對稱矩陣 - 维基百科，自由的百科全书

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href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-例子" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#例子"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>例子</span> </div> </a> <ul id="toc-例子-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-性质" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#性质"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>性质</span> </div> </a> <ul id="toc-性质-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-分解" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#分解"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>分解</span> </div> </a> <ul id="toc-分解-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-實對稱矩陣" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#實對稱矩陣"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>實對稱矩陣</span> </div> </a> <ul id="toc-實對稱矩陣-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-黑塞矩阵" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#黑塞矩阵"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>黑塞矩阵</span> </div> </a> <ul id="toc-黑塞矩阵-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-可对称化矩阵" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#可对称化矩阵"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>可对称化矩阵</span> </div> </a> <ul id="toc-可对称化矩阵-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-与不等式的关系" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#与不等式的关系"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>与不等式的关系</span> </div> </a> <ul id="toc-与不等式的关系-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>参见</span> </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-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="目录" 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="开关目录" > <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">开关目录</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">對稱矩陣</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="前往另一种语言写成的文章。40种语言可用" > <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-40" 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">40种语言</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%85%D8%AA%D9%85%D8%A7%D8%AB%D9%84%D8%A9" title="مصفوفة متماثلة – 阿拉伯语" lang="ar" hreflang="ar" data-title="مصفوفة متماثلة" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%96%D0%BC%D0%B5%D1%82%D1%80%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0" title="Сіметрычная матрыца – 白俄罗斯语" lang="be" hreflang="be" data-title="Сіметрычная матрыца" data-language-autonym="Беларуская" data-language-local-name="白俄罗斯语" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Симетрична матрица – 保加利亚语" lang="bg" hreflang="bg" data-title="Симетрична матрица" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matriu_sim%C3%A8trica" title="Matriu simètrica – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Matriu simètrica" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" 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/Symetrick%C3%A1_matice" title="Symetrická matice – 捷克语" lang="cs" hreflang="cs" data-title="Symetrická matice" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D0%BB%D0%BB%C4%95_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Симметриллĕ матрица – 楚瓦什语" lang="cv" hreflang="cv" data-title="Симметриллĕ матрица" data-language-autonym="Чӑвашла" data-language-local-name="楚瓦什语" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Symmetrisk_matrix" title="Symmetrisk matrix – 丹麦语" lang="da" hreflang="da" data-title="Symmetrisk matrix" data-language-autonym="Dansk" data-language-local-name="丹麦语" 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/Symmetrische_Matrix" title="Symmetrische Matrix – 德语" lang="de" hreflang="de" data-title="Symmetrische Matrix" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BC%CE%BC%CE%B5%CF%84%CF%81%CE%B9%CE%BA%CF%8C%CF%82_%CF%80%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82" title="Συμμετρικός πίνακας – 希腊语" lang="el" hreflang="el" data-title="Συμμετρικός πίνακας" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Symmetric_matrix" title="Symmetric matrix – 英语" lang="en" hreflang="en" data-title="Symmetric matrix" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Simetria_matrico" title="Simetria matrico – 世界语" lang="eo" hreflang="eo" data-title="Simetria matrico" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matriz_sim%C3%A9trica" title="Matriz simétrica – 西班牙语" lang="es" hreflang="es" data-title="Matriz simétrica" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/S%C3%BCmmeetriline_maatriks" title="Sümmeetriline maatriks – 爱沙尼亚语" lang="et" hreflang="et" data-title="Sümmeetriline maatriks" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Matrize_simetriko" title="Matrize simetriko – 巴斯克语" lang="eu" hreflang="eu" data-title="Matrize simetriko" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</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%85%D8%AA%D9%82%D8%A7%D8%B1%D9%86" title="ماتریس متقارن – 波斯语" lang="fa" hreflang="fa" data-title="ماتریس متقارن" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Symmetrinen_matriisi" title="Symmetrinen matriisi – 芬兰语" lang="fi" hreflang="fi" data-title="Symmetrinen matriisi" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Matrice_sym%C3%A9trique" title="Matrice symétrique – 法语" lang="fr" hreflang="fr" data-title="Matrice symétrique" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Matriz_sim%C3%A9trica" title="Matriz simétrica – 加利西亚语" lang="gl" hreflang="gl" data-title="Matriz simétrica" data-language-autonym="Galego" data-language-local-name="加利西亚语" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%94_%D7%A1%D7%99%D7%9E%D7%98%D7%A8%D7%99%D7%AA" title="מטריצה סימטרית – 希伯来语" lang="he" hreflang="he" data-title="מטריצה סימטרית" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Szimmetrikus_m%C3%A1trix" title="Szimmetrikus mátrix – 匈牙利语" lang="hu" hreflang="hu" data-title="Szimmetrikus mátrix" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Matrice_symmetric" title="Matrice symmetric – 国际语" lang="ia" hreflang="ia" data-title="Matrice symmetric" data-language-autonym="Interlingua" data-language-local-name="国际语" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Matriks_simetrik" title="Matriks simetrik – 印度尼西亚语" lang="id" hreflang="id" data-title="Matriks simetrik" data-language-autonym="Bahasa Indonesia" data-language-local-name="印度尼西亚语" 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_simmetrica" title="Matrice simmetrica – 意大利语" lang="it" hreflang="it" data-title="Matrice simmetrica" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%AF%BE%E7%A7%B0%E8%A1%8C%E5%88%97" title="対称行列 – 日语" lang="ja" hreflang="ja" data-title="対称行列" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%8C%80%EC%B9%AD%ED%96%89%EB%A0%AC" title="대칭행렬 – 韩语" lang="ko" hreflang="ko" data-title="대칭행렬" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Symmetrische_matrix" title="Symmetrische matrix – 荷兰语" lang="nl" hreflang="nl" data-title="Symmetrische matrix" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Symmetrisk_matrise" title="Symmetrisk matrise – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Symmetrisk matrise" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Symmetrisk_matrise" title="Symmetrisk matrise – 书面挪威语" lang="nb" hreflang="nb" data-title="Symmetrisk matrise" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Macierz_symetryczna" title="Macierz symetryczna – 波兰语" lang="pl" hreflang="pl" data-title="Macierz symetryczna" data-language-autonym="Polski" data-language-local-name="波兰语" 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_sim%C3%A9trica" title="Matriz simétrica – 葡萄牙语" lang="pt" hreflang="pt" data-title="Matriz simétrica" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Matrice_simetric%C4%83" title="Matrice simetrică – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Matrice simetrică" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Симметричная матрица – 俄语" lang="ru" hreflang="ru" data-title="Симметричная матрица" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Simetri%C4%8Dna_matrika" title="Simetrična matrika – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Simetrična matrika" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Symmetrisk_matris" title="Symmetrisk matris – 瑞典语" lang="sv" hreflang="sv" data-title="Symmetrisk matris" data-language-autonym="Svenska" data-language-local-name="瑞典语" 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%9A%E0%AE%AE%E0%AE%9A%E0%AF%8D%E0%AE%9A%E0%AF%80%E0%AE%B0%E0%AF%8D_%E0%AE%85%E0%AE%A3%E0%AE%BF" title="சமச்சீர் அணி – 泰米尔语" lang="ta" hreflang="ta" data-title="சமச்சீர் அணி" data-language-autonym="தமிழ்" data-language-local-name="泰米尔语" 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%B8%AA%E0%B8%A1%E0%B8%A1%E0%B8%B2%E0%B8%95%E0%B8%A3" title="เมทริกซ์สมมาตร – 泰语" lang="th" hreflang="th" data-title="เมทริกซ์สมมาตร" data-language-autonym="ไทย" data-language-local-name="泰语" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Simetrik_matris" title="Simetrik matris – 土耳其语" lang="tr" hreflang="tr" data-title="Simetrik matris" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%B8%D0%BC%D0%B5%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F" title="Симетрична матриця – 乌克兰语" lang="uk" hreflang="uk" data-title="Симетрична матриця" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%85%D8%AA%D9%86%D8%A7%D8%B8%D8%B1_%D9%85%DB%8C%D9%B9%D8%B1%DA%A9%D8%B3" title="متناظر میٹرکس – 乌尔都语" lang="ur" hreflang="ur" data-title="متناظر میٹرکس" data-language-autonym="اردو" data-language-local-name="乌尔都语" 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margin-top:1em;"> <tbody><tr> <th style="font-size:90%; background:#DCF0FF"><a href="/wiki/%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="线性代数">线性代数</a> </th></tr> <tr> <td><div style="padding-top: 7px; padding-bottom: 4px;"><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 {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">A</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {A} 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style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a></span> </td></tr> <tr> <td> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF; font-size: 90%;">向量 </th></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E6%A0%87%E9%87%8F_(%E6%95%B0%E5%AD%A6)" title="标量 (数学)">标量</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">向量空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%82%B9%E7%A7%AF" title="点积">向量投影</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%A4%96%E7%A7%AF" class="mw-disambig" title="外积">外积</a>（<a href="/wiki/%E5%8F%89%E7%A7%AF" title="叉积">向量积</a> ·</span> <span class="nowrap"><a href="/wiki/%E4%B8%83%E7%BB%B4%E5%8F%89%E7%A7%AF" title="七维叉积">七维向量积</a>） ·</span> <span class="nowrap"><a href="/wiki/%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4" title="内积空间">内积</a>（<a href="/wiki/%E7%82%B9%E7%A7%AF" title="点积">数量积</a>） ·</span> <span class="nowrap"><a href="/wiki/%E4%BA%8C%E9%87%8D%E5%90%91%E9%87%8F" title="二重向量">二重向量</a></span> </td></tr></tbody></table> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF; font-size: 90%;">矩阵与行列式 </th></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式">行列式</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%96%B9%E7%A8%8B%E7%BB%84" title="线性方程组">线性方程组</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%A7%A9_(%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0)" title="秩 (线性代数)">秩</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%9B%B6%E7%A9%BA%E9%97%B4" title="零空间">核</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%B7%A1" title="跡">跡</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%96%AE%E4%BD%8D%E7%9F%A9%E9%99%A3" title="單位矩陣">單位矩陣</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%88%9D%E7%AD%89%E7%9F%A9%E9%98%B5" title="初等矩阵">初等矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%96%B9%E5%9D%97%E7%9F%A9%E9%98%B5" title="方块矩阵">方块矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%88%86%E5%A1%8A%E7%9F%A9%E9%99%A3" title="分塊矩陣">分块矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E4%B8%89%E8%A7%92%E7%9F%A9%E9%98%B5" title="三角矩阵">三角矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%9D%9E%E5%A5%87%E5%BC%82%E6%96%B9%E9%98%B5" title="非奇异方阵">非奇异方阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%BD%AC%E7%BD%AE%E7%9F%A9%E9%98%B5" title="转置矩阵">转置矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%80%86%E7%9F%A9%E9%98%B5" title="逆矩阵">逆矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%B0%8D%E8%A7%92%E7%9F%A9%E9%99%A3" title="對角矩陣">对角矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%8F%AF%E5%AF%B9%E8%A7%92%E5%8C%96%E7%9F%A9%E9%98%B5" title="可对角化矩阵">可对角化矩阵</a> ·</span> <span class="nowrap"><a class="mw-selflink selflink">对称矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%8F%8D%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3" title="反對稱矩陣">反對稱矩陣</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5" title="正交矩阵">正交矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%85%89%E7%9F%A9%E9%98%B5" title="酉矩阵">幺正矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E7%9F%A9%E9%98%B5" title="埃尔米特矩阵">埃尔米特矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%96%9C%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E7%9F%A9%E9%98%B5" title="斜埃尔米特矩阵">反埃尔米特矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E8%A7%84%E7%9F%A9%E9%98%B5" title="正规矩阵">正规矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E4%BC%B4%E9%9A%8F%E7%9F%A9%E9%98%B5" title="伴随矩阵">伴随矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E9%A4%98%E5%9B%A0%E5%AD%90%E7%9F%A9%E9%99%A3" title="餘因子矩陣">余因子矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%85%B1%E8%BD%AD%E8%BD%AC%E7%BD%AE" title="共轭转置">共轭转置</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5" title="正定矩阵">正定矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%B9%82%E9%9B%B6%E7%9F%A9%E9%98%B5" title="幂零矩阵">幂零矩阵</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3" title="矩阵分解">矩阵分解</a> （<a href="/wiki/LU%E5%88%86%E8%A7%A3" title="LU分解">LU分解</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%A5%87%E5%BC%82%E5%80%BC%E5%88%86%E8%A7%A3" title="奇异值分解">奇异值分解</a> ·</span> <span class="nowrap"><a href="/wiki/QR%E5%88%86%E8%A7%A3" title="QR分解">QR分解</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%9E%81%E5%88%86%E8%A7%A3" title="极分解">极分解</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%89%B9%E5%BE%81%E5%88%86%E8%A7%A3" title="特征分解">特征分解</a>） ·</span> <span class="nowrap"><a href="/wiki/%E5%AD%90%E5%BC%8F%E5%92%8C%E4%BD%99%E5%AD%90%E5%BC%8F" title="子式和余子式">子式和余子式</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%B1%95%E5%BC%80" title="拉普拉斯展开">拉普拉斯展開</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%85%8B%E7%BD%97%E5%86%85%E5%85%8B%E7%A7%AF" title="克罗内克积">克罗内克积</a></span> </td></tr></tbody></table> <table