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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">矩阵分解</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。19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-19" 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">19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/%D8%AA%D9%81%D9%83%D9%8A%D9%83_%D9%85%D8%B5%D9%81%D9%88%D9%81%D8%A9" title="تفكيك مصفوفة – 阿拉伯语" lang="ar" hreflang="ar" data-title="تفكيك مصفوفة" 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-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Descomposici%C3%B3_de_matrius" title="Descomposició de matrius – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Descomposició de matrius" 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-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_%D1%81%D0%B0%D1%80%C4%83%D0%BC%C4%95" title="Матрица сарăмĕ – 楚瓦什语" lang="cv" hreflang="cv" data-title="Матрица сарăмĕ" 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-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Matrix_decomposition" title="Matrix decomposition – 英语" lang="en" hreflang="en" data-title="Matrix decomposition" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Factorizaci%C3%B3n_de_matrices" title="Factorización de matrices – 西班牙语" lang="es" hreflang="es" data-title="Factorización de matrices" 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-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%AC%D8%B2%DB%8C%D9%87_%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3" title="تجزیه ماتریس – 波斯语" lang="fa" hreflang="fa" data-title="تجزیه ماتریس" 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-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9composition_d%27une_matrice" title="Décomposition d'une matrice – 法语" lang="fr" hreflang="fr" data-title="Décomposition d'une matrice" 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/Descomposici%C3%B3n_de_matrices" title="Descomposición de matrices – 加利西亚语" lang="gl" hreflang="gl" data-title="Descomposición de matrices" 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-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fylkjali%C3%B0un" title="Fylkjaliðun – 冰岛语" lang="is" hreflang="is" data-title="Fylkjaliðun" data-language-autonym="Íslenska" data-language-local-name="冰岛语" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Decomposizione_di_una_matrice" title="Decomposizione di una matrice – 意大利语" lang="it" hreflang="it" data-title="Decomposizione di una matrice" 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/%E8%A1%8C%E5%88%97%E3%81%AE%E5%88%86%E8%A7%A3" title="行列の分解 – 日语" lang="ja" hreflang="ja" data-title="行列の分解" 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/%ED%96%89%EB%A0%AC_%EB%B6%84%ED%95%B4" title="행렬 분해 – 韩语" lang="ko" hreflang="ko" data-title="행렬 분해" 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/Matrixdecompositie" title="Matrixdecompositie – 荷兰语" lang="nl" hreflang="nl" data-title="Matrixdecompositie" 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-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Rozk%C5%82ad_macierzy" title="Rozkład macierzy – 波兰语" lang="pl" hreflang="pl" data-title="Rozkład macierzy" 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/Decomposi%C3%A7%C3%A3o_matricial" title="Decomposição matricial – 葡萄牙语" lang="pt" hreflang="pt" data-title="Decomposição matricial" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D0%BB%D0%BE%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8B" title="Разложение матрицы – 俄语" lang="ru" hreflang="ru" data-title="Разложение матрицы" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matrisfaktorisering" title="Matrisfaktorisering – 瑞典语" lang="sv" hreflang="sv" data-title="Matrisfaktorisering" 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-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D0%BA%D0%BB%D0%B0%D0%B4_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%96" title="Розклад матриці – 乌克兰语" lang="uk" hreflang="uk" data-title="Розклад матриці" 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-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ph%C3%A2n_r%C3%A3_ma_tr%E1%BA%ADn" title="Phân rã ma trận – 越南语" lang="vi" hreflang="vi" data-title="Phân rã ma trận" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1361088#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> 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id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><table class="infobox noprint" style="width:210px; float: right; clear: right; text-align:center; 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} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a31efc33ac33577d719a3ccd162a9bf21e4847ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.972ex; height:6.176ex;" alt="{\displaystyle \mathbf {A} ={\begin{bmatrix}1&2\\3&4\end{bmatrix}}}"></span></div> </td></tr> <tr style="font-size: 90%; line-height: 150%;"> <td><span class="nowrap"><a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a><span style="white-space:nowrap; font-weight:bold;"> ·</span> <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 style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E5%9F%BA_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="基 (線性代數)">基底</a> <span style="white-space:nowrap; font-weight:bold;"> ·</span> <a href="/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式">行列式</a> <span 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 href="/wiki/%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3" title="對稱矩陣">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 class="mw-selflink selflink">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 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<p><b>矩阵分解</b>（<b><span lang="en">decomposition</span></b>, <b><span lang="en">factorization</span></b>）是将一个<a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a>拆解为数个矩阵的<a href="/wiki/%E7%9F%A9%E9%98%B5%E4%B9%98%E6%B3%95" class="mw-redirect" title="矩阵乘法">乘积</a>的运算。其依使用目的的不同，可分为几类。 </p> <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=%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%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>在<a href="/wiki/%E6%95%B0%E5%80%BC%E5%88%86%E6%9E%90" title="数值分析">数值分析</a>，矩阵分解常常用来实现一些矩阵运算的快速<a href="/wiki/%E7%AE%97%E6%B3%95" title="算法">算法</a>。 </p><p>例如，当对<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 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\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span> 进行求解时，矩阵<i>A</i>可以通过<a href="/wiki/LU%E5%88%86%E8%A7%A3" title="LU分解">LU分解</a>进行分解。LU分解将矩阵分解为<a href="/wiki/%E4%B8%8B%E4%B8%89%E8%A7%92%E7%9F%A9%E9%98%B5" class="mw-redirect" title="下三角矩阵">下三角矩阵</a><i>L</i>和<a href="/wiki/%E4%B8%8A%E4%B8%89%E8%A7%92%E7%9F%A9%E9%98%B5" class="mw-redirect" title="上三角矩阵">上三角矩阵</a><i>U</i>。相比于原方程<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\mathbf {x} =\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45d894430af69e29d6dda5aacbf4bb19336226a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.738ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\mathbf {b} }"></span>，方程组<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 L(U\mathbf {x} )=\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L(U\mathbf {x} )=\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55c9cfdfe772f24e24bb4ba0307adc167681d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.169ex; height:2.843ex;" alt="{\displaystyle