행렬 - 위키백과, 우리 모두의 백과사전

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id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=ko.wikipedia.org&uselang=ko" class=""><span>기부</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%ED%96%89%EB%A0%AC" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다" class=""><span>계정 만들기</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%ED%96%89%EB%A0%AC" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o" class=""><span>로그인</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="더 많은 옵션" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="개인 도구" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-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-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">개인 도구</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="사용자 메뉴" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=ko.wikipedia.org&uselang=ko"><span>기부</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EA%B3%84%EC%A0%95%EB%A7%8C%EB%93%A4%EA%B8%B0&returnto=%ED%96%89%EB%A0%AC" title="계정을 만들고 로그인하는 것이 좋습니다. 하지만 필수는 아닙니다"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>계정 만들기</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=%ED%8A%B9%EC%88%98:%EB%A1%9C%EA%B7%B8%EC%9D%B8&returnto=%ED%96%89%EB%A0%AC" title="위키백과에 로그인하면 여러가지 편리한 기능을 사용할 수 있습니다. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>로그인</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"> 로그아웃한 편집자를 위한 문서 <a href="/wiki/%EB%8F%84%EC%9B%80%EB%A7%90:%EC%86%8C%EA%B0%9C" aria-label="편집에 관해 더 알아보기"><span>더 알아보기</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/%ED%8A%B9%EC%88%98:%EB%82%B4%EA%B8%B0%EC%97%AC" title="이 IP 주소의 편집 목록 [y]" accesskey="y"><span>기여</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/%ED%8A%B9%EC%88%98:%EB%82%B4%EC%82%AC%EC%9A%A9%EC%9E%90%ED%86%A0%EB%A1%A0" title="현재 사용하는 IP 주소에 대한 토론 문서 [n]" accesskey="n"><span>토론</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="사이트"> <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="목차" 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">목차</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">사이드바로 이동</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">숨기기</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">처음 위치</div> </a> </li> <li id="toc-정의" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#정의"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>정의</span> </div> </a> <button aria-controls="toc-정의-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>정의 하위섹션 토글하기</span> </button> <ul id="toc-정의-sublist" class="vector-toc-list"> <li id="toc-크기" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#크기"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>크기</span> </div> </a> <ul id="toc-크기-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-연산" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#연산"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>연산</span> </div> </a> <button aria-controls="toc-연산-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>연산 하위섹션 토글하기</span> </button> <ul id="toc-연산-sublist" class="vector-toc-list"> <li id="toc-덧셈과_스칼라배" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#덧셈과_스칼라배"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>덧셈과 스칼라배</span> </div> </a> <ul id="toc-덧셈과_스칼라배-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-곱셈" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#곱셈"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>곱셈</span> </div> </a> <ul id="toc-곱셈-sublist" class="vector-toc-list"> <li id="toc-교환_법칙과_소거_법칙의_실패" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#교환_법칙과_소거_법칙의_실패"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>교환 법칙과 소거 법칙의 실패</span> </div> </a> <ul id="toc-교환_법칙과_소거_법칙의_실패-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-역행렬" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#역행렬"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.2</span> <span>역행렬</span> </div> </a> <ul id="toc-역행렬-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-전치_행렬" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#전치_행렬"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>전치 행렬</span> </div> </a> <ul id="toc-전치_행렬-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-대각합" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#대각합"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.4</span> <span>대각합</span> </div> </a> <ul id="toc-대각합-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-행렬식" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#행렬식"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.5</span> <span>행렬식</span> </div> </a> <ul id="toc-행렬식-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-부분_행렬과_소행렬식" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#부분_행렬과_소행렬식"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.6</span> <span>부분 행렬과 소행렬식</span> </div> </a> <ul id="toc-부분_행렬과_소행렬식-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-예" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#예"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>예</span> </div> </a> <ul id="toc-예-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-역사와_어원" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#역사와_어원"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>역사와 어원</span> </div> </a> <ul id="toc-역사와_어원-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-같이_보기" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#같이_보기"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>같이 보기</span> </div> </a> <ul id="toc-같이_보기-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-참고_문헌" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#참고_문헌"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>참고 문헌</span> </div> </a> <ul id="toc-참고_문헌-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-외부_링크" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#외부_링크"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>외부 링크</span> </div> </a> <ul id="toc-외부_링크-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="목차" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="목차" > <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="목차 토글" > <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">목차 토글</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">행렬</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="다른 언어로 문서를 방문합니다. 93개 언어로 읽을 수 있습니다" > <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-93" 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">93개 언어</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Matriks" title="Matriks – 아프리칸스어" lang="af" hreflang="af" data-title="Matriks" data-language-autonym="Afrikaans" data-language-local-name="아프리칸스어" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-am mw-list-item"><a href="https://am.wikipedia.org/wiki/%E1%88%9B%E1%89%B5%E1%88%AA%E1%8A%AD%E1%88%B5" title="ማትሪክስ – 암하라어" lang="am" hreflang="am" data-title="ማትሪክስ" data-language-autonym="አማርኛ" data-language-local-name="암하라어" class="interlanguage-link-target"><span>አማርኛ</span></a></li><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_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="مصفوفة (رياضيات) – 아랍어" lang="ar" hreflang="ar" data-title="مصفوفة (رياضيات)" data-language-autonym="العربية" data-language-local-name="아랍어" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ary mw-list-item"><a href="https://ary.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%D9%8A%D8%B3" title="ماتريس – 모로코 아랍어" lang="ary" hreflang="ary" data-title="ماتريس" data-language-autonym="الدارجة" data-language-local-name="모로코 아랍어" class="interlanguage-link-target"><span>الدارجة</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Matris" title="Matris – 아제르바이잔어" lang="az" hreflang="az" data-title="Matris" data-language-autonym="Azərbaycanca" data-language-local-name="아제르바이잔어" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D1%8D%D0%BC%D0%B0%D1%82%D1%8B%D0%BA%D0%B0)" title="Матрыца (матэматыка) – 벨라루스어" lang="be" hreflang="be" data-title="Матрыца (матэматыка)" data-language-autonym="Беларуская" data-language-local-name="벨라루스어" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D1%8B%D1%86%D0%B0" title="Матрыца – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Матрыца" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – 불가리아어" lang="bg" hreflang="bg" data-title="Матрица (математика)" data-language-autonym="Български" data-language-local-name="불가리아어" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AE%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%9F%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%95%E0%A7%8D%E0%A6%B8" title="ম্যাট্রিক্স – 벵골어" lang="bn" hreflang="bn" data-title="ম্যাট্রিক্স" data-language-autonym="বাংলা" data-language-local-name="벵골어" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Matrica_(matematika)" title="Matrica (matematika) – 보스니아어" lang="bs" hreflang="bs" data-title="Matrica (matematika)" data-language-autonym="Bosanski" data-language-local-name="보스니아어" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Matriu_(matem%C3%A0tiques)" title="Matriu (matemàtiques) – 카탈로니아어" lang="ca" hreflang="ca" data-title="Matriu (matemàtiques)" data-language-autonym="Català" data-language-local-name="카탈로니아어" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%DA%A9%D8%B3" title="ماتریکس – 소라니 쿠르드어" lang="ckb" hreflang="ckb" data-title="ماتریکس" data-language-autonym="کوردی" data-language-local-name="소라니 쿠르드어" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matice" title="Matice – 체코어" lang="cs" hreflang="cs" data-title="Matice" data-language-autonym="Čeština" data-language-local-name="체코어" 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%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – 추바시어" lang="cv" hreflang="cv" data-title="Матрица (математика)" data-language-autonym="Чӑвашла" data-language-local-name="추바시어" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Matrics" title="Matrics – 웨일스어" lang="cy" hreflang="cy" data-title="Matrics" data-language-autonym="Cymraeg" data-language-local-name="웨일스어" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Matrix" title="Matrix – 덴마크어" lang="da" hreflang="da" data-title="Matrix" data-language-autonym="Dansk" data-language-local-name="덴마크어" 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/Matrix_(Mathematik)" title="Matrix (Mathematik) – 독일어" lang="de" hreflang="de" data-title="Matrix (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="독일어" 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%A0%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82_(%CE%BC%CE%B1%CE%B8%CE%B7%CE%BC%CE%B1%CF%84%CE%B9%CE%BA%CE%AC)" title="Πίνακας (μαθηματικά) – 그리스어" lang="el" hreflang="el" data-title="Πίνακας (μαθηματικά)" data-language-autonym="Ελληνικά" data-language-local-name="그리스어" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en badge-Q17437798 badge-goodarticle mw-list-item" title="좋은 글"><a href="https://en.wikipedia.