class="collapsible collapsed" width="100%"> <tbody><tr> <th style="text-align: left; background: #DCF0FF; font-size: 90%;">线性空间与线性变换 </th></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F%E7%A9%BA%E9%97%B4" title="向量空间">线性空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射">线性变换</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E5%AD%90%E7%A9%BA%E9%97%B4" title="线性子空间">线性子空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E7%94%9F%E6%88%90%E7%A9%BA%E9%97%B4" title="线性生成空间">线性生成空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="基 (線性代數)">基</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%98%A0%E5%B0%84" title="线性映射">线性映射</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%8A%95%E5%BD%B1" class="mw-disambig" title="投影">线性投影</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%B7%9A%E6%80%A7%E7%84%A1%E9%97%9C" title="線性無關">線性無關</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E7%BB%84%E5%90%88" title="线性组合">线性组合</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%B3%9B%E5%87%BD" class="mw-redirect" title="线性泛函">线性泛函</a> ·</span> <span class="nowrap"><a href="/wiki/%E8%A1%8C%E7%A9%BA%E9%97%B4%E4%B8%8E%E5%88%97%E7%A9%BA%E9%97%B4" title="行空间与列空间">行空间与列空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E5%AF%B9%E5%81%B6%E7%A9%BA%E9%97%B4" title="对偶空间">对偶空间</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%AD%A3%E4%BA%A4" title="正交">正交</a> ·</span> <span class="nowrap"><a href="/wiki/%E7%89%B9%E5%BE%81%E5%80%BC%E5%92%8C%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F" title="特征值和特征向量">特征向量</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%9C%80%E5%B0%8F%E4%BA%8C%E4%B9%98%E6%B3%95" title="最小二乘法">最小二乘法</a> ·</span> <span class="nowrap"><a href="/wiki/%E6%A0%BC%E6%8B%89%E5%A7%86-%E6%96%BD%E5%AF%86%E7%89%B9%E6%AD%A3%E4%BA%A4%E5%8C%96" title="格拉姆-施密特正交化">格拉姆-施密特正交化</a></span> </td></tr></tbody></table> </td></tr> <tr style="text-align: center; font-size: 90%;"> <td><style data-mw-deduplicate="TemplateStyles:r84265675">.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 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.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 a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:110%;margin:0 8em}.mw-parser-output .navbar-ct-mini{font-size:110%;margin:0 5em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="Template:线性代数"><abbr title="查看该模板">查</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Template_talk:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0&action=edit&redlink=1" class="new" title="Template talk:线性代数（页面不存在）"><abbr title="讨论该模板">论</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="Special:编辑页面/Template:线性代数"><abbr title="编辑该模板">编</abbr></a></li></ul></div> </td></tr></tbody></table> <p>在<a href="/wiki/%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8" class="mw-redirect" title="線性代數">線性代數</a>中，<b>對稱矩陣</b>（英語：<span lang="en">symmetric matrix</span>）指<a href="/wiki/%E8%BD%89%E7%BD%AE%E7%9F%A9%E9%99%A3" class="mw-redirect" title="轉置矩陣">轉置矩陣</a>和自身相等<a href="/wiki/%E6%96%B9%E5%BD%A2%E7%9F%A9%E9%99%A3" class="mw-redirect" title="方形矩陣">方形矩陣</a>。 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=A^{\textrm {T}}\ \!}"> <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"> <mtext>T</mtext> </mrow> </mrow> </msup> <mtext> </mtext> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=A^{\textrm {T}}\ \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1b0639778d31e3eee2c8582f8fa33a4bf320fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.197ex; height:2.676ex;" alt="{\displaystyle A=A^{\textrm {T}}\ \!}"></span></dd></dl> <p>對稱矩陣中的右上至左下方向元素以<a href="/wiki/%E4%B8%BB%E5%B0%8D%E8%A7%92%E7%B7%9A" title="主對角線">主對角線</a>（左上至右下）為軸進行對稱。若將其寫作<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_{ij})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(a_{ij})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/296ad42d9541f8285979ce822ccb661da56111ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.358ex; height:3.009ex;" alt="{\displaystyle A=(a_{ij})}"></span>，則对所有的<i>i</i>和<i>j</i>， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}=a_{ji}\ \!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <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>j</mi> <mi>i</mi> </mrow> </msub> <mtext> </mtext> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}=a_{ji}\ \!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/345ee164a8ed5ddbfaf43a3dabb821bcbed54324" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.706ex; height:2.343ex;" alt="{\displaystyle a_{ij}=a_{ji}\ \!