L(U\mathbf {x} )=\mathbf {b} }"></span> 与<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\mathbf {x} =L^{-1}\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U\mathbf {x} =L^{-1}\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/254570218cd039cd14aaf3bb030de8a21e8124c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.693ex; height:2.676ex;" alt="{\displaystyle U\mathbf {x} =L^{-1}\mathbf {b} }"></span>仅需更少的相加和乘法来求解，然而在不精确的算术（如 <a href="/wiki/%E6%B5%AE%E7%82%B9%E6%95%B0" class="mw-redirect" title="浮点数">浮点数</a>）中可能需要更多的数字。 </p><p>类似的，<a href="/wiki/QR%E5%88%86%E8%A7%A3" title="QR分解">QR分解</a>将矩阵<i>A</i>分解为两个矩阵的乘积<i>QR</i>，其中<i>Q</i>是<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>， <i>R</i>是上三角矩阵。方程<i>Q</i>(<i>R</i><b>x</b>) = <b>b</b>可以通过<i>R</i><b>x</b> = <i>Q</i><sup>T</sup><b>b</b> = <b>c</b>求解；方程<i>R</i><b>x</b> = <b>c</b>可以通过<a href="/wiki/%E4%B8%89%E8%A7%92%E7%9F%A9%E9%98%B5#向后替换" title="三角矩阵">回带</a>求解。该方法所需的额外的加法和乘法大概是LU分解法的两倍，但在不精确的算术中不要求额外的数字，因为QR分解是<a href="/wiki/%E6%95%B0%E5%80%BC%E7%A8%B3%E5%AE%9A%E6%80%A7" title="数值稳定性">数值稳定</a>的。 </p> <div class="mw-heading mw-heading2"><h2 id="与线性方程解法相关的矩阵分解"><span id=".E4.B8.8E.E7.BA.BF.E6.80.A7.E6.96.B9.E7.A8.8B.E8.A7.A3.E6.B3.95.E7.9B.B8.E5.85.B3.E7.9A.84.E7.9F.A9.E9.98.B5.E5.88.86.E8.A7.A3"></span>与线性方程解法相关的矩阵分解</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3&action=edit&section=2" title="编辑章节：与线性方程解法相关的矩阵分解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/LU%E5%88%86%E8%A7%A3" title="LU分解">LU分解</a></li> <li><a href="/wiki/%E7%A7%91%E5%88%97%E6%96%AF%E5%9F%BA%E5%88%86%E8%A7%A3" title="科列斯基分解">科列斯基分解</a></li> <li><a href="/wiki/QR%E5%88%86%E8%A7%A3" title="QR分解">QR分解</a></li> <li><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></li></ul> <div class="mw-heading mw-heading2"><h2 id="基于特征值和相关概念的分解"><span id=".E5.9F.BA.E4.BA.8E.E7.89.B9.E5.BE.81.E5.80.BC.E5.92.8C.E7.9B.B8.E5.85.B3.E6.A6.82.E5.BF.B5.E7.9A.84.E5.88.86.E8.A7.A3"></span>基于特征值和相关概念的分解</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3&action=edit&section=3" title="编辑章节：基于特征值和相关概念的分解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><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></li> <li><a href="/wiki/%E8%8B%A5%E5%B0%94%E5%BD%93%E6%A0%87%E5%87%86%E5%9E%8B" title="若尔当标准型">若尔当标准型</a></li> <li><a href="/wiki/%E8%88%92%E5%B0%94%E5%88%86%E8%A7%A3" title="舒尔分解">舒尔分解</a></li> <li><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></li></ul> <div class="mw-heading mw-heading2"><h2 id="其他分解"><span id=".E5.85.B6.E4.BB.96.E5.88.86.E8.A7.A3"></span>其他分解</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3&action=edit&section=4" title="编辑章节：其他分解"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E6%9E%81%E5%88%86%E8%A7%A3" title="极分解">极分解</a></li> <li><a href="/wiki/%E7%9F%A9%E9%98%B5%E7%9A%84%E5%B9%B3%E6%96%B9%E6%A0%B9" title="矩阵的平方根">矩阵的平方根</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="外部链接"><span id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5"></span>外部链接</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3&action=edit&section=5" title="编辑章节：外部链接"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081212221215/http://www.bluebit.gr/matrix-calculator/">Online Matrix Calculator</a></li> <li><a rel="nofollow" class="external text" href="http://eom.springer.de/M/m120140.htm">Springer Encyclopaedia of Mathematics » Matrix factorization</a> （<a rel="nofollow" class="external text" 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style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%A0%87%E9%87%8F_(%E6%95%B0%E5%AD%A6)" title="标量 (数学)">标量</a></li> <li><a