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics) – 영어" lang="en" hreflang="en" data-title="Matrix (mathematics)" data-language-autonym="English" data-language-local-name="영어" 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/Matrico" title="Matrico – 에스페란토어" lang="eo" hreflang="eo" data-title="Matrico" data-language-autonym="Esperanto" data-language-local-name="에스페란토어" 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_(matem%C3%A1tica)" title="Matriz (matemática) – 스페인어" lang="es" hreflang="es" data-title="Matriz (matemática)" data-language-autonym="Español" data-language-local-name="스페인어" 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/Maatriks" title="Maatriks – 에스토니아어" lang="et" hreflang="et" data-title="Maatriks" data-language-autonym="Eesti" data-language-local-name="에스토니아어" 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" title="Matrize – 바스크어" lang="eu" hreflang="eu" data-title="Matrize" data-language-autonym="Euskara" data-language-local-name="바스크어" 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" title="ماتریس – 페르시아어" lang="fa" hreflang="fa" data-title="ماتریس" data-language-autonym="فارسی" data-language-local-name="페르시아어" 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/Matriisi" title="Matriisi – 핀란드어" lang="fi" hreflang="fi" data-title="Matriisi" data-language-autonym="Suomi" data-language-local-name="핀란드어" 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_(math%C3%A9matiques)" title="Matrice (mathématiques) – 프랑스어" lang="fr" hreflang="fr" data-title="Matrice (mathématiques)" data-language-autonym="Français" data-language-local-name="프랑스어" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-frr mw-list-item"><a href="https://frr.wikipedia.org/wiki/Maatriks" title="Maatriks – 북부 프리지아어" lang="frr" hreflang="frr" data-title="Maatriks" data-language-autonym="Nordfriisk" data-language-local-name="북부 프리지아어" class="interlanguage-link-target"><span>Nordfriisk</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Maitr%C3%ADs" title="Maitrís – 아일랜드어" lang="ga" hreflang="ga" data-title="Maitrís" data-language-autonym="Gaeilge" data-language-local-name="아일랜드어" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gan mw-list-item"><a href="https://gan.wikipedia.org/wiki/%E8%A1%8C%E5%88%97" title="行列 – 간어" lang="gan" hreflang="gan" data-title="行列" data-language-autonym="贛語" data-language-local-name="간어" class="interlanguage-link-target"><span>贛語</span></a></li><li class="interlanguage-link interwiki-gcr mw-list-item"><a href="https://gcr.wikipedia.org/wiki/Matris_(mat%C3%A9matik)" title="Matris (matématik) – Guianan Creole" lang="gcr" hreflang="gcr" data-title="Matris (matématik)" data-language-autonym="Kriyòl gwiyannen" data-language-local-name="Guianan Creole" class="interlanguage-link-target"><span>Kriyòl gwiyannen</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Matriz_(matem%C3%A1ticas)" title="Matriz (matemáticas) – 갈리시아어" lang="gl" hreflang="gl" data-title="Matriz (matemáticas)" data-language-autonym="Galego" data-language-local-name="갈리시아어" 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" title="מטריצה – 히브리어" lang="he" hreflang="he" data-title="מטריצה" data-language-autonym="עברית" data-language-local-name="히브리어" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%86%E0%A4%B5%E0%A5%8D%E0%A4%AF%E0%A5%82%E0%A4%B9" title="आव्यूह – 힌디어" lang="hi" hreflang="hi" data-title="आव्यूह" data-language-autonym="हिन्दी" data-language-local-name="힌디어" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matrica_(matematika)" title="Matrica (matematika) – 크로아티아어" lang="hr" hreflang="hr" data-title="Matrica (matematika)" data-language-autonym="Hrvatski" data-language-local-name="크로아티아어" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/M%C3%A1trix_(matematika)" title="Mátrix (matematika) – 헝가리어" lang="hu" hreflang="hu" data-title="Mátrix (matematika)" data-language-autonym="Magyar" data-language-local-name="헝가리어" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%BF%D6%80%D5%AB%D6%81" title="Մատրից – 아르메니아어" lang="hy" hreflang="hy" data-title="Մատրից" data-language-autonym="Հայերեն" data-language-local-name="아르메니아어" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Matrice_(mathematica)" title="Matrice (mathematica) – 인터링구아" lang="ia" hreflang="ia" data-title="Matrice (mathematica)" data-language-autonym="Interlingua" data-language-local-name="인터링구아" 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_(matematika)" title="Matriks (matematika) – 인도네시아어" lang="id" hreflang="id" data-title="Matriks (matematika)" data-language-autonym="Bahasa Indonesia" data-language-local-name="인도네시아어" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Matrico_(matematiko)" title="Matrico (matematiko) – 이도어" lang="io" hreflang="io" data-title="Matrico (matematiko)" data-language-autonym="Ido" data-language-local-name="이도어" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Fylki_(st%C3%A6r%C3%B0fr%C3%A6%C3%B0i)" title="Fylki (stærðfræði) – 아이슬란드어" lang="is" hreflang="is" data-title="Fylki (stærðfræði)" data-language-autonym="Íslenska" data-language-local-name="아이슬란드어" 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/Matrice" title="Matrice – 이탈리아어" lang="it" hreflang="it" data-title="Matrice" data-language-autonym="Italiano" data-language-local-name="이탈리아어" 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" title="行列 – 일본어" lang="ja" hreflang="ja" data-title="行列" data-language-autonym="日本語" data-language-local-name="일본어" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%9B%E1%83%90%E1%83%A2%E1%83%A0%E1%83%98%E1%83%AA%E1%83%90_(%E1%83%9B%E1%83%90%E1%83%97%E1%83%94%E1%83%9B%E1%83%90%E1%83%A2%E1%83%98%E1%83%99%E1%83%90)" title="მატრიცა (მათემატიკა) – 조지아어" lang="ka" hreflang="ka" data-title="მატრიცა (მათემატიკა)" data-language-autonym="ქართული" data-language-local-name="조지아어" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – 카자흐어" lang="kk" hreflang="kk" data-title="Матрица (математика)" data-language-autonym="Қазақша" data-language-local-name="카자흐어" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AE%E0%B2%BE%E0%B2%A4%E0%B3%83%E0%B2%95%E0%B3%86%E0%B2%97%E0%B2%B3%E0%B3%81" title="ಮಾತೃಕೆಗಳು – 칸나다어" lang="kn" hreflang="kn" data-title="ಮಾತೃಕೆಗಳು" data-language-autonym="ಕನ್ನಡ" data-language-local-name="칸나다어" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Matrix_(mathematica)" title="Matrix (mathematica) – 라틴어" lang="la" hreflang="la" data-title="Matrix (mathematica)" data-language-autonym="Latina" data-language-local-name="라틴어" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lmo mw-list-item"><a href="https://lmo.wikipedia.org/wiki/Matris" title="Matris – 롬바르드어" lang="lmo" hreflang="lmo" data-title="Matris" data-language-autonym="Lombard" data-language-local-name="롬바르드어" class="interlanguage-link-target"><span>Lombard</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BA%A1%E0%BA%B2%E0%BA%95%E0%BA%A3%E0%BA%B4%E0%BA%81" title="ມາຕຣິກ – 라오어" lang="lo" hreflang="lo" data-title="ມາຕຣິກ" data-language-autonym="ລາວ" data-language-local-name="라오어" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Matrica_(matematika)" title="Matrica (matematika) – 리투아니아어" lang="lt" hreflang="lt" data-title="Matrica (matematika)" data-language-autonym="Lietuvių" data-language-local-name="리투아니아어" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Matrica" title="Matrica – 라트비아어" lang="lv" hreflang="lv" data-title="Matrica" data-language-autonym="Latviešu" data-language-local-name="라트비아어" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-mhr mw-list-item"><a href="https://mhr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B5" title="Матрице – Eastern Mari" lang="mhr" hreflang="mhr" data-title="Матрице" data-language-autonym="Олык марий" data-language-local-name="Eastern Mari" class="interlanguage-link-target"><span>Олык марий</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – 마케도니아어" lang="mk" hreflang="mk" data-title="Матрица (математика)" data-language-autonym="Македонски" data-language-local-name="마케도니아어" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%AE%E0%B4%BE%E0%B4%9F%E0%B5%8D%E0%B4%B0%E0%B4%BF%E0%B4%95%E0%B5%8D%E0%B4%B8%E0%B5%8D" title="മാട്രിക്സ് – 말라얄람어" lang="ml" hreflang="ml" data-title="മാട്രിക്സ്" data-language-autonym="മലയാളം" data-language-local-name="말라얄람어" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Matriks_(matematik)" title="Matriks (matematik) – 말레이어" lang="ms" hreflang="ms" data-title="Matriks (matematik)" data-language-autonym="Bahasa Melayu" data-language-local-name="말레이어" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-my mw-list-item"><a href="https://my.wikipedia.org/wiki/%E1%80%80%E1%80%AD%E1%80%94%E1%80%BA%E1%80%B8%E1%80%A1%E1%80%AF%E1%80%B6" title="ကိန်းအုံ – 버마어" lang="my" hreflang="my" data-title="ကိန်းအုံ" data-language-autonym="မြန်မာဘာသာ" data-language-local-name="버마어" class="interlanguage-link-target"><span>မြန်မာဘာသာ</span></a></li><li class="interlanguage-link interwiki-ne mw-list-item"><a href="https://ne.wikipedia.org/wiki/%E0%A4%AE%E0%A5%87%E0%A4%9F%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%8D%E0%A4%B8" title="मेट्रिक्स – 네팔어" lang="ne" hreflang="ne" data-title="मेट्रिक्स" data-language-autonym="नेपाली" data-language-local-name="네팔어" class="interlanguage-link-target"><span>नेपाली</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Matrix_(wiskunde)" title="Matrix (wiskunde) – 네덜란드어" lang="nl" hreflang="nl" data-title="Matrix (wiskunde)" data-language-autonym="Nederlands" data-language-local-name="네덜란드어" 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/Matrise" title="Matrise – 노르웨이어(니노르스크)" lang="nn" hreflang="nn" data-title="Matrise" data-language-autonym="Norsk nynorsk" data-language-local-name="노르웨이어(니노르스크)" 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/Matrise" title="Matrise – 노르웨이어(보크말)" lang="nb" hreflang="nb" data-title="Matrise" data-language-autonym="Norsk bokmål" data-language-local-name="노르웨이어(보크말)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Tareentaa_(Maatiriksii)" title="Tareentaa (Maatiriksii) – 오로모어" lang="om" hreflang="om" data-title="Tareentaa (Maatiriksii)" data-language-autonym="Oromoo" data-language-local-name="오로모어" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-or mw-list-item"><a href="https://or.wikipedia.org/wiki/%E0%AC%AE%E0%AC%BE%E0%AC%9F%E0%AD%8D%E0%AC%B0%E0%AC%BF%E0%AC%95%E0%AD%8D%E0%AC%B8" title="ମାଟ୍ରିକ୍ସ – 오리야어" lang="or" hreflang="or" data-title="ମାଟ୍ରିକ୍ସ" data-language-autonym="ଓଡ଼ିଆ" data-language-local-name="오리야어" class="interlanguage-link-target"><span>ଓଡ଼ିଆ</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AE%E0%A9%88%E0%A8%9F%E0%A9%8D%E0%A8%B0%E0%A8%BF%E0%A8%95%E0%A8%B8_(%E0%A8%97%E0%A8%A3%E0%A8%BF%E0%A8%A4)" title="ਮੈਟ੍ਰਿਕਸ (ਗਣਿਤ) – 펀잡어" lang="pa" hreflang="pa" data-title="ਮੈਟ੍ਰਿਕਸ (ਗਣਿਤ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="펀잡어" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Macierz" title="Macierz – 폴란드어" lang="pl" hreflang="pl" data-title="Macierz" data-language-autonym="Polski" data-language-local-name="폴란드어" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pms mw-list-item"><a href="https://pms.wikipedia.org/wiki/Matris" title="Matris – Piedmontese" lang="pms" hreflang="pms" data-title="Matris" data-language-autonym="Piemontèis" data-language-local-name="Piedmontese" class="interlanguage-link-target"><span>Piemontèis</span></a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%85%D8%A7%D9%B9%D8%B1%DA%A9%D8%B3_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA)" title="ماٹرکس (ریاضیات) – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ماٹرکس (ریاضیات)" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target"><span>پنجابی</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Matriz_(matem%C3%A1tica)" title="Matriz (matemática) – 포르투갈어" lang="pt" hreflang="pt" data-title="Matriz (matemática)" data-language-autonym="Português" data-language-local-name="포르투갈어" 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" title="Matrice – 루마니아어" lang="ro" hreflang="ro" data-title="Matrice" data-language-autonym="Română" data-language-local-name="루마니아어" 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%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – 러시아어" lang="ru" hreflang="ru" data-title="Матрица (математика)" data-language-autonym="Русский" data-language-local-name="러시아어" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sah mw-list-item"><a href="https://sah.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – 야쿠트어" lang="sah" hreflang="sah" data-title="Матрица (математика)" data-language-autonym="Саха тыла" data-language-local-name="야쿠트어" class="interlanguage-link-target"><span>Саха тыла</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Matrici_(matim%C3%A0tica)" title="Matrici (matimàtica) – 시칠리아어" lang="scn" hreflang="scn" data-title="Matrici (matimàtica)" data-language-autonym="Sicilianu" data-language-local-name="시칠리아어" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics) – 스코틀랜드어" lang="sco" hreflang="sco" data-title="Matrix (mathematics)" data-language-autonym="Scots" data-language-local-name="스코틀랜드어" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Matrica_(matematika)" title="Matrica (matematika) – 세르비아-크로아티아어" lang="sh" hreflang="sh" data-title="Matrica (matematika)" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="세르비아-크로아티아어" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-si badge-Q17437798 badge-goodarticle mw-list-item" title="좋은 글"><a href="https://si.wikipedia.org/wiki/%E0%B6%B1%E0%B7%8A%E2%80%8D%E0%B6%BA%E0%B7%8F%E0%B7%83_(%E0%B6%9C%E0%B6%AB%E0%B7%92%E0%B6%AD%E0%B6%BA)" title="න්‍යාස (ගණිතය) – 싱할라어" lang="si" hreflang="si" data-title="න්‍යාස (ගණිතය)" data-language-autonym="සිංහල" data-language-local-name="싱할라어" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Matrix_(mathematics)" title="Matrix (mathematics) – Simple English" lang="en-simple" hreflang="en-simple" data-title="Matrix (mathematics)" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Matica_(matematika)" title="Matica (matematika) – 슬로바키아어" lang="sk" hreflang="sk" data-title="Matica (matematika)" data-language-autonym="Slovenčina" data-language-local-name="슬로바키아어" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Matrika" title="Matrika – 슬로베니아어" lang="sl" hreflang="sl" data-title="Matrika" data-language-autonym="Slovenščina" data-language-local-name="슬로베니아어" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Taxane" title="Taxane – 소말리아어" lang="so" hreflang="so" data-title="Taxane" data-language-autonym="Soomaaliga" data-language-local-name="소말리아어" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Matrica" title="Matrica – 알바니아어" lang="sq" hreflang="sq" data-title="Matrica" data-language-autonym="Shqip" data-language-local-name="알바니아어" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матрица (математика) – 세르비아어" lang="sr" hreflang="sr" data-title="Матрица (математика)" data-language-autonym="Српски / srpski" data-language-local-name="세르비아어" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matris" title="Matris – 스웨덴어" lang="sv" hreflang="sv" data-title="Matris" data-language-autonym="Svenska" data-language-local-name="스웨덴어" 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%85%E0%AE%A3%E0%AE%BF_(%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4%E0%AE%AE%E0%AF%8D)" title="அணி (கணிதம்) – 타밀어" lang="ta" hreflang="ta" data-title="அணி (கணிதம்)" data-language-autonym="தமிழ்" data-language-local-name="타밀어" 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%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C)" title="เมทริกซ์ (คณิตศาสตร์) – 태국어" lang="th" hreflang="th" data-title="เมทริกซ์ (คณิตศาสตร์)" data-language-autonym="ไทย" data-language-local-name="태국어" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Matris_(matematika)" title="Matris (matematika) – 타갈로그어" lang="tl" hreflang="tl" data-title="Matris (matematika)" data-language-autonym="Tagalog" data-language-local-name="타갈로그어" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matris_(matematik)" title="Matris (matematik) – 튀르키예어" lang="tr" hreflang="tr" data-title="Matris (matematik)" data-language-autonym="Türkçe" data-language-local-name="튀르키예어" 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%9C%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0)" title="Матриця (математика) – 우크라이나어" lang="uk" hreflang="uk" data-title="Матриця (математика)" data-language-autonym="Українська" data-language-local-name="우크라이나어" 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%DB%8C%D9%B9%D8%B1%DA%A9%D8%B3_(%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C)" title="میٹرکس (ریاضی) – 우르두어" lang="ur" hreflang="ur" data-title="میٹرکس (ریاضی)" data-language-autonym="اردو" data-language-local-name="우르두어" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matritsa_matematikada" title="Matritsa matematikada – 우즈베크어" lang="uz" hreflang="uz" data-title="Matritsa matematikada" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="우즈베크어" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Ma_tr%E1%BA%ADn_(to%C3%A1n_h%E1%BB%8Dc)" title="Ma trận (toán học) – 베트남어" lang="vi" hreflang="vi" data-title="Ma trận (toán học)" data-language-autonym="Tiếng Việt" data-language-local-name="베트남어" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E7%9F%A9%E9%98%B5" title="矩阵 – 우어" lang="wuu" hreflang="wuu" data-title="矩阵" data-language-autonym="吴语" data-language-local-name="우어" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh badge-Q17437798 badge-goodarticle mw-list-item" title="좋은 글"><a href="https://zh.wikipedia.org/wiki/%E7%9F%A9%E9%98%B5" title="矩阵 – 중국어" lang="zh" hreflang="zh" data-title="矩阵" data-language-autonym="中文" data-language-local-name="중국어" class="interlanguage-link-target"><span>中文</span></a></li><li class="interlanguage-link interwiki-zh-classical mw-list-item"><a href="https://zh-classical.wikipedia.org/wiki/%E7%9F%A9%E9%99%A3" title="矩陣 – Literary Chinese" lang="lzh" hreflang="lzh" data-title="矩陣" data-language-autonym="文言" data-language-local-name="Literary Chinese" class="interlanguage-link-target"><span>文言</span></a></li><li class="interlanguage-link interwiki-zh-min-nan mw-list-item"><a href="https://zh-min-nan.wikipedia.org/wiki/H%C3%A2ng-lia%CC%8Dt" title="Hâng-lia̍t – 민난어" lang="nan" hreflang="nan" data-title="Hâng-lia̍t" data-language-autonym="閩南語 / Bân-lâm-gú" data-language-local-name="민난어" class="interlanguage-link-target"><span>閩南語 / Bân-lâm-gú</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E7%9F%A9%E9%99%A3" title="矩陣 – 광둥어" lang="yue" hreflang="yue" data-title="矩陣" data-language-autonym="粵語" data-language-local-name="광둥어" class="interlanguage-link-target"><span>粵語</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/Q44337#sitelinks-wikipedia" title="언어 간 링크 편집" class="wbc-editpage">링크 편집</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="이름공간"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%ED%96%89%EB%A0%AC" 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<b>행렬론</b>은 여기로 연결됩니다. 이론물리학 용어에 대해서는 <a href="/wiki/%ED%96%89%EB%A0%AC_%EC%9D%B4%EB%A1%A0" title="행렬 이론">행렬 이론</a> 문서를 참고하십시오.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Matrix_ko.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Matrix_ko.svg/220px-Matrix_ko.svg.png" decoding="async" width="220" height="199" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Matrix_ko.svg/330px-Matrix_ko.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4d/Matrix_ko.svg/440px-Matrix_ko.svg.png 2x" data-file-width="243" data-file-height="220" /></a><figcaption>행렬의 각 성분은 보통 그 행과 열의 번째수를 나타내는 첨자로 표기한다. 예를 들어, 행렬 <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>의 3번째 행의 2번째 열에 있는 성분은 <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_{32}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{32}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7963f39d82cffee051a6599222e1ab49aff45c0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.106ex; height:2.009ex;" alt="{\displaystyle a_{32}}"></span>이다.</figcaption></figure> <p><a href="/wiki/%EC%88%98%ED%95%99" title="수학">수학</a>에서 <b>행렬</b>(行列, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">matrix</span>)은 <a href="/wiki/%EC%88%98_(%EC%88%98%ED%95%99)" title="수 (수학)">수</a> 또는 <a href="/wiki/%EB%8B%A4%ED%95%AD%EC%8B%9D" title="다항식">다항식</a> 등을 <a href="/wiki/%EC%A7%81%EC%82%AC%EA%B0%81%ED%98%95" title="직사각형">직사각형</a> 모양으로 <a href="/wiki/%EB%B0%B0%EC%97%B4" title="배열">배열</a>한 것이다.<sup id="cite_ref-Lang_1-0" class="reference"><a href="#cite_note-Lang-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Kharab_2-0" class="reference"><a href="#cite_note-Kharab-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 예를 들어, 실수 1, 9, −13, 20, 5, −16을 2×3 직사각형 위에 배열한 행렬은 다음과 같다. </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{pmatrix}1&9&-13\\20&5&-16\end{pmatrix}}}"> <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>9</mn> </mtd> <mtd> <mo>−</mo> <mn>13</mn> </mtd> </mtr> <mtr> <mtd> <mn>20</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mo>−</mo> <mn>16</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&9&-13\\20&5&-16\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71076d72176f8e48cbb51ad5b14b637c6ee7832b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.438ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&9&-13\\20&5&-16\end{pmatrix}}}"></span></dd></dl> <p>행렬에는 덧셈과 <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_%EA%B3%B1%EC%85%88" title="스칼라 곱셈">스칼라배</a>, 곱셈 연산이 존재한다. 