}"></span></dd></dl> <p>下列是3×3的對稱矩陣： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&2&3\\2&4&-5\\3&-5&6\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>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>5</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&2&3\\2&4&-5\\3&-5&6\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e60305cc200c2ca3432be53fc0c983f8914d2c59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:15.601ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&2&3\\2&4&-5\\3&-5&6\end{bmatrix}}}"></span></dd></dl> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="例子"><span id=".E4.BE.8B.E5.AD.90"></span>例子</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3&action=edit&section=1" title="编辑章节：例子"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 {\begin{bmatrix}a&b&c\\b&d&e\\c&e&f\end{bmatrix}},{\begin{bmatrix}1&5\\5&7\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> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi>d</mi> </mtd> <mtd> <mi>e</mi> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> </mtd> <mtd> <mi>e</mi> </mtd> <mtd> <mi>f</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a&b&c\\b&d&e\\c&e&f\end{bmatrix}},{\begin{bmatrix}1&5\\5&7\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7e804b73a13abb407f7c829c9138d4aebc68e13" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:21.756ex; height:9.509ex;" alt="{\displaystyle {\begin{bmatrix}a&b&c\\b&d&e\\c&e&f\end{bmatrix}},{\begin{bmatrix}1&5\\5&7\end{bmatrix}},}"></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 {\begin{bmatrix}1&3&0\\3&1&6\\0&6&1\end{bmatrix}},{\begin{bmatrix}a&b\\b&c\end{bmatrix}},{\begin{bmatrix}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>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mi>b</mi> </mtd> </mtr> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mi>c</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&3&0\\3&1&6\\0&6&1\end{bmatrix}},{\begin{bmatrix}a&b\\b&c\end{bmatrix}},{\begin{bmatrix}2\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee668c9520a381736a116bcffc60e84e8aba5b16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:25.026ex; height:9.176ex;" alt="{\displaystyle {\begin{bmatrix}1&3&0\\3&1&6\\0&6&1\end{bmatrix}},{\begin{bmatrix}a&b\\b&c\end{bmatrix}},{\begin{bmatrix}2\end{bmatrix}}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="性质"><span id=".E6.80.A7.E8.B4.A8"></span>性质</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3&action=edit&section=2" title="编辑章节：性质"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>對於任何方形矩陣<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></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 X+X^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>+</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X+X^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608d01396919787afc8c0f2188e61d910bb1dda8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.207ex; height:2.843ex;" alt="{\displaystyle X+X^{T}}"></span>是對稱矩陣。</li> <li><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>為<a href="/wiki/%E6%96%B9%E5%BD%A2%E7%9F%A9%E9%99%A3" class="mw-redirect" title="方形矩陣">方形矩陣</a>是<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>為對稱矩陣的必要條件，即對稱矩陣行數必等於列數。</li> <li><a href="/wiki/%E5%B0%8D%E8%A7%92%E7%9F%A9%E9%99%A3" title="對角矩陣">對角矩陣</a>都是對稱矩陣。</li> <li><a href="/wiki/%E8%8B%A5%E4%B8%94%E5%94%AF%E8%8B%A5" class="mw-redirect" title="若且唯若">若且唯若</a>兩者的<a href="/wiki/%E4%B9%98%E6%B3%95" title="乘法">乘法</a>可<a href="/wiki/%E4%BA%A4%E6%8F%9B%E5%BE%8B" title="交換律">交換</a>（即<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AB=BA}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mi>B</mi> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB=BA}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992c8ea49fdd26b491180036c5a4d879fec77442" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.113ex; height:2.176ex;" alt="{\displaystyle AB=BA}"></span>）時，兩個對稱矩陣的積（<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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b04153f9681e5b06066357774475c04aaef3a8bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.507ex; height:2.176ex;" alt="{\displaystyle AB}"></span>）是對稱矩陣。<style data-mw-deduplicate="TemplateStyles:r83946278">.mw-parser-output .template-facttext{background-color:var(--background-color-neutral,#eaecf0);color:inherit;margin:-.3em 0;padding:.3em 0}</style><mark class="template-facttext" title="需要提供文献来源">兩個實對稱矩陣乘法可交換<a href="/wiki/%E8%8B%A5%E4%B8%94%E5%94%AF%E8%8B%A5" class="mw-redirect" title="若且唯若">若且唯若</a>兩者的<a href="/wiki/%E7%89%B9%E5%BE%81%E7%A9%BA%E9%97%B4" class="mw-redirect" title="特征空间">特徵空間</a>相同。</mark><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求开始于2019年7月19日。">[來源請求]</span></a></sup></li> <li>任何方形矩陣<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>，如果它的元素屬於一個<a href="/wiki/%E7%89%B9%E5%BE%B5_(%E4%BB%A3%E6%95%B8)" class="mw-redirect" title="特徵 (代數)">特徵</a>不為2的域（例如<a href="/wiki/%E5%AF%A6%E6%95%B8" class="mw-redirect" title="實數">實數</a>），可以用剛好一種方法寫成一個對稱矩陣和一個<a href="/wiki/%E6%96%9C%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3" class="mw-redirect" title="斜對稱矩陣">斜對稱矩陣</a>之和：</li></ul> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X={\frac {1}{2}}(X+X^{T})+{\frac {1}{2}}(X-X^{T})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>+</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>−</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X={\frac {1}{2}}(X+X^{T})+{\frac {1}{2}}(X-X^{T})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff4ac7044b6f6dbaec5f8ef3d74ec4d553972c78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:31.948ex; height:5.176ex;" alt="{\displaystyle X={\frac {1}{2}}(X+X^{T})+{\frac {1}{2}}(X-X^{T})}"></span></dd></dl></dd></dl> <ul><li>每個實方形矩陣都可寫作兩個實對稱矩陣的積，每個複方形矩陣都可寫作兩個複對稱矩陣的積。