" title="Duplication and elimination matrices">Duplication and elimination</a></li> <li><a href="/wiki/Euclidean_distance_matrix" title="Euclidean distance matrix">Euclidean distance</a></li> <li><a href="/wiki/Fundamental_matrix_(linear_differential_equation)" title="Fundamental matrix (linear differential equation)">Fundamental (linear differential equation)</a></li> <li><a href="/wiki/Generator_matrix" title="Generator matrix">Generator</a></li> <li><a href="/wiki/Gram_matrix" title="Gram matrix">Gram</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li> <li><a href="/wiki/Householder_transformation" title="Householder transformation">Householder</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Moment_matrix" title="Moment matrix">Moment</a></li> <li><a href="/wiki/Payoff_matrix" class="mw-redirect" title="Payoff matrix">Payoff</a></li> <li><a href="/wiki/Pick_matrix" class="mw-redirect" title="Pick matrix">Pick</a></li> <li><a href="/wiki/Random_matrix" title="Random matrix">Random</a></li> <li><a href="/wiki/Rotation_matrix" title="Rotation matrix">Rotation</a></li> <li><a href="/wiki/Routh%E2%80%93Hurwitz_matrix" title="Routh–Hurwitz matrix">Routh-Hurwitz</a></li> <li><a href="/wiki/Seifert_matrix" class="mw-redirect" title="Seifert matrix">Seifert</a></li> <li><a href="/wiki/Shear_matrix" class="mw-redirect" title="Shear matrix">Shear</a></li> <li><a href="/wiki/Similarity_matrix" class="mw-redirect" title="Similarity matrix">Similarity</a></li> <li><a href="/wiki/Symplectic_matrix" title="Symplectic matrix">Symplectic</a></li> <li><a href="/wiki/Totally_positive_matrix" title="Totally positive matrix">Totally positive</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="/wiki/Statistics" title="Statistics">statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centering_matrix" title="Centering matrix">Centering</a></li> <li><a href="/wiki/Correlation_matrix" class="mw-redirect" title="Correlation matrix">Correlation</a></li> <li><a href="/wiki/Covariance_matrix" title="Covariance matrix">Covariance</a></li> <li><a href="/wiki/Design_matrix" title="Design matrix">Design</a></li> <li><a href="/wiki/Doubly_stochastic_matrix" title="Doubly stochastic matrix">Doubly stochastic</a></li> <li><a href="/wiki/Fisher_information_matrix" class="mw-redirect" title="Fisher information matrix">Fisher information</a></li> <li><a href="/wiki/Projection_matrix" title="Projection matrix">Hat</a></li> <li><a href="/wiki/Precision_(statistics)" title="Precision (statistics)">Precision</a></li> <li><a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Stochastic</a></li> <li><a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Transition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjacency_matrix" title="Adjacency matrix">Adjacency</a></li> <li><a href="/wiki/Biadjacency_matrix" class="mw-redirect" title="Biadjacency matrix">Biadjacency</a></li> <li><a href="/wiki/Degree_matrix" title="Degree matrix">Degree</a></li> <li><a href="/wiki/Edmonds_matrix" title="Edmonds matrix">Edmonds</a></li> <li><a href="/wiki/Incidence_matrix" title="Incidence matrix">Incidence</a></li> <li><a href="/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacian</a></li> <li><a href="/wiki/Seidel_adjacency_matrix" title="Seidel adjacency matrix">Seidel adjacency</a></li> <li><a href="/wiki/Tutte_matrix" title="Tutte matrix">Tutte</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in science and engineering</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density</a></li> <li><a href="/wiki/Fundamental_matrix_(computer_vision)" title="Fundamental matrix (computer vision)">Fundamental (computer vision)</a></li> <li><a href="/wiki/Fuzzy_associative_matrix" title="Fuzzy associative matrix">Fuzzy associative</a></li> <li><a href="/wiki/Gamma_matrices" title="Gamma matrices">Gamma</a></li> <li><a href="/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann</a></li> <li><a href="/wiki/Hamiltonian_matrix" title="Hamiltonian matrix">Hamiltonian</a></li> <li><a href="/wiki/Irregular_matrix" title="Irregular matrix">Irregular</a></li> <li><a href="/wiki/Overlap_matrix" class="mw-redirect" title="Overlap matrix">Overlap</a></li> <li><a href="/wiki/S-matrix" title="S-matrix">S</a></li> <li><a href="/wiki/State-transition_matrix" title="State-transition matrix">State transition</a></li> <li><a href="/wiki/Substitution_matrix" title="Substitution matrix">Substitution</a></li> <li><a href="/wiki/Z-matrix_(chemistry)" title="Z-matrix (chemistry)">Z (chemistry)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related terms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Matrix_exponential" title="Matrix exponential">Matrix exponential</a></li> <li><a href="/wiki/Matrix_representation_of_conic_sections" title="Matrix representation of conic sections">Matrix representation of conic sections</a></li> <li><a href="/wiki/Perfect_matrix" title="Perfect matrix">Perfect matrix</a></li> <li><a href="/wiki/Pseudoinverse" class="mw-redirect" title="Pseudoinverse">Pseudoinverse</a></li> <li><a href="/wiki/Row_echelon_form" title="Row echelon form">Row echelon form</a></li> <li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" 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