크기가 같은 두 행렬은 같은 위치의 성분별로 더할 수 있으며, 첫째 행렬의 열과 둘째 행렬의 행의 수가 같은 두 행렬은 첫째 행렬의 각 행벡터와 둘째 행렬의 각 열벡터의 <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" title="스칼라곱">스칼라곱</a>을 통해 곱할 수 있다. 곱셈의 <a href="/wiki/%EA%B5%90%ED%99%98_%EB%B2%95%EC%B9%99" class="mw-redirect" title="교환 법칙">교환 법칙</a>이나 <a href="/wiki/%EC%86%8C%EA%B1%B0_%EB%B2%95%EC%B9%99" title="소거 법칙">소거 법칙</a> 등 <a href="/wiki/%EB%B3%B5%EC%86%8C%EC%88%98" title="복소수">복소수</a>의 일부 성질들은 행렬 연산에서 더 이상 성립하지 않는다. <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a> 위의 유한 차원 <a href="/wiki/%EC%9E%90%EC%9C%A0_%EA%B0%80%EA%B5%B0" title="자유 가군">자유 가군</a>(특히, <a href="/wiki/%EC%B2%B4_(%EC%88%98%ED%95%99)" title="체 (수학)">체</a> 위의 유한 차원 <a href="/wiki/%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="벡터 공간">벡터 공간</a>)의 <a href="/wiki/%EC%84%A0%ED%98%95_%EB%B3%80%ED%99%98" title="선형 변환">선형 변환</a>을 행렬로 유일하게 표현할 수 있으며, 이는 행렬의 중요한 응용이다. 예를 들어, 3차원 <a href="/wiki/%EC%9C%A0%ED%81%B4%EB%A6%AC%EB%93%9C_%EA%B3%B5%EA%B0%84" title="유클리드 공간">유클리드 공간</a>의 <a href="/wiki/%ED%9A%8C%EC%A0%84" title="회전">회전</a>은 <a href="/wiki/%ED%9A%8C%EC%A0%84_%ED%96%89%EB%A0%AC" class="mw-redirect" title="회전 행렬">회전 행렬</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>을 각 열벡터 <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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}"></span>에 곱하여 새 열벡터 <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 Rv}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Rv}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0805cf0a391155521e3b2c41b59e2405ccda49a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.892ex; height:2.176ex;" alt="{\displaystyle Rv}"></span>를 얻는 함수이다. 행렬의 덧셈과 스칼라배는 선형 변환의 점별 덧셈과 점별 스칼라배, 행렬의 곱셈은 선형 변환의 <a href="/wiki/%ED%95%A8%EC%88%98%EC%9D%98_%ED%95%A9%EC%84%B1" title="함수의 합성">합성</a>에 대응한다. 행렬은 <a href="/wiki/%EA%B0%80%EC%9A%B0%EC%8A%A4_%EC%86%8C%EA%B1%B0%EB%B2%95" title="가우스 소거법">가우스 소거법</a> 등 <a href="/wiki/%EC%97%B0%EB%A6%BD_%EC%9D%BC%EC%B0%A8_%EB%B0%A9%EC%A0%95%EC%8B%9D" title="연립 일차 방정식">연립 일차 방정식</a>의 풀이에도 응용된다.<sup id="cite_ref-Kharab_2-1" class="reference"><a href="#cite_note-Kharab-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:97</sup></span> <a href="/wiki/%EC%A0%95%EC%82%AC%EA%B0%81_%ED%96%89%EB%A0%AC" class="mw-redirect" title="정사각 행렬">정사각 행렬</a>과 그 선형 변환의 일부 성질들은 그 <a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a> 또는 <a href="/wiki/%EA%B3%A0%EC%9C%B3%EA%B0%92" class="mw-redirect" title="고윳값">고윳값</a>과 <a href="/wiki/%EA%B3%A0%EC%9C%A0_%EB%B2%A1%ED%84%B0" class="mw-redirect" title="고유 벡터">고유 벡터</a>에서 반영된다. 예를 들어, <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a>의 원소를 성분으로 하는 행렬이 <a href="/wiki/%EC%97%AD%ED%96%89%EB%A0%AC" class="mw-redirect" title="역행렬">역행렬</a>을 가질 <a href="/wiki/%ED%95%84%EC%9A%94_%EC%B6%A9%EB%B6%84_%EC%A1%B0%EA%B1%B4" class="mw-redirect" title="필요 충분 조건">필요 충분 조건</a>은 행렬식이 <a href="/wiki/%EA%B0%80%EC%97%AD%EC%9B%90" title="가역원">가역원</a>인 것이며, 특히 <a href="/wiki/%EC%B2%B4_(%EC%88%98%ED%95%99)" title="체 (수학)">체</a>의 경우 필요 충분 조건은 행렬식이 0이 아닌 것이다. </p><p>행렬은 과학과 수학의 수많은 분야에서 다양한 응용이 있다. <a href="/wiki/%EB%AC%BC%EB%A6%AC%ED%95%99" title="물리학">물리학</a>의 <a href="/wiki/%EC%A0%84%EA%B8%B0_%ED%9A%8C%EB%A1%9C" title="전기 회로">전기 회로</a> 이론, <a href="/wiki/%EA%B3%A0%EC%A0%84%EC%97%AD%ED%95%99" title="고전역학">고전역학</a>, <a href="/wiki/%EA%B4%91%ED%95%99" title="광학">광학</a>, <a href="/wiki/%EC%A0%84%EC%9E%90%EA%B8%B0%ED%95%99" title="전자기학">전자기학</a>, <a href="/wiki/%EC%96%91%EC%9E%90%EC%97%AD%ED%95%99" title="양자역학">양자역학</a>, <a href="/wiki/%EC%96%91%EC%9E%90_%EC%A0%84%EA%B8%B0%EC%97%AD%ED%95%99" title="양자 전기역학">양자 전기역학</a> 등 분야에서 응용되며, <a href="/wiki/%EC%BB%B4%ED%93%A8%ED%84%B0_%EA%B7%B8%EB%9E%98%ED%94%BD%EC%8A%A4" title="컴퓨터 그래픽스">컴퓨터 그래픽스</a>에서 3차원 이미지를 2차원 평면에 투영하거나 사실적인 움직임을 그려내기 위해 사용한다. <a href="/wiki/%ED%99%95%EB%A5%A0%EB%A1%A0" title="확률론">확률론</a>과 <a href="/wiki/%ED%86%B5%EA%B3%84%ED%95%99" title="통계학">통계학</a>의 <a href="/wiki/%EB%A7%88%EB%A5%B4%EC%BD%94%ED%94%84_%ED%96%89%EB%A0%AC" title="마르코프 행렬">마르코프 행렬</a>과 <a href="/wiki/%EB%8B%A4%EB%B3%80%EC%88%98_%EB%AF%B8%EC%A0%81%EB%B6%84%ED%95%99" class="mw-redirect" title="다변수 미적분학">다변수 미적분학</a>의 <a href="/wiki/%ED%97%A4%EC%84%B8_%ED%96%89%EB%A0%AC" title="헤세 행렬">헤세 행렬</a> 등 역시 행렬의 응용이다. 행렬 계산은 <a href="/wiki/%EC%88%98%EC%B9%98%ED%95%B4%EC%84%9D%ED%95%99" title="수치해석학">수치해석학</a>의 중요한 문제 중 하나이다. <a href="/wiki/%ED%96%89%EB%A0%AC_%EB%B6%84%ED%95%B4" title="행렬 분해">행렬 분해</a>는 행렬 계산을 이론과 실제 응용에서 모두 단순화할 수 있다. <a href="/wiki/%ED%9D%AC%EC%86%8C%ED%96%89%EB%A0%AC" class="mw-redirect" title="희소행렬">희소행렬</a>, <a href="/wiki/%EB%9D%A0%ED%96%89%EB%A0%AC" title="띠행렬">띠행렬</a> 등 널리 사용되는 특수한 구조의 행렬들의 경우 특화된 고속 알고리즘들이 존재한다. <a href="/wiki/%EC%B2%9C%EC%B2%B4%EB%AC%BC%EB%A6%AC%ED%95%99" title="천체물리학">천체물리학</a>과 <a href="/wiki/%EC%96%91%EC%9E%90%EB%AC%BC%EB%A6%AC%ED%95%99" class="mw-redirect" title="양자물리학">양자물리학</a> 등 분야에서는 무한 행렬도 등장한다. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="정의"><span id=".EC.A0.95.EC.9D.98"></span>정의</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=1" title="부분 편집: 정의"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> <b>행렬</b>은 각 행 <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\in \{1,\dotsc ,m\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\dotsc ,m\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7a25137e9baaf70273ac4d1c5fa1a23b13b4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.349ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\dotsc ,m\}}"></span> 및 열 <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 j\in \{1,\dotsc ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\in \{1,\dotsc ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa124a2f23e6c9ce001436dc1fc4bfb05ebd09e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:13.886ex; height:2.843ex;" alt="{\displaystyle j\in \{1,\dotsc ,n\}}"></span>의 <a href="/wiki/%EC%88%9C%EC%84%9C%EC%8C%8D" title="순서쌍">순서쌍</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (i,j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ef21910f980c6fca2b15bee102a7a0d861ed712" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.604ex; height:2.843ex;" alt="{\displaystyle (i,j)}"></span>에 환의 원소 <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}\in R}"> <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> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{ij}\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9136c3c68965eb06d8aa4ecef2172afaa494be9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.825ex; height:2.843ex;" alt="{\displaystyle A_{ij}\in R}"></span>를 대응시키는 <a href="/wiki/%ED%95%A8%EC%88%98" title="함수">함수</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=(A_{ij})_{i,j}}"> <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> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=(A_{ij})_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1a57db6e9297f99cc484327c8e3481707d9e60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.806ex; height:3.009ex;" alt="{\displaystyle A=(A_{ij})_{i,j}}"></span>이다.<sup id="cite_ref-Kharab_2-2" class="reference"><a href="#cite_note-Kharab-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:98</sup></span> </p><p>행렬 <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>는 모든 성분을 직사각형으로 배열한 다음 <a href="/wiki/%EC%86%8C%EA%B4%84%ED%98%B8" class="mw-redirect" title="소괄호">소괄호</a> 또는 <a href="/wiki/%EB%8C%80%EA%B4%84%ED%98%B8" class="mw-redirect" title="대괄호">대괄호</a>를 추가하여 </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{pmatrix}A_{11}&A_{12}&A_{13}&\cdots &A_{1n}\\A_{21}&A_{22}&A_{23}&\cdots &A_{2n}\\A_{31}&A_{32}&A_{33}&\cdots &A_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\A_{m1}&A_{m2}&A_{m3}&\cdots &A_{mn}\end{pmatrix}}}"> <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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}A_{11}&A_{12}&A_{13}&\cdots &A_{1n}\\A_{21}&A_{22}&A_{23}&\cdots &A_{2n}\\A_{31}&A_{32}&A_{33}&\cdots &A_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\A_{m1}&A_{m2}&A_{m3}&\cdots &A_{mn}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5612d8946cc5adad19b5383379dfea7506d5581d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:34.212ex; height:17.509ex;" alt="{\displaystyle {\begin{pmatrix}A_{11}&A_{12}&A_{13}&\cdots &A_{1n}\\A_{21}&A_{22}&A_{23}&\cdots &A_{2n}\\A_{31}&A_{32}&A_{33}&\cdots &A_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\A_{m1}&A_{m2}&A_{m3}&\cdots &A_{mn}\end{pmatrix}}}"></span></dd></dl> <p>또는 </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}A_{11}&A_{12}&A_{13}&\cdots &A_{1n}\\A_{21}&A_{22}&A_{23}&\cdots &A_{2n}\\A_{31}&A_{32}&A_{33}&\cdots &A_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\A_{m1}&A_{m2}&A_{m3}&\cdots &A_{mn}\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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>13</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>23</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>31</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>32</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>33</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}A_{11}&A_{12}&A_{13}&\cdots &A_{1n}\\A_{21}&A_{22}&A_{23}&\cdots &A_{2n}\\A_{31}&A_{32}&A_{33}&\cdots &A_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\A_{m1}&A_{m2}&A_{m3}&\cdots &A_{mn}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cacfe518d80e091252cda178bc62a8fedb1df7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.171ex; width:33.246ex; height:17.509ex;" alt="{\displaystyle {\begin{bmatrix}A_{11}&A_{12}&A_{13}&\cdots &A_{1n}\\A_{21}&A_{22}&A_{23}&\cdots &A_{2n}\\A_{31}&A_{32}&A_{33}&\cdots &A_{3n}\\\vdots &\vdots &\vdots &\ddots &\vdots \\A_{m1}&A_{m2}&A_{m3}&\cdots &A_{mn}\end{bmatrix}}}"></span></dd></dl> <p>와 같이 표기한다. </p><p>각 <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8272b28f5aae6dbb8d6f829d58bab353b21bde20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.22ex; height:2.843ex;" alt="{\displaystyle A_{ij}}"></span>를 <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>의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>번째 행 <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>번째 열의 <b>성분</b>(成分, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">entry</span>) 또는 <b>원소</b>(元素, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">element</span>) 또는 <b>계수</b>(係數, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">coefficient</span>)라고 한다. 