</li> <li>若對稱矩陣<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>的每個元素均為實數，<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>是<a href="/wiki/%E5%AE%9E%E5%AF%B9%E7%A7%B0%E7%9F%A9%E9%98%B5" class="mw-redirect" title="实对称矩阵">實對稱矩陣</a>。</li> <li>一個矩陣同時為對稱矩陣及斜對稱矩陣若且唯若所有元素都是零。</li> <li>如果X是對稱矩陣，那麼 <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 AXA^{\textrm {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>X</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AXA^{\textrm {T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dba3887e4fb855a7779e452dcfa3548bce46be2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.885ex; height:2.676ex;" alt="{\displaystyle AXA^{\textrm {T}}}"></span> 也是對稱矩陣.</li></ul> <div class="mw-heading mw-heading2"><h2 id="分解"><span id=".E5.88.86.E8.A7.A3"></span>分解</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3&action=edit&section=3" title="编辑章节：分解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>利用<a href="/wiki/%E8%8B%A5%E5%B0%94%E5%BD%93%E6%A0%87%E5%87%86%E5%BD%A2" class="mw-redirect" title="若尔当标准形">若尔当标准形</a>，我们可以证明每一个实方阵都可以写成两个实对称矩阵的乘积，而每一个复方阵都可以写成两个复对称矩阵的乘积。<sup id="cite_ref-Bosch1986_1-0" class="reference"><a href="#cite_note-Bosch1986-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>每一个实<a href="/wiki/%E9%9D%9E%E5%A5%87%E5%BC%82%E7%9F%A9%E9%98%B5" class="mw-redirect" title="非奇异矩阵">非奇异矩阵</a>都可以唯一分解成一个<a href="/wiki/%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5" title="正交矩阵">正交矩阵</a>和一个对称<a href="/wiki/%E6%AD%A3%E5%AE%9A%E7%9F%A9%E9%98%B5" title="正定矩阵">正定矩阵</a>的乘积，这称为<a href="/wiki/%E6%9E%81%E5%88%86%E8%A7%A3" title="极分解">极分解</a>。奇异矩阵也可以分解，但不是唯一的。 </p><p><a href="/wiki/Cholesky%E5%88%86%E8%A7%A3" class="mw-redirect" title="Cholesky分解">Cholesky分解</a>说明每一个实正定对称矩阵都是一个上三角矩阵和它的转置的乘积。 </p> <div class="mw-heading mw-heading2"><h2 id="實對稱矩陣"><span id=".E5.AF.A6.E5.B0.8D.E7.A8.B1.E7.9F.A9.E9.99.A3"></span>實對稱矩陣</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3&action=edit&section=4" title="编辑章节：實對稱矩陣"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>实對稱矩陣</b>是一個元素都为<a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</a>的对称矩陣，用<,>表示<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^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2ebce0f18894435d56a8fe182e22f135aa8ed07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.982ex; height:2.343ex;" alt="{\displaystyle R^{n}}"></span>上的<a href="/wiki/%E5%85%A7%E7%A9%8D" class="mw-redirect mw-disambig" title="內積">內積</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 n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <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/59d2b4cb72e304526cf5b5887147729ea259da78" 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>的實矩陣<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>是對稱的，<a href="/wiki/%E8%8B%A5%E4%B8%94%E5%94%AF%E8%8B%A5" class="mw-redirect" title="若且唯若">若且唯若</a>對於所有<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x,y\in \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x,y\in \mathbb {R} ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/968ade084462d5ec7193a762f77904c61ab42822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.256ex; height:2.676ex;" alt="{\displaystyle x,y\in \mathbb {R} ^{n}}"></span>， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mi>A</mi> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mo fence="false" stretchy="false">⟨</mo> <mi>x</mi> <mo>,</mo> <mi>A</mi> <mi>y</mi> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b148dd9277ab2b18762deba9d034b58832bb025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.241ex; height:2.843ex;" alt="{\displaystyle \langle Ax,y\rangle =\langle x,Ay\rangle }"></span>。</dd></dl> <p>实對稱矩陣有以下的性质： </p> <ul><li>实对称矩阵A的不同<a href="/wiki/%E7%89%B9%E5%BE%81%E5%80%BC" class="mw-redirect" title="特征值">特征值</a>所对应的<a href="/wiki/%E7%89%B9%E5%BE%81%E5%90%91%E9%87%8F" class="mw-redirect" title="特征向量">特征向量</a>是<a href="/wiki/%E6%AD%A3%E4%BA%A4" title="正交">正交</a>的。</li> <li>实对称矩阵A的特征值都是实数，特征向量都是实向量。</li> <li>n阶实对称矩阵A必可<a href="/wiki/%E5%AF%B9%E8%A7%92%E5%8C%96" class="mw-redirect" title="对角化">对角化</a>。</li> <li>可用<a href="/wiki/%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5" title="正交矩阵">正交矩阵</a>对角化。</li> <li>K重特征值必有K个线性无关的特征向量，或者说必有<a href="/wiki/%E7%A7%A9" class="mw-disambig" title="秩">秩</a>r(λE-A)=n-k。