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阶梯形矩阵 - 维基百科，自由的百科全书
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class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?utm_source=donate&utm_medium=sidebar&utm_campaign=spontaneous&uselang=zh-hans"><span>资助维基百科</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a 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"><!-- CentralNotice --></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-简化-{zh-hans:列;_zh-hant:列;}-阶梯形矩阵" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#简化-{zh-hans:列;_zh-hant:列;}-阶梯形矩阵"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>简化列阶梯形矩阵</span> </div> </a> <ul id="toc-简化-{zh-hans:列;_zh-hant:列;}-阶梯形矩阵-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-矩阵变换到-{zh-hans:行;_zh-hant:列;}-阶梯形矩阵" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#矩阵变换到-{zh-hans:行;_zh-hant:列;}-阶梯形矩阵"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>矩阵变换到行阶梯形矩阵</span> </div> </a> <ul id="toc-矩阵变换到-{zh-hans:行;_zh-hant:列;}-阶梯形矩阵-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-线性方程组" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#线性方程组"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>线性方程组</span> </div> </a> <ul id="toc-线性方程组-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-一些示例" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#一些示例"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>一些示例</span> </div> </a> <ul id="toc-一些示例-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">5</span> <span>参见</span> </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考来源" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考来源"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>参考来源</span> </div> </a> <ul id="toc-参考来源-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">阶梯形矩阵</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。18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-18" 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">18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-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matriu_esglaonada" title="Matriu esglaonada – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Matriu esglaonada" 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%9A%D0%B0%D1%80%D1%82%D0%BB%D0%B0%C3%A7%D0%BB%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Картлаçла матрица – 楚瓦什语" lang="cv" hreflang="cv" data-title="Картлаçла 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-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Lineares_Gleichungssystem#Stufenform,_Treppenform" title="Lineares Gleichungssystem – 德语" lang="de" hreflang="de" data-title="Lineares Gleichungssystem" 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%9A%CE%BB%CE%B9%CE%BC%CE%B1%CE%BA%CF%89%CF%84%CE%AE_%CE%BC%CE%BF%CF%81%CF%86%CE%AE" title="Κλιμακωτή μορφή – 希腊语" lang="el" hreflang="el" data-title="Κλιμακωτή μορφή" 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/Row_echelon_form" title="Row echelon form – 英语" lang="en" hreflang="en" data-title="Row echelon form" 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/Matriz_escalonada" title="Matriz escalonada – 西班牙语" lang="es" hreflang="es" data-title="Matriz escalonada" 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%B4%DA%A9%D9%84_%D8%B3%D8%B7%D8%B1%DB%8C_%D9%BE%D9%84%DA%A9%D8%A7%D9%86%DB%8C" title="شکل سطری پلکانی – 波斯语" lang="fa" hreflang="fa" data-title="شکل سطری پلکانی" 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/Matrice_%C3%A9chelonn%C3%A9e" title="Matrice échelonnée – 法语" lang="fr" hreflang="fr" data-title="Matrice