행렬 <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>의 각 성분은 행과 열의 번째수를 첨수로 사용하여 <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}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8272b28f5aae6dbb8d6f829d58bab353b21bde20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.22ex; height:2.843ex;" alt="{\displaystyle A_{ij}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f379f42961dd620c7a05dc1c538117ec105877d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.678ex; height:2.843ex;" alt="{\displaystyle A_{i,j}}"></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 a_{ij}}"> <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> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebea6cd2813c330c798921a2894b358f7b643917" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ij}}"></span>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{i,j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i,j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bb5a346f58c6568306a02596dd318d1b7e6b2c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.164ex; height:2.343ex;" alt="{\displaystyle a_{i,j}}"></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 A(i,j)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(i,j)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93ba217815c0cdc100fe745358075af5bea1c025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.347ex; height:2.843ex;" alt="{\displaystyle A(i,j)}"></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 A[i,j]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">[</mo> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A[i,j]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a9bfca6057f52b4e5c3844ecd704962b7e426c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.831ex; height:2.843ex;" alt="{\displaystyle A[i,j]}"></span> 등과 같이 나타낸다. 행과 열의 번째수가 같은 성분 <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_{ii}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{ii}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af863a03eae1616a6d1ef9bfc42c31820098622" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.11ex; height:2.509ex;" alt="{\displaystyle A_{ii}}"></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 i\in \{1,\dotsc ,\min\{m,n\}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mo movablelimits="true" form="prefix">min</mo> <mo fence="false" stretchy="false">{</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\dotsc ,\min\{m,n\}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b3be2ee807eda5db6551610fca8da1905f83628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.978ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\dotsc ,\min\{m,n\}\}}"></span>)을 <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>의 <b>대각 성분</b>(對角成分, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">diagonal entry</span>) 또는 <b>대각 원소</b>(對角元素, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">diagonal element</span>) 또는 <b>대각 요소</b>(對角要素) 또는 <b><a href="/wiki/%EC%A3%BC%EB%8C%80%EA%B0%81%EC%84%A0" title="주대각선">주대각선</a></b> 성분이라고 한다.<sup id="cite_ref-Kharab_2-3" class="reference"><a href="#cite_note-Kharab-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:99</sup></span> </p><p><a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> 행렬의 집합은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac711c4052d893e23d11c15600663ed800018f82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.274ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mat} (m,n;R)}"></span> 또는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {M} _{m,n}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {M} _{m,n}(R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c8acc3badf484f598be6292e2e422c02bb173de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.823ex; height:3.009ex;" alt="{\displaystyle \operatorname {M} _{m,n}(R)}"></span>로 표기한다. </p> <div class="mw-heading mw-heading3"><h3 id="크기"><span id=".ED.81.AC.EA.B8.B0"></span>크기</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=2" title="부분 편집: 크기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>행렬 <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>의 <b>크기</b>(<span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">size</span>)는 행과 열의 수의 <a href="/wiki/%EC%88%9C%EC%84%9C%EC%8C%8D" title="순서쌍">순서쌍</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 (m,n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (m,n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/274d4857135a7d28a94ba9ee8135779615084d43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.278ex; height:2.843ex;" alt="{\displaystyle (m,n)}"></span> 또는 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span>을 뜻한다. 일부 특수한 크기의 행렬들은 특별한 이름으로 불린다. </p> <ul><li>만약 행과 열의 수가 같다면 (<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 m=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69c9d8e54796e7de7d4738510cc10bc3fc55d48e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.534ex; height:1.676ex;" alt="{\displaystyle m=n}"></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 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>를 <b>정사각 행렬</b>(正四角行列, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">square matrix</span>) 또는 <b>정방 행렬</b>(正方行列)이라고 한다. <a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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> 정사각 행렬의 집합은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e9f24a35d629b7bc279a9565e553b1893523d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.2ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mat} (n;R)}"></span> 또는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {M} _{n}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {M} _{n}(R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed7afdc57984c7fa44c42fcfae7be672af77ef87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.923ex; height:2.843ex;" alt="{\displaystyle \operatorname {M} _{n}(R)}"></span>로 표기한다.</li> <li>만약 <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 m=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6100c5ebd48c6fd848709f2be624465203eb173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.301ex; height:2.176ex;" alt="{\displaystyle m=1}"></span>이라면, <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>를 <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 1\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bce5f6a6d0d32834484048c16f3b39f9c23d076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle 1\times n}"></span> <b>행벡터</b>(行-, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">row vector</span>)라고 한다.</li> <li>만약 <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=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9ec7e1edc2e6d98f5aec2a39ae5f1c99d1e1425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n=1}"></span>이라면, <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>를 <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 m\times 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7ea91d81567531f5ef6d3b669be211ff953e6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.043ex; height:2.176ex;" alt="{\displaystyle m\times 1}"></span> <b>열벡터</b>(列-, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">column vector</span>)라고 한다.</li></ul> <p>특히, 행렬 <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>의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>번째 행벡터와 <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>번째 열벡터는 각각 </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_{i,-}={\begin{pmatrix}A_{i1}&A_{i2}&\cdots A_{in}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mo>−</mo> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i,-}={\begin{pmatrix}A_{i1}&A_{i2}&\cdots A_{in}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22c396a4d19125f3a8cc1109dc68e28723fe1dd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.952ex; height:3.009ex;" alt="{\displaystyle A_{i,-}={\begin{pmatrix}A_{i1}&A_{i2}&\cdots A_{in}\end{pmatrix}}}"></span></dd></dl> <p>와 </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_{-,j}{\begin{pmatrix}A_{1j}\\A_{2j}\\\vdots \\A_{mj}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mi>j</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{-,j}{\begin{pmatrix}A_{1j}\\A_{2j}\\\vdots \\A_{mj}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/773af90684262608d3f229375d0339eb370ea6da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:13.303ex; height:14.509ex;" alt="{\displaystyle A_{-,j}{\begin{pmatrix}A_{1j}\\A_{2j}\\\vdots \\A_{mj}\end{pmatrix}}}"></span></dd></dl> <p>이며, 이를 통해 행렬을 다음과 같이 나타낼 수 있다. </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={\begin{pmatrix}A_{1,-}\\A_{2,-}\\\vdots \\A_{m,-}\end{pmatrix}}={\begin{pmatrix}A_{-,1}&A_{-,2}&\cdots &A_{-,n}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mo>−</mo> </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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}A_{1,-}\\A_{2,-}\\\vdots \\A_{m,-}\end{pmatrix}}={\begin{pmatrix}A_{-,1}&A_{-,2}&\cdots &A_{-,n}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f510e915ff4fd8484f88a8a1b9be7eb3fcd54d62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:43.928ex; height:14.509ex;" alt="{\displaystyle A={\begin{pmatrix}A_{1,-}\\A_{2,-}\\\vdots \\A_{m,-}\end{pmatrix}}={\begin{pmatrix}A_{-,1}&A_{-,2}&\cdots &A_{-,n}\end{pmatrix}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="연산"><span id=".EC.97.B0.EC.82.B0"></span>연산</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=3" title="부분 편집: 연산"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>행렬들에 대하여 덧셈, 스칼라배, 곱셈, <a href="/wiki/%EC%A0%84%EC%B9%98_%ED%96%89%EB%A0%AC" title="전치 행렬">전치 행렬</a> 등의 연산을 정의할 수 있으며, 정사각 행렬은 <a href="/wiki/%EC%97%AD%ED%96%89%EB%A0%AC" class="mw-redirect" title="역행렬">역행렬</a>, <a href="/wiki/%EB%8C%80%EA%B0%81%ED%95%A9" title="대각합">대각합</a>, <a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a> 등 연산이 추가로 정의된다. 덧셈은 같은 크기의 두 행렬에 대해서만 정의되며, 곱셈은 오직 첫 번째 행렬의 열의 수와 두 번째 행렬의 행의 수가 같은 경우에만 정의된다.<sup id="cite_ref-Kharab_2-4" class="reference"><a href="#cite_note-Kharab-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:99</sup></span> <a href="/wiki/%EC%97%AD%ED%96%89%EB%A0%AC" class="mw-redirect" title="역행렬">역행렬</a>은 <a href="/wiki/%EA%B0%80%EC%97%AD_%ED%96%89%EB%A0%AC" class="mw-redirect" title="가역 행렬">가역</a> 정사각 행렬에 대하여 정의되며, <a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a>은 <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a> 위의 정사각 행렬에 대하여 정의된다. </p> <div class="mw-heading mw-heading3"><h3 id="덧셈과_스칼라배"><span id=".EB.8D.A7.EC.85.88.EA.B3.BC_.EC.8A.A4.EC.B9.BC.EB.9D.BC.EB.B0.B0"></span>덧셈과 스칼라배</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=4" title="부분 편집: 덧셈과 스칼라배"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r34311305">.mw-parser-output .hatnote{}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}</style><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/w/index.php?title=%ED%96%89%EB%A0%AC_%EB%8D%A7%EC%85%88&action=edit&redlink=1" class="new" title="행렬 덧셈 (없는 문서)">행렬 덧셈</a>입니다.