</li></ul> <div class="mw-heading mw-heading2"><h2 id="黑塞矩阵"><span id=".E9.BB.91.E5.A1.9E.E7.9F.A9.E9.98.B5"></span>黑塞矩阵</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3&action=edit&section=5" title="编辑章节：黑塞矩阵"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r84833064">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/Hessian%E7%9F%A9%E9%98%B5" class="mw-redirect" title="Hessian矩阵">Hessian矩阵</a></div> <p>实对称<i>n</i> × <i>n</i>矩阵出现在二阶连续可微的<i>n</i>元函数的<a href="/wiki/%E9%BB%91%E5%A1%9E%E7%9F%A9%E9%98%B5" class="mw-redirect" title="黑塞矩阵">黑塞矩阵</a>之中。 </p><p><b>R</b><sup><i>n</i></sup>上的每一个<a href="/wiki/%E4%BA%8C%E6%AC%A1%E5%9E%8B" title="二次型">二次型</a><i>q</i>都可以唯一写成<i>q</i>(<b>x</b>) = <b>x</b><sup>T</sup><i>A</i><b>x</b>的形式，其中<i>A</i>是对称的<i>n</i> × <i>n</i>矩阵。于是，根据<a href="/wiki/%E8%B0%B1%E5%AE%9A%E7%90%86" title="谱定理">谱定理</a>，可以说每一个二次型，不考虑<b>R</b><sup><i>n</i></sup>的正交基的选择，“看起来像”： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}\lambda _{i}x_{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}\lambda _{i}x_{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30b1ef164ce0cefe4fcb9c02e1ab66e916564956" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.369ex; height:6.843ex;" alt="{\displaystyle q(x_{1},\ldots ,x_{n})=\sum _{i=1}^{n}\lambda _{i}x_{i}^{2}}"></span></dd></dl> <p>其中λ<sub><i>i</i></sub>是实数。这大大简化了二次型的研究，以及水平集{<b>x</b> : <i>q</i>(<b>x</b>) = 1}的研究，它们是<a href="/wiki/%E5%9C%86%E9%94%A5%E6%9B%B2%E7%BA%BF" title="圆锥曲线">圆锥曲线</a>的推广。 </p><p>这是很重要的，部分是由于每一个光滑的多元函数的二阶表现，都由属于该函数的黑塞矩阵的二次型描述；这是<a href="/wiki/%E6%B3%B0%E5%8B%92%E5%AE%9A%E7%90%86" class="mw-redirect" title="泰勒定理">泰勒定理</a>的一个结果。 </p> <div class="mw-heading mw-heading2"><h2 id="可对称化矩阵"><span id=".E5.8F.AF.E5.AF.B9.E7.A7.B0.E5.8C.96.E7.9F.A9.E9.98.B5"></span>可对称化矩阵</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3&action=edit&section=6" title="编辑章节：可对称化矩阵"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>矩阵A称为<b>可对称化</b>的，如果存在一个可逆对角矩阵D和一个对称矩阵S，使得： </p> <dl><dd>A = DS.</dd></dl> <p>可对称化矩阵的转置也是可对称化的，因为<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 (DS)^{T}=SD=D^{-1}DSD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>D</mi> <mi>S</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mo>=</mo> <mi>S</mi> <mi>D</mi> <mo>=</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>D</mi> <mi>S</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (DS)^{T}=SD=D^{-1}DSD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb7ab948d09566e7c8f2a4e01096a01625868651" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.847ex; height:3.176ex;" alt="{\displaystyle (DS)^{T}=SD=D^{-1}DSD}"></span>，而<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 DSD}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>S</mi> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle DSD}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb1fcbf8536c0fd7f3aa3b49a13bc8c53c85f3ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.348ex; height:2.176ex;" alt="{\displaystyle DSD}"></span>是对称的。 </p><p><br /> 当且仅当<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_{jk}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>k</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=[a_{jk}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9614eb9d204266404f005a4c30a9619935bdac86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.131ex; height:3.009ex;" alt="{\displaystyle A=[a_{jk}]}"></span>满足以下的条件时，矩阵可对称化： </p> <ol><li>如果<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_{ij}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a565e93211d1ad5c06a571ff8952ab3dfcff3638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.968ex; height:2.843ex;" alt="{\displaystyle a_{ij}=0}"></span>，那么<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_{ji}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ji}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d059b9aad48129a1907d2354eeeaa865847952e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.968ex; height:2.843ex;" alt="{\displaystyle a_{ji}=0}"></span>；</li> <li>对于任何有限序列<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_{1},i_{2},...,i_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i_{1},i_{2},...,i_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/719605ba8194f8211c3828d0a9192f68f7fa8c44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.808ex; height:2.509ex;" alt="{\displaystyle i_{1},i_{2},...,i_{k}}"></span>，都有<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_{i_{1}i_{2}}a_{i_{2}i_{3}}...a_{i_{k}i_{1}}=a_{i_{2}i_{1}}a_{i_{3}i_{2}}...a_{i_{1}i_{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>.</mo> <mo>.</mo> <mo>.</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i_{1}i_{2}}a_{i_{2}i_{3}}...a_{i_{k}i_{1}}=a_{i_{2}i_{1}}a_{i_{3}i_{2}}...a_{i_{1}i_{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7a9ba619cf571ffb05f5b6d68529109889b8324" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:34.918ex; height:2.343ex;" alt="{\displaystyle a_{i_{1}i_{2}}a_{i_{2}i_{3}}...a_{i_{k}i_{1}}=a_{i_{2}i_{1}}a_{i_{3}i_{2}}...a_{i_{1}i_{k}}}"></span>。