échelonnée" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Stalla%C3%B0_form" title="Stallað form – 冰岛语" lang="is" hreflang="is" data-title="Stallað form" 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-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A1%8C%E9%9A%8E%E6%AE%B5%E5%BD%A2" title="行階段形 – 日语" lang="ja" hreflang="ja" data-title="行階段形" 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/%EC%82%AC%EB%8B%A4%EB%A6%AC%EA%BC%B4%ED%96%89%EB%A0%AC" title="사다리꼴행렬 – 韩语" lang="ko" hreflang="ko" data-title="사다리꼴행렬" 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/Echelonvorm" title="Echelonvorm – 荷兰语" lang="nl" hreflang="nl" data-title="Echelonvorm" 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/Macierz_schodkowa" title="Macierz schodkowa – 波兰语" lang="pl" hreflang="pl" data-title="Macierz schodkowa" 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-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D1%83%D0%BF%D0%B5%D0%BD%D1%87%D0%B0%D1%82%D1%8B%D0%B9_%D0%B2%D0%B8%D0%B4_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8B" title="Ступенчатый вид матрицы – 俄语" lang="ru" hreflang="ru" data-title="Ступенчатый вид 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/Trappstegsform" title="Trappstegsform – 瑞典语" lang="sv" hreflang="sv" data-title="Trappstegsform" 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-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Sat%C4%B1r_basamak_formu" title="Satır basamak formu – 土耳其语" lang="tr" hreflang="tr" data-title="Satır basamak formu" 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%A0%D1%8F%D0%B4%D0%BA%D0%BE%D0%B2%D0%B0_%D1%81%D1%82%D1%83%D0%BF%D1%96%D0%BD%D1%87%D0%B0%D1%81%D1%82%D0%B0_%D1%84%D0%BE%D1%80%D0%BC%D0%B0" title="Рядкова ступінчаста форма – 乌克兰语" lang="uk" hreflang="uk" data-title="Рядкова ступінчаста форма" 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/D%E1%BA%A1ng_h%C3%A0ng_b%E1%BA%ADc_thang" title="Dạng hàng bậc thang – 越南语" lang="vi" hreflang="vi" data-title="Dạng hàng bậc thang" 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 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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 id="mw-indicator-noteTA-307af9e9" class="mw-indicator"><div class="mw-parser-output"><span class="skin-invert" typeof="mw:File"><span title="本页使用了标题或全文手工转换"><img alt="本页使用了标题或全文手工转换" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/35px-Zh_conversion_icon_m.svg.png" decoding="async" width="35" height="22" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/53px-Zh_conversion_icon_m.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Zh_conversion_icon_m.svg/70px-Zh_conversion_icon_m.svg.png 2x" data-file-width="32" data-file-height="20" /></span></span></div></div> </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"><div id="noteTA-307af9e9" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Math"></div></div></div> <p>线性代数中，一個<a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a>如果符合下列條件的話，我們稱之為<b>行阶梯形矩阵</b>或<b>行梯形式矩阵</b>（英語：<span lang="en">Row Echelon Form</span>）： </p> <ul><li>若某行有个非零元素，則必在任何全零行之上。</li> <li>某行最左边的（即第一個）非零元素稱為首项系数（<span lang="en">leading coefficient</span>）。某列的首项系数必定比上一列的首项系数更靠右（某些版本會要求非零行的首项系数必須是1<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>）。</li></ul> <p>因為首项系数要不是最靠右的，要不就是左邊都是零，所以根據上面二點，在首项系数所在的列中，在首项系数下面的元素都會是零。 </p><p>这个3×4矩阵是行阶梯形矩阵： </p><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 \left[{\begin{array}{ccc|c}1&a_{1}&a_{2}&b_{1}\\0&2&a_{3}&b_{2}\\0&0&1&b_{3}\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{ccc|c}1&a_{1}&a_{2}&b_{1}\\0&2&a_{3}&b_{2}\\0&0&1&b_{3}\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/badc06dd95232274f69fa392481f1f1292842886" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.46ex; height:10.176ex;" alt="{\displaystyle \left[{\begin{array}{ccc|c}1&a_{1}&a_{2}&b_{1}\\0&2&a_{3}&b_{2}\\0&0&1&b_{3}\end{array}}\right]}"></span> </p><p>有時候，<a href="/wiki/%E5%A2%9E%E5%B9%BF%E7%9F%A9%E9%98%B5" title="增广矩阵">增廣矩陣</a>右邊的直線也會省略。 </p><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 \left[{\begin{array}{cccc}1&a_{1}&a_{2}&b_{1}\\0&2&a_{3}&b_{2}\\0&0&1&b_{3}\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{cccc}1&a_{1}&a_{2}&b_{1}\\0&2&a_{3}&b_{2}\\0&0&1&b_{3}\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/350b679f2bac92c41ee31058bd36ba7542172a3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:18.602ex; height:9.176ex;" alt="{\displaystyle \left[{\begin{array}{cccc}1&a_{1}&a_{2}&b_{1}\\0&2&a_{3}&b_{2}\\0&0&1&b_{3}\end{array}}\right]}"></span> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="简化-{zh-hans:列;_zh-hant:列;}-阶梯形矩阵"><span id=".E7.AE.80.E5.8C.96-.7Bzh-hans:.E5.88.97.3B_zh-hant:.E5.88.97.3B.7D-.E9.98.B6.E6.A2.AF.E5.BD.A2.E7.9F.A9.E9.98.B5"></span>简化列阶梯形矩阵</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5&action=edit&section=1" title="编辑章节：简化列阶梯形矩阵"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>简化列阶梯形矩阵</b>或<b>簡約行梯形式矩陣</b>（<span lang="en">reduced row echelon form</span>），也称作<b>行规范形矩阵</b>（<span lang="en">row canonical form</span>），如果满足额外的条件： </p> <ul><li>每个首项系数是1，且是其所在列的唯一的非零元素。例如：</li></ul> <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 \left[{\begin{array}{cccc|c}1&0&a_{1}&0&b_{1}\\0&1&a_{2}&0&b_{2}\\0&0&0&1&b_{3}\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center center" rowspacing="4pt" columnspacing="1em" columnlines="none none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{cccc|c}1&0&a_{1}&0&b_{1}\\0&1&a_{2}&0&b_{2}\\0&0&0&1&b_{3}\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/530b9a218da41b816d92865b882e6b89e595311e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:22.824ex; height:10.176ex;" alt="{\displaystyle \left[{\begin{array}{cccc|c}1&0&a_{1}&0&b_{1}\\0&1&a_{2}&0&b_{2}\\0&0&0&1&b_{3}\end{array}}\right]}"></span> </p><p>注意，这并不意味着简化列阶梯形矩阵的左部总是单位阵。例如，如下的矩阵是简化行阶梯形矩阵： </p><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 \left[{\begin{array}{cccc|c}1&0&1/2&0&b_{1}\\0&1&-1/3&0&b_{2}\\0&0&0&1&b_{3}\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center center" rowspacing="4pt" columnspacing="1em" columnlines="none none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>b</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{cccc|c}1&0&1/2&0&b_{1}\\0&1&-1/3&0&b_{2}\\0&0&0&1&b_{3}\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/640d36cafff94579ed7aac7a7751b870ce7e9dff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:25.835ex; height:10.509ex;" alt="{\displaystyle \left[{\begin{array}{cccc|c}1&0&1/2&0&b_{1}\\0&1&-1/3&0&b_{2}\\0&0&0&1&b_{3}\end{array}}\right]}"></span> </p><p>因为第3列并不包含任何列的首项系数。 </p> <div class="mw-heading mw-heading2"><h2 id="矩阵变换到-{zh-hans:行;_zh-hant:列;}-阶梯形矩阵"><span id=".E7.9F.A9.E9.98.B5.E5.8F.98.E6.8D.A2.E5.88.B0-.7Bzh-hans:.E8.A1.8C.3B_zh-hant:.E5.88.97.3B.7D-.E9.98.B6.E6.A2.AF.E5.BD.A2.E7.9F.A9.E9.98.B5"></span>矩阵变换到行阶梯形矩阵</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5&action=edit&section=2" title="编辑章节：矩阵变换到行阶梯形矩阵"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>通过有限步的<a href="/wiki/%E5%88%9D%E7%AD%89%E7%9F%A9%E9%98%B5" title="初等矩阵">行初等变换</a>，任何矩阵可以变换为行阶梯形矩阵。