</div> <p><a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 두 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span>의 행렬 <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,B\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3783625affda4332d80837c42888684c5e0ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.656ex; height:2.843ex;" alt="{\displaystyle A,B\in \operatorname {Mat} (m,n;R)}"></span>의 합 <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+B\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>+</mo> <mi>B</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A+B\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c2b59f37026952ae3468cb3b2d9d479217ffee8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.463ex; height:2.843ex;" alt="{\displaystyle A+B\in \operatorname {Mat} (m,n;R)}"></span>은 두 행렬을 성분별로 합한 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> 행렬이다. 즉, 각 행과 열 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>, <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>에 대하여, </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+B)_{ij}=A_{ij}+B_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>+</mo> <mi>B</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72e0b9e2fa0cab10d008321d379900b05fefd076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.034ex; height:3.009ex;" alt="{\displaystyle (A+B)_{ij}=A_{ij}+B_{ij}}"></span></dd></dl> <p>이다. </p><p>실수 행렬의 예는 다음과 같다. </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{pmatrix}1&3&7\\1&0&0\end{pmatrix}}+{\begin{pmatrix}0&0&5\\7&5&0\end{pmatrix}}={\begin{pmatrix}1+0&3+0&7+5\\1+7&0+5&0+0\end{pmatrix}}={\begin{pmatrix}1&3&12\\8&5&0\end{pmatrix}}}"> <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>7</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</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> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</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> <mn>1</mn> <mo>+</mo> <mn>0</mn> </mtd> <mtd> <mn>3</mn> <mo>+</mo> <mn>0</mn> </mtd> <mtd> <mn>7</mn> <mo>+</mo> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <mo>+</mo> <mn>7</mn> </mtd> <mtd> <mn>0</mn> <mo>+</mo> <mn>5</mn> </mtd> <mtd> <mn>0</mn> <mo>+</mo> <mn>0</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> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>12</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mn>5</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&3&7\\1&0&0\end{pmatrix}}+{\begin{pmatrix}0&0&5\\7&5&0\end{pmatrix}}={\begin{pmatrix}1+0&3+0&7+5\\1+7&0+5&0+0\end{pmatrix}}={\begin{pmatrix}1&3&12\\8&5&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d67d50e0392780513f3f6f2e13751cd06c4efa63" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:71.429ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&3&7\\1&0&0\end{pmatrix}}+{\begin{pmatrix}0&0&5\\7&5&0\end{pmatrix}}={\begin{pmatrix}1+0&3+0&7+5\\1+7&0+5&0+0\end{pmatrix}}={\begin{pmatrix}1&3&12\\8&5&0\end{pmatrix}}}"></span></dd></dl> <p><a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span>의 행렬 <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\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0a7b74fc96ebe6b858c776506c1065f6932bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.858ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (m,n;R)}"></span> 및 환의 원소 <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\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca49c66b5e9b5f32249a737e4429c3df136c33f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.653ex; height:2.176ex;" alt="{\displaystyle r\in R}"></span>에 대하여, 왼쪽·오른쪽 스칼라배 <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 rA,Ar\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mi>A</mi> <mo>,</mo> <mi>A</mi> <mi>r</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle rA,Ar\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b10e97b16b71d762d18f0687b44bb90123857292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.732ex; height:2.843ex;" alt="{\displaystyle rA,Ar\in \operatorname {Mat} (m,n;R)}"></span>는 각각 행렬의 각 성분의 왼쪽·오른쪽에 스칼라를 곱한 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> 행렬이다. </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 (rA)_{ij}=rA_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>r</mi> <mi>A</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>r</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (rA)_{ij}=rA_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a0b4663763e36f8806a03630ced13b8ac900286" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.446ex; height:3.009ex;" alt="{\displaystyle (rA)_{ij}=rA_{ij}}"></span></dd> <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 (Ar)_{ij}=A_{ij}r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>r</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Ar)_{ij}=A_{ij}r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7726222aedbf3810295d185af381b8943c1c59dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.446ex; height:3.009ex;" alt="{\displaystyle (Ar)_{ij}=A_{ij}r}"></span></dd></dl> <p>만약 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>가 <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a>일 경우, 이 두 연산은 일치하며, 이를 스칼라배라고 부른다. </p><p>실수 행렬의 예는 다음과 같다. </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 2{\begin{pmatrix}1&8&-3\\4&-2&5\end{pmatrix}}={\begin{pmatrix}2\cdot 1&2\cdot 8&2\cdot -3\\2\cdot 4&2\cdot -2&2\cdot 5\end{pmatrix}}={\begin{pmatrix}2&16&-6\\8&-4&10\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>5</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> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mn>8</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mo>−</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mo>⋅</mo> <mn>4</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>2</mn> <mo>⋅</mo> <mn>5</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> <mn>2</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mo>−</mo> <mn>4</mn> </mtd> <mtd> <mn>10</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\begin{pmatrix}1&8&-3\\4&-2&5\end{pmatrix}}={\begin{pmatrix}2\cdot 1&2\cdot 8&2\cdot -3\\2\cdot 4&2\cdot -2&2\cdot 5\end{pmatrix}}={\begin{pmatrix}2&16&-6\\8&-4&10\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45eb6d2f96911df9557857d4f500548eb9e36404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.648ex; height:6.176ex;" alt="{\displaystyle 2{\begin{pmatrix}1&8&-3\\4&-2&5\end{pmatrix}}={\begin{pmatrix}2\cdot 1&2\cdot 8&2\cdot -3\\2\cdot 4&2\cdot -2&2\cdot 5\end{pmatrix}}={\begin{pmatrix}2&16&-6\\8&-4&10\end{pmatrix}}}"></span></dd></dl> <p><a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> 행렬의 집합 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac711c4052d893e23d11c15600663ed800018f82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.274ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mat} (m,n;R)}"></span>은 위 덧셈과 왼쪽·오른쪽 스칼라배에 따라 <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,R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R,R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c541d71440092290621d4785f95797a1859fce7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.371ex; height:2.843ex;" alt="{\displaystyle (R,R)}"></span>-<a href="/wiki/%EC%8C%8D%EA%B0%80%EA%B5%B0" title="쌍가군">쌍가군</a>을 이룬다. 만약 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>가 <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a>일 경우, 이는 (덧셈과 스칼라배에 따른) <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-<a href="/wiki/%EA%B0%80%EA%B5%B0" title="가군">가군</a>이 되며, 특히 만약 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>가 <a href="/wiki/%EC%B2%B4_(%EC%88%98%ED%95%99)" title="체 (수학)">체</a>일 경우 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-<a href="/wiki/%EB%B2%A1%ED%84%B0_%EA%B3%B5%EA%B0%84" title="벡터 공간">벡터 공간</a>이다. 이 <a href="/wiki/%EC%8C%8D%EA%B0%80%EA%B5%B0" title="쌍가군">쌍가군</a>의 덧셈 <a href="/wiki/%ED%95%AD%EB%93%B1%EC%9B%90" title="항등원">항등원</a>은 <b><a href="/wiki/%EC%98%81%ED%96%89%EB%A0%AC" title="영행렬">영행렬</a></b>(즉, 모든 성분이 0인 행렬) </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 0_{m\times n}={\begin{pmatrix}0&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>0</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0_{m\times n}={\begin{pmatrix}0&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22495ca1f20ffa025d80c8f6b968270786aee971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:42.568ex; height:14.176ex;" alt="{\displaystyle 0_{m\times n}={\begin{pmatrix}0&0&\cdots &0\\0&0&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &0\end{pmatrix}}\in \operatorname {Mat} (m,n;R)}"></span></dd></dl> <p>이며, 각 행렬 <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\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0a7b74fc96ebe6b858c776506c1065f6932bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.858ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (m,n;R)}"></span>의 덧셈 <a href="/wiki/%EC%97%AD%EC%9B%90" title="역원">역원</a>은 성분별 덧셈 역원 </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\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -A\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27138c03572b0bde919c42c1229982e8fbd3faa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.666ex; height:2.843ex;" alt="{\displaystyle -A\in \operatorname {Mat} (m,n;R)}"></span></dd> <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_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>A</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (-A)_{ij}=-A_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc0f0818dd75cf2f336113432461f792db6ba149" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.965ex; height:3.009ex;" alt="{\displaystyle (-A)_{ij}=-A_{ij}}"></span></dd></dl> <p>이다. </p><p>특히, 두 행렬 <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,B\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3783625affda4332d80837c42888684c5e0ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.656ex; height:2.843ex;" alt="{\displaystyle A,B\in \operatorname {Mat} (m,n;R)}"></span>의 차를 다음과 같이 정의할 수 있다. </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-B=A+(-B)\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mi>B</mi> <mo>=</mo> <mi>A</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-B=A+(-B)\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d6637580954eb3ff9e37032c0ed21d18bcb7b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.526ex; height:2.843ex;" alt="{\displaystyle A-B=A+(-B)\in \operatorname {Mat} (m,n;R)}"></span></dd></dl> <p><a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> 행렬의 <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,R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R,R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c541d71440092290621d4785f95797a1859fce7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.371ex; height:2.843ex;" alt="{\displaystyle (R,R)}"></span>-<a