href="/wiki/%E5%90%91%E9%87%8F" title="向量">向量</a></li> <li><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></li> <li><a href="/wiki/%E7%BA%BF%E6%80%A7%E5%AD%90%E7%A9%BA%E9%97%B4" title="线性子空间">向量子空间</a></li></ul> <ul><li><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></li> <li><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></li> <li><a href="/wiki/%E6%8A%95%E5%BD%B1" class="mw-disambig" title="投影">投影</a></li> <li><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></li> <li><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></li></ul> <ul><li><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></li> <li><a href="/wiki/%E6%A8%99%E8%A8%98_(%E7%B7%9A%E6%80%A7%E4%BB%A3%E6%95%B8)" title="標記 (線性代數)">標記</a></li> <li><a href="/wiki/%E5%88%97%E7%A9%BA%E9%97%B4" class="mw-redirect" title="列空间">列空间</a></li> <li><a href="/wiki/%E8%A1%8C%E7%A9%BA%E9%97%B4" class="mw-redirect" title="行空间">行空间</a></li> <li><a href="/wiki/%E9%9B%B6%E7%A9%BA%E9%97%B4" title="零空间">零空间</a></li> <li><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></li> <li><a href="/wiki/%E6%AD%A3%E4%BA%A4" title="正交">正交</a></li> <li><a href="/wiki/%E7%89%B9%E5%BE%81%E5%80%BC" class="mw-redirect" title="特征值">特征值</a></li> <li><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></li></ul> <ul><li><a href="/wiki/%E7%82%B9%E7%A7%AF" title="点积">数量积</a></li> <li><a href="/wiki/%E5%86%85%E7%A7%AF%E7%A9%BA%E9%97%B4" title="内积空间">内积空间</a></li> <li><a href="/wiki/%E7%82%B9%E4%B9%98" class="mw-redirect" title="点乘">点乘</a></li> <li><a href="/wiki/%E8%BD%89%E7%BD%AE" class="mw-redirect" title="轉置">轉置</a></li> <li><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></li> <li><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></li> <li><a href="/wiki/%E5%85%8B%E8%90%8A%E5%A7%86%E6%B3%95%E5%89%87" title="克萊姆法則">克萊姆法則</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">矩阵</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a></li> <li><a href="/wiki/%E7%9F%A9%E9%99%A3%E4%B9%98%E6%B3%95" title="矩陣乘法">矩陣乘法</a></li> <li><a class="mw-selflink selflink">矩阵分解</a></li> <li><a href="/wiki/%E8%A1%8C%E5%88%97%E5%BC%8F" title="行列式">行列式</a></li> <li><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></li> <li><a href="/wiki/%E7%A7%A9_(%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0)" title="秩 (线性代数)">矩阵的秩</a></li> <li><a href="/wiki/%E5%85%8B%E8%90%8A%E5%A7%86%E6%B3%95%E5%89%87" title="克萊姆法則">克萊姆法則</a></li> <li><a href="/wiki/%E9%80%86%E7%9F%A9%E9%98%B5" title="逆矩阵">逆矩阵</a></li> <li><a href="/wiki/%E9%AB%98%E6%96%AF%E6%B6%88%E5%8E%BB%E6%B3%95" title="高斯消去法">高斯消去法</a></li> <li><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></li> <li><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></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E6%95%B0%E5%80%BC%E7%BA%BF%E6%80%A7%E4%BB%A3%E6%95%B0" title="数值线性代数">数值线性代数</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%B5%AE%E7%82%B9%E6%95%B0%E8%BF%90%E7%AE%97" title="浮点数运算">浮点数</a></li> <li><a href="/wiki/%E6%95%B0%E5%80%BC%E7%A8%B3%E5%AE%9A%E6%80%A7" title="数值稳定性">数值稳定性</a></li> <li><a href="/wiki/BLAS" title="BLAS">基础线性代数程序集</a></li> <li><a href="/wiki/%E7%A8%80%E7%96%8F%E7%9F%A9%E9%98%B5" title="稀疏矩阵">稀疏矩阵</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" href="https://zh.wikipedia.org/w/index.php?title=矩阵分解&oldid=81928005">https://zh.wikipedia.org/w/index.php?title=矩阵分解&oldid=81928005</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB" title="Special:页面分类">分类</a>：<ul><li><a href="/wiki/Category:%E7%9F%A9%E9%98%B5%E5%88%86%E8%A7%A3" title="Category:矩阵分解">矩阵分解</a></li><li><a href="/w/index.php?title=Category:%E4%BF%A1%E6%81%AF%E8%8E%B7%E5%8F%96%E6%8A%80%E6%9C%AF&action=edit&redlink=1" 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