由于行初等变换保持了矩阵的<a href="/wiki/%E8%A1%8C%E7%A9%BA%E9%96%93%E8%88%87%E5%88%97%E7%A9%BA%E9%96%93" class="mw-redirect" title="行空間與列空間">行空间</a>，因此行阶梯形矩阵的行空间与变换前的原矩阵的行空间相同。 </p><p>行阶梯形矩阵的结果并不是唯一的。例如，行阶梯形矩阵乘以一个标量系数仍然是行阶梯形矩阵。但是，可以证明一个矩阵的简化行阶梯形矩阵是唯一的。 </p> <div class="mw-heading mw-heading2"><h2 id="线性方程组"><span id=".E7.BA.BF.E6.80.A7.E6.96.B9.E7.A8.8B.E7.BB.84"></span>线性方程组</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5&action=edit&section=3" title="编辑章节：线性方程组"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>如果一个<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>的<a href="/wiki/%E5%A2%9E%E5%B9%BF%E7%9F%A9%E9%98%B5" title="增广矩阵">增广矩阵</a>是行阶梯形矩阵，則其係數矩陣也是<b>行阶梯形矩阵</b>。类似的，如果一个线性方程组的增广矩阵是简化行阶梯形矩阵，則其係數矩陣也是<b>简化行阶梯形矩阵</b>。 </p> <div class="mw-heading mw-heading2"><h2 id="一些示例"><span id=".E4.B8.80.E4.BA.9B.E7.A4.BA.E4.BE.8B"></span>一些示例</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5&action=edit&section=4" title="编辑章节：一些示例"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>定义： <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{array}{ccc|c}1&a_{1}&a_{2}&a_{3}\\0&1&a_{4}&a_{5}\\0&0&1&a_{6}\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>5</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>6</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{ccc|c}1&a_{1}&a_{2}&a_{3}\\0&1&a_{4}&a_{5}\\0&0&1&a_{6}\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecdbc2f2d71e8c1e9bc1e63c755ab4bc8f240343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:20.693ex; height:10.176ex;" alt="{\displaystyle \left[{\begin{array}{ccc|c}1&a_{1}&a_{2}&a_{3}\\0&1&a_{4}&a_{5}\\0&0&1&a_{6}\end{array}}\right]}"></span> </p><p>例子： <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{array}{cccc}1&8&9\\0&1&2\\0&0&1\end{array}}\right]\left[{\begin{array}{cccc}1&-6&33\\0&1&0\\0&0&0\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>9</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>6</mn> </mtd> <mtd> <mn>33</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{cccc}1&8&9\\0&1&2\\0&0&1\end{array}}\right]\left[{\begin{array}{cccc}1&-6&33\\0&1&0\\0&0&0\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52074a1d6c94f669609bc7937a98536ebc7f39d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:27.327ex; height:9.176ex;" alt="{\displaystyle \left[{\begin{array}{cccc}1&8&9\\0&1&2\\0&0&1\end{array}}\right]\left[{\begin{array}{cccc}1&-6&33\\0&1&0\\0&0&0\end{array}}\right]}"></span> </p><p>错误示例： <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{array}{ccc|c}1&0&0&-8\\0&1&0&0\\0&4&1&26\end{array}}\right]\left[{\begin{array}{ccc}1&-29&3\\0&0&0\\0&0&1\end{array}}\right]\left[{\begin{array}{ccc}0&1&2\\1&6&7\\0&0&1\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center" rowspacing="4pt" columnspacing="1em" columnlines="none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>8</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>26</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>29</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>6</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{ccc|c}1&0&0&-8\\0&1&0&0\\0&4&1&26\end{array}}\right]\left[{\begin{array}{ccc}1&-29&3\\0&0&0\\0&0&1\end{array}}\right]\left[{\begin{array}{ccc}0&1&2\\1&6&7\\0&0&1\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb26dc6ddea29e9db3dc03641ed371682a633d5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:46.85ex; height:10.176ex;" alt="{\displaystyle \left[{\begin{array}{ccc|c}1&0&0&-8\\0&1&0&0\\0&4&1&26\end{array}}\right]\left[{\begin{array}{ccc}1&-29&3\\0&0&0\\0&0&1\end{array}}\right]\left[{\begin{array}{ccc}0&1&2\\1&6&7\\0&0&1\end{array}}\right]}"></span> </p><p>注： </p> <ul><li>矩阵1：第二列的第一非零项1的下方的列项不全为零（有非零项4），见定义第二条，所以不是行阶梯型矩阵。