</li></ol> <div class="mw-heading mw-heading2"><h2 id="与不等式的关系"><span id=".E4.B8.8E.E4.B8.8D.E7.AD.89.E5.BC.8F.E7.9A.84.E5.85.B3.E7.B3.BB"></span>与不等式的关系</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3&action=edit&section=7" title="编辑章节：与不等式的关系"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>对称阵 Z 分解为3行3列: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}Z_{11}&Z_{12}&Z_{13}\\Z_{12}^{T}&Z_{22}&Z_{23}\\Z_{13}^{T}&Z_{23}^{T}&Z_{33}\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> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}Z_{11}&Z_{12}&Z_{13}\\Z_{12}^{T}&Z_{22}&Z_{23}\\Z_{13}^{T}&Z_{23}^{T}&Z_{33}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7ae2226038a21acc6162198b254b3735101be4b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:18.888ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}Z_{11}&Z_{12}&Z_{13}\\Z_{12}^{T}&Z_{22}&Z_{23}\\Z_{13}^{T}&Z_{23}^{T}&Z_{33}\end{bmatrix}}}"></span></dd></dl> <p>当且仅当 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}Z_{11}&Z_{12}\\Z_{12}^{T}&Z_{22}\end{bmatrix}},{\begin{bmatrix}Z_{11}&Z_{13}\\Z_{13}^{T}&Z_{33}\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> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}Z_{11}&Z_{12}\\Z_{12}^{T}&Z_{22}\end{bmatrix}},{\begin{bmatrix}Z_{11}&Z_{13}\\Z_{13}^{T}&Z_{33}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d1eb2123353c20b0f49414fc9a322695fbfe41a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:25.947ex; height:6.509ex;" alt="{\displaystyle {\begin{bmatrix}Z_{11}&Z_{12}\\Z_{12}^{T}&Z_{22}\end{bmatrix}},{\begin{bmatrix}Z_{11}&Z_{13}\\Z_{13}^{T}&Z_{33}\end{bmatrix}}}"></span></dd></dl> <p>时, 存在 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=Z_{13}^{T}Z_{11}^{-1}Z_{12}-Z_{23}^{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>−</mo> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=Z_{13}^{T}Z_{11}^{-1}Z_{12}-Z_{23}^{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eef4ce399ae6f3c3c6839dcea1d1559156094fe1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.351ex; height:3.343ex;" alt="{\displaystyle X=Z_{13}^{T}Z_{11}^{-1}Z_{12}-Z_{23}^{T}}"></span>, 使得 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}Z_{11}&Z_{12}&Z_{13}\\Z_{12}^{T}&Z_{22}&Z_{23}+X^{T}\\Z_{13}^{T}&Z_{23}^{T}+X&Z_{33}\end{bmatrix}}<0}"> <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> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> </mtd> <mtd> <msubsup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mo>+</mo> <mi>X</mi> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}Z_{11}&Z_{12}&Z_{13}\\Z_{12}^{T}&Z_{22}&Z_{23}+X^{T}\\Z_{13}^{T}&Z_{23}^{T}+X&Z_{33}\end{bmatrix}}<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e979e23a2900779994b4368d084b3274f4735ed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:34.196ex; height:10.176ex;" alt="{\displaystyle {\begin{bmatrix}Z_{11}&Z_{12}&Z_{13}\\Z_{12}^{T}&Z_{22}&Z_{23}+X^{T}\\Z_{13}^{T}&Z_{23}^{T}+X&Z_{33}\end{bmatrix}}<0}"></span></dd></dl> <p>成立。 </p> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3&action=edit&section=8" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E5%8F%8D%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3" title="反對稱矩陣">反对称阵</a></li> <li><a href="/wiki/%E5%BE%AA%E7%8E%AF%E7%9F%A9%E9%98%B5" title="循环矩阵">循环矩阵</a></li> <li><a href="/wiki/%E6%B1%89%E5%85%8B%E5%B0%94%E7%9F%A9%E9%98%B5" title="汉克尔矩阵">汉克尔矩阵</a></li> <li><a href="/w/index.php?title=%E7%89%B9%E6%99%AE%E5%88%A9%E8%8C%A8%E7%9F%A9%E9%98%B5&action=edit&redlink=1" class="new" title="特普利茨矩阵（页面不存在）">特普利茨矩阵</a></li> <li><a href="/w/index.php?title=%E4%B8%AD%E5%BF%83%E5%AF%B9%E7%A7%B0%E7%9F%A9%E9%98%B5&action=edit&redlink=1" class="new" title="中心对称矩阵（页面不存在）">中心对称矩阵</a></li> <li><a href="/wiki/%E5%B8%8C%E5%B0%94%E4%BC%AF%E7%89%B9%E7%9F%A9%E9%98%B5" title="希尔伯特矩阵">希尔伯特矩阵</a></li> <li><a href="/w/index.php?title=%E8%80%83%E5%85%8B%E6%96%AF%E7%89%B9%E7%9F%A9%E9%98%B5&action=edit&redlink=1" class="new" title="考克斯特矩阵（页面不存在）">考克斯特矩阵</a></li> <li><a href="/wiki/%E5%8D%8F%E6%96%B9%E5%B7%AE%E7%9F%A9%E9%98%B5" title="协方差矩阵">协方差矩阵</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3&action=edit&section=9" title="编辑章节：参考文献"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-Bosch1986-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-Bosch1986_1-0">^</a></b></span> <span class="reference-text"><cite class="citation journal">A. J. Bosch. The factorization of a square matrix into two symmetric matrices. American Mathematical Monthly. 1986, <b>93</b>: 462–464. <a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2323471"><span title="數位物件識別號">doi:10.2307/2323471</span></a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3&rft.atitle=The+factorization+of+a+square+matrix+into+two+symmetric+matrices&rft.au=A.+J.+Bosch&rft.date=1986&rft.genre=article&rft.jtitle=American+Mathematical+Monthly&rft.pages=462-464&rft.volume=93&rft_id=info%3Adoi%2F10.2307%2F2323471&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" 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