href="/wiki/%EC%8C%8D%EA%B0%80%EA%B5%B0" title="쌍가군">쌍가군</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,B\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c3783625affda4332d80837c42888684c5e0ef7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.656ex; height:2.843ex;" alt="{\displaystyle A,B\in \operatorname {Mat} (m,n;R)}"></span>는 왼쪽 가군으로서 <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 mn}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mn}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348cd26a0b7a0034f57a951e2cf5f637dd47c1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:1.676ex;" alt="{\displaystyle mn}"></span>차원 왼쪽 <a href="/wiki/%EC%9E%90%EC%9C%A0_%EA%B0%80%EA%B5%B0" title="자유 가군">자유 가군</a>을 이루며, 오른쪽 가군으로서 <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 mn}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mn}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348cd26a0b7a0034f57a951e2cf5f637dd47c1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:1.676ex;" alt="{\displaystyle mn}"></span>차원 오른쪽 <a href="/wiki/%EC%9E%90%EC%9C%A0_%EA%B0%80%EA%B5%B0" title="자유 가군">자유 가군</a>을 이룬다. <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>가 <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a>일 경우 <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 mn}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle mn}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/348cd26a0b7a0034f57a951e2cf5f637dd47c1ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:1.676ex;" alt="{\displaystyle mn}"></span>차원 자유 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-가군이다. 그 한 <a href="/wiki/%EA%B8%B0%EC%A0%80_(%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99)" title="기저 (선형대수학)">기저</a>는 다음과 같다. </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 E_{ij}\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{ij}\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca098ddc17d7d1a2b24a39b3d8c9d50eb5ca49a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.307ex; height:3.009ex;" alt="{\displaystyle E_{ij}\in \operatorname {Mat} (m,n;R)}"></span></dd> <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 (E_{ij})_{kl}=\delta _{ik}\delta _{jl}={\begin{cases}1&i=k\land j=l\\0&i\neq k\lor j\neq l\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>l</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>i</mi> <mo>=</mo> <mi>k</mi> <mo>∧</mo> <mi>j</mi> <mo>=</mo> <mi>l</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>i</mi> <mo>≠</mo> <mi>k</mi> <mo>∨</mo> <mi>j</mi> <mo>≠</mo> <mi>l</mi> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (E_{ij})_{kl}=\delta _{ik}\delta _{jl}={\begin{cases}1&i=k\land j=l\\0&i\neq k\lor j\neq l\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9412e31f2f1093e50911d7236765a11c2677b8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.323ex; height:6.176ex;" alt="{\displaystyle (E_{ij})_{kl}=\delta _{ik}\delta _{jl}={\begin{cases}1&i=k\land j=l\\0&i\neq k\lor j\neq l\end{cases}}}"></span></dd> <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 i\in \{1,\dotsc ,m\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\dotsc ,m\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d7a25137e9baaf70273ac4d1c5fa1a23b13b4f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.349ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\dotsc ,m\}}"></span></dd> <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 j\in \{1,\dotsc ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\in \{1,\dotsc ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efa124a2f23e6c9ce001436dc1fc4bfb05ebd09e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:13.886ex; height:2.843ex;" alt="{\displaystyle j\in \{1,\dotsc ,n\}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="곱셈"><span id=".EA.B3.B1.EC.85.88"></span>곱셈</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=5" title="부분 편집: 곱셈"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%ED%96%89%EB%A0%AC_%EA%B3%B1%EC%85%88" title="행렬 곱셈">행렬 곱셈</a>입니다.</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/%ED%8C%8C%EC%9D%BC:Matrix_multiplication_diagram.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Matrix_multiplication_diagram.svg/220px-Matrix_multiplication_diagram.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/11/Matrix_multiplication_diagram.svg/330px-Matrix_multiplication_diagram.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/11/Matrix_multiplication_diagram.svg/440px-Matrix_multiplication_diagram.svg.png 2x" data-file-width="188" data-file-height="188" /></a><figcaption>행렬 곱셈</figcaption></figure> <p><a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> 행렬 <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\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0a7b74fc96ebe6b858c776506c1065f6932bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.858ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (m,n;R)}"></span>와 <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}"></span> 행렬 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in \operatorname {Mat} (n,p;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\in \operatorname {Mat} (n,p;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4160920bf95c7dbdedc36b83f6c78bc84ed0b779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.008ex; height:2.843ex;" alt="{\displaystyle B\in \operatorname {Mat} (n,p;R)}"></span>의 곱 <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\in \operatorname {Mat} (m,p;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>p</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB\in \operatorname {Mat} (m,p;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3cb9e2d44a90ad0ab01d731673b05a5edd7a82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.397ex; height:2.843ex;" alt="{\displaystyle AB\in \operatorname {Mat} (m,p;R)}"></span>는 <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 m\times p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e459f087ef822dc6fba54b953c60de61be69c42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.05ex; height:2.009ex;" alt="{\displaystyle m\times p}"></span> 행렬이며, 그 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>번째 행 <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>번째 열 성분은 <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>의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>번째 행벡터와 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>의 <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>번째 열벡터의 ‘<a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC%EA%B3%B1" title="스칼라곱">스칼라곱</a>’이다 (둘 모두 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>차원 벡터이므로 ‘스칼라곱’이 정의된다). </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 (AB)_{ij}=\sum _{k=1}^{n}A_{ik}B_{kj}=A_{i1}B_{1j}+A_{i2}B_{2j}+\cdots A_{in}B_{nj}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (AB)_{ij}=\sum _{k=1}^{n}A_{ik}B_{kj}=A_{i1}B_{1j}+A_{i2}B_{2j}+\cdots A_{in}B_{nj}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d38a0988d8f7b46aef1ef64578db181eaf022a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.363ex; height:6.843ex;" alt="{\displaystyle (AB)_{ij}=\sum _{k=1}^{n}A_{ik}B_{kj}=A_{i1}B_{1j}+A_{i2}B_{2j}+\cdots A_{in}B_{nj}}"></span></dd></dl> <p>다음은 실수 행렬의 예다. </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{pmatrix}1&0&2\\-1&3&1\\\end{pmatrix}}{\begin{pmatrix}3&1\\2&1\\1&0\end{pmatrix}}={\begin{pmatrix}1\cdot 3+0\cdot 2+2\cdot 1&1\cdot 1+0\cdot 1+2\cdot 0\\-1\cdot 3+3\cdot 2+1\cdot 1&-1\cdot 1+3\cdot 1+1\cdot 0\end{pmatrix}}={\begin{pmatrix}5&1\\4&2\end{pmatrix}}}"> <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>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>3</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</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> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <mn>2</mn> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mn>0</mn> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mn>2</mn> <mo>⋅</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mn>3</mn> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mn>2</mn> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mn>3</mn> <mo>⋅</mo> <mn>1</mn> <mo>+</mo> <mn>1</mn> <mo>⋅</mo> <mn>0</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> <mn>5</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&0&2\\-1&3&1\\\end{pmatrix}}{\begin{pmatrix}3&1\\2&1\\1&0\end{pmatrix}}={\begin{pmatrix}1\cdot 3+0\cdot 2+2\cdot 1&1\cdot 1+0\cdot 1+2\cdot 0\\-1\cdot 3+3\cdot 2+1\cdot 1&-1\cdot 1+3\cdot 1+1\cdot 0\end{pmatrix}}={\begin{pmatrix}5&1\\4&2\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d1043b1cad84fa36a5a5e41e360777cabf86c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:84.093ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}1&0&2\\-1&3&1\\\end{pmatrix}}{\begin{pmatrix}3&1\\2&1\\1&0\end{pmatrix}}={\begin{pmatrix}1\cdot 3+0\cdot 2+2\cdot 1&1\cdot 1+0\cdot 1+2\cdot 0\\-1\cdot 3+3\cdot 2+1\cdot 1&-1\cdot 1+3\cdot 1+1\cdot 0\end{pmatrix}}={\begin{pmatrix}5&1\\4&2\end{pmatrix}}}"></span></dd></dl> <p>행벡터와 열벡터 </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={\begin{pmatrix}A_{1,-}\\A_{2,-}\\\vdots \\A_{m,-}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mo>−</mo> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{pmatrix}A_{1,-}\\A_{2,-}\\\vdots \\A_{m,-}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af1faffe26ed112076583b44b842b6f2747c3f9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:14.814ex; height:14.509ex;" alt="{\displaystyle A={\begin{pmatrix}A_{1,-}\\A_{2,-}\\\vdots \\A_{m,-}\end{pmatrix}}}"></span></dd> <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 B={\begin{pmatrix}B_{-,1}&B_{-,2}&\cdots &B_{-,n}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\begin{pmatrix}B_{-,1}&B_{-,2}&\cdots &B_{-,n}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d1d3d702f48df6e39594622c4399405671d9b9b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.941ex; height:3.176ex;" alt="{\displaystyle B={\begin{pmatrix}B_{-,1}&B_{-,2}&\cdots &B_{-,n}\end{pmatrix}}}"></span></dd></dl> <p>를 통해 행렬 곱셈을 다음과 같이 나타낼 수 있다. </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 AB={\begin{pmatrix}A_{1,-}B\\A_{2,-}B\\\vdots \\A_{m,-}B\end{pmatrix}}={\begin{pmatrix}AB_{-,1}&AB_{-,2}&\cdots &AB_{-,n}\end{pmatrix}}={\begin{pmatrix}A_{1,-}B_{-,1}&A_{1,-}B_{-,2}&\cdots &A_{1,-}B_{-,p}\\A_{2,-}B_{-,1}&A_{2,-}B_{-,2}&\cdots &A_{2,-}B_{-,p}\\\vdots &\vdots &\ddots &\vdots \\A_{m,-}B_{-,1}&A_{m,-}B_{-,2}&\cdots &A_{m,-}B_{-,p}\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> <mi>B</mi> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mo>−</mo> </mrow> </msub> <mi>B</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> <mi>A</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi>A</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mi>A</mi> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mi>n</mi> </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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mi>p</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mo>,</mo> <mo>−</mo> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mi>p</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mo>−</mo> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mo>−</mo> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>,</mo> <mo>−</mo> </mrow> </msub> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mo>,</mo> <mi>p</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AB={\begin{pmatrix}A_{1,-}B\\A_{2,-}B\\\vdots \\A_{m,-}B\end{pmatrix}}={\begin{pmatrix}AB_{-,1}&AB_{-,2}&\cdots &AB_{-,n}\end{pmatrix}}={\begin{pmatrix}A_{1,-}B_{-,1}&A_{1,-}B_{-,2}&\cdots &A_{1,-}B_{-,p}\\A_{2,-}B_{-,1}&A_{2,-}B_{-,2}&\cdots &A_{2,-}B_{-,p}\\\vdots &\vdots &\ddots &\vdots \\A_{m,-}B_{-,1}&A_{m,-}B_{-,2}&\cdots &A_{m,-}B_{-,p}\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8082e1364780e4908dba24bf96eca6287bbb2e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:99.741ex; height:14.843ex;" alt="{\displaystyle AB={\begin{pmatrix}A_{1,-}B\\A_{2,-}B\\\vdots \\A_{m,-}B\end{pmatrix}}={\begin{pmatrix}AB_{-,1}&AB_{-,2}&\cdots &AB_{-,n}\end{pmatrix}}={\begin{pmatrix}A_{1,-}B_{-,1}&A_{1,-}B_{-,2}&\cdots &A_{1,-}B_{-,p}\\A_{2,-}B_{-,1}&A_{2,-}B_{-,2}&\cdots &A_{2,-}B_{-,p}\\\vdots &\vdots &\ddots &\vdots \\A_{m,-}B_{-,1}&A_{m,-}B_{-,2}&\cdots &A_{m,-}B_{-,p}\end{pmatrix}}}"></span></dd></dl> <p>행렬 곱셈은 <a href="/wiki/%EA%B2%B0%ED%95%A9_%EB%B2%95%EC%B9%99" class="mw-redirect" title="결합 법칙">결합 법칙</a>을 만족시킨다. 즉, <a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 임의의 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> 행렬 <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\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0a7b74fc96ebe6b858c776506c1065f6932bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.858ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (m,n;R)}"></span> 및 <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}"></span> 행렬 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in \operatorname {Mat} (n,p;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\in \operatorname {Mat} (n,p;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4160920bf95c7dbdedc36b83f6c78bc84ed0b779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.008ex; height:2.843ex;" alt="{\displaystyle B\in \operatorname {Mat} (n,p;R)}"></span> 및 <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 p\times q}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>×</mo> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\times q}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2da588b05b98e9d72312b8fae68ebb25b34db18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.169ex; height:2.009ex;" alt="{\displaystyle p\times q}"></span> 행렬 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C\in \operatorname {Mat} (p,q;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C\in \operatorname {Mat} (p,q;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7cb9475fa420b22139b713f9f82bf4cff470bc7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.685ex; height:2.843ex;" alt="{\displaystyle C\in \operatorname {Mat} (p,q;R)}"></span>에 대하여, </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 (AB)C=A(BC)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mi>C</mi> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>B</mi> <mi>C</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (AB)C=A(BC)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e041d54540f5581ab13f2ef4ca2b757033bdc37e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.264ex; height:2.843ex;" alt="{\displaystyle (AB)C=A(BC)}"></span></dd></dl> <p>가 성립한다. </p><p>행렬 곱셈은 함수 </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 \operatorname {Mat} (m,n;R)\oplus \operatorname {Mat} (n,p;R)\to \operatorname {Mat} (m,p;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>⊕</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>p</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} (m,n;R)\oplus \operatorname {Mat} (n,p;R)\to \operatorname {Mat} (m,p;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3357479970ecaa0cc16a604ab6fdb0fac0b31407" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.181ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mat} (m,n;R)\oplus \operatorname {Mat} (n,p;R)\to \operatorname {Mat} (m,p;R)}"></span></dd></dl> <p>로서 <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,R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R,R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c541d71440092290621d4785f95797a1859fce7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.371ex; height:2.843ex;" alt="{\displaystyle (R,R)}"></span>-쌍선형 함수를 이룬다. </p><p>특히, <a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 정사각 행렬들의 <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,R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R,R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c541d71440092290621d4785f95797a1859fce7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.371ex; height:2.843ex;" alt="{\displaystyle (R,R)}"></span>-<a href="/wiki/%EC%8C%8D%EA%B0%80%EA%B5%B0" title="쌍가군">쌍가군</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 \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e9f24a35d629b7bc279a9565e553b1893523d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.2ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mat} (n;R)}"></span>는 그 위의 행렬 곱셈에 따라 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-<a href="/wiki/%EA%B2%B0%ED%95%A9_%EB%8C%80%EC%88%98" title="결합 대수">결합 대수</a>를 이룬다. 특히 <a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</a>을 이루며, <b><a href="/w/index.php?title=%ED%96%89%EB%A0%AC%ED%99%98&action=edit&redlink=1" class="new" title="행렬환 (없는 문서)">행렬환</a></b>(行列環, <span style="font-size: smaller;"><a href="/wiki/%EC%98%81%EC%96%B4" title="영어">영어</a>: </span><span lang="en">matrix ring</span>)이라고 한다. 행렬환의 곱셈 <a href="/wiki/%ED%95%AD%EB%93%B1%EC%9B%90" title="항등원">항등원</a>은 <b><a href="/wiki/%EB%8B%A8%EC%9C%84_%ED%96%89%EB%A0%AC" class="mw-redirect" title="단위 행렬">단위 행렬</a></b>(즉, 모든 대각 성분이 1, 그 밖의 성분이 0인 행렬) </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 1_{n\times n}={\begin{pmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{pmatrix}}\in \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1_{n\times n}={\begin{pmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{pmatrix}}\in \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cf0d1e3792c4dc3dbf022dac01c168c54906d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:39.037ex; height:14.176ex;" alt="{\displaystyle 1_{n\times n}={\begin{pmatrix}1&0&\cdots &0\\0&1&\cdots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\cdots &1\end{pmatrix}}\in \operatorname {Mat} (n;R)}"></span></dd></dl> <p>이다. </p> <div class="mw-heading mw-heading4"><h4 id="교환_법칙과_소거_법칙의_실패"><span id=".EA.B5.90.ED.99.98_.EB.B2.95.EC.B9.99.EA.B3.BC_.EC.86.8C.EA.B1.B0_.EB.B2.95.EC.B9.99.EC.9D.98_.EC.8B.A4.ED.8C.A8"></span>교환 법칙과 소거 법칙의 실패</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=6" title="부분 편집: 교환 법칙과 소거 법칙의 실패"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>행렬환은 일반적으로 <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a>이 아니다. 즉, 행렬 곱셈의 <a href="/wiki/%EA%B5%90%ED%99%98_%EB%B2%95%EC%B9%99" class="mw-redirect" title="교환 법칙">교환 법칙</a>은 (<a href="/wiki/%EC%B2%B4_(%EC%88%98%ED%95%99)" title="체 (수학)">체</a>의 경우에도) 일반적으로 성립하지 않는다. 예를 들어, 실수 2×2 행렬의 경우 </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{pmatrix}1&0\\0&0\end{pmatrix}}{\begin{pmatrix}1&2\\0&3\end{pmatrix}}={\begin{pmatrix}1&2\\0&0\end{pmatrix}}}"> <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>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <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>0</mn> </mtd> <mtd> <mn>3</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> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&0\\0&0\end{pmatrix}}{\begin{pmatrix}1&2\\0&3\end{pmatrix}}={\begin{pmatrix}1&2\\0&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47bc8a6ae16cc8cdd7965532a44bd49a9285d9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.559ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&0\\0&0\end{pmatrix}}{\begin{pmatrix}1&2\\0&3\end{pmatrix}}={\begin{pmatrix}1&2\\0&0\end{pmatrix}}}"></span></dd></dl> <p>이지만 </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{pmatrix}1&2\\0&3\end{pmatrix}}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}}}"> <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> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</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> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&2\\0&3\end{pmatrix}}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27703afb1ef0a51df06379d646c5913476e52569" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.559ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}1&2\\0&3\end{pmatrix}}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}={\begin{pmatrix}1&0\\0&0\end{pmatrix}}}"></span></dd></dl> <p>이다. </p><p>물론 <a href="/wiki/%EA%B5%90%ED%99%98%EB%B2%95%EC%B9%99" title="교환법칙">가환</a>하는 두 행렬도 존재한다. 예를 들어, <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a> 위의 <a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_%ED%96%89%EB%A0%AC" title="스칼라 행렬">스칼라 행렬</a>은 (같은 크기의) 모든 행렬과 가환한다. 또한, <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 및 정사각 행렬 <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\in \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bae03fe1013ece5fefa7e970c4bd28c3493109e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.784ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (n;R)}"></span>에 대하여, </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 R[A]=\{p(A)\colon p\in R[x]\}\subseteq \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <mo stretchy="false">[</mo> <mi>A</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mi>p</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>:</mo> <mi>p</mi> <mo>∈</mo> <mi>R</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R[A]=\{p(A)\colon p\in R[x]\}\subseteq \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b38178f10a53f84623b7238d16e4646161d27563" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.676ex; height:2.843ex;" alt="{\displaystyle R[A]=\{p(A)\colon p\in R[x]\}\subseteq \operatorname {Mat} (n;R)}"></span></dd></dl> <p>는 <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a>이다. </p><p>행렬환은 일반적으로 0이 아닌 왼쪽·오른쪽 <a href="/wiki/%EC%98%81%EC%9D%B8%EC%9E%90" title="영인자">영인자</a>를 