</li></ul> <ul><li>矩阵2：全为零的行应该在非全为零行的下方，见定义第三条，所以不是行阶梯型矩阵。</li></ul> <ul><li>矩阵3：k+1行比k行的第一个非零项之前的0少，见定义第三条，所以不是行阶梯型矩阵。</li></ul> <p>简化行阶梯形矩阵的例子： <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}}\right]\left[{\begin{array}{ccccc|c}1&9&0&5&0&17\\0&0&1&0&0&-3\\0&0&0&0&1&32\\0&0&0&0&0&0\end{array}}\right]\left[{\begin{array}{cc}0&0\\0&0\\\end{array}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center center center" rowspacing="4pt" columnspacing="1em" columnlines="none none none none solid"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>9</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>17</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>32</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}}\right]\left[{\begin{array}{ccccc|c}1&9&0&5&0&17\\0&0&1&0&0&-3\\0&0&0&0&1&32\\0&0&0&0&0&0\end{array}}\right]\left[{\begin{array}{cc}0&0\\0&0\\\end{array}}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/568cc39fba1fc09caec583d36289a9f9b1c0c3c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:46.719ex; height:13.509ex;" alt="{\displaystyle \left[{\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}}\right]\left[{\begin{array}{ccccc|c}1&9&0&5&0&17\\0&0&1&0&0&-3\\0&0&0&0&1&32\\0&0&0&0&0&0\end{array}}\right]\left[{\begin{array}{cc}0&0\\0&0\\\end{array}}\right]}"></span> </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=%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5&action=edit&section=5" 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/%E9%AB%98%E6%96%AF%E6%B6%88%E5%85%83%E6%B3%95" class="mw-redirect" title="高斯消元法">高斯消元法</a></li> <li><a href="/wiki/%E9%AB%98%E6%96%AF-%E8%8B%A5%E7%88%BE%E7%95%B6%E6%B6%88%E5%85%83%E6%B3%95" title="高斯-若爾當消元法">高斯-若爾當消元法</a></li> <li><a href="/wiki/%E5%88%9D%E7%AD%89%E8%AE%8A%E6%8F%9B" class="mw-redirect" title="初等變換">初等变换</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="参考来源"><span id=".E5.8F.82.E8.80.83.E6.9D.A5.E6.BA.90"></span>参考来源</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5&action=edit&section=6" title="编辑章节：参考来源"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><cite id="CITEREFLeon2009" class="citation">Leon, Steve, Linear Algebra with Applications 8th, Pearson: 13, 2009, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0136009290" title="Special:网络书源/978-0136009290"><span title="国际标准书号">ISBN</span> 978-0136009290</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E9%98%B6%E6%A2%AF%E5%BD%A2%E7%9F%A9%E9%98%B5&rft.aufirst=Steve&rft.aulast=Leon&rft.btitle=Linear+Algebra+with+Applications&rft.date=2009&rft.edition=8th&rft.genre=book&rft.isbn=978-0136009290&rft.pages=13&rft.pub=Pearson&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120928023406/http://www.pyrtvu.cn/jinjishuxue/lesson/ch09/0903.html">矩阵的初等行变换、阶梯形矩阵与矩阵的秩</a></li> <li><a rel="nofollow" class="external text" href="http://people.revoledu.com/kardi/tutorial/LinearAlgebra/RREF.html">Interactive Row Echelon Form with rational output</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20120111061057/http://people.revoledu.com/kardi/tutorial/LinearAlgebra/RREF.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li></ul> <style data-mw-deduplicate="TemplateStyles:r82655521">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox" style="font-size:small;"><style data-mw-deduplicate="TemplateStyles:r82655520">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/40px-Wikibooks-logo.svg.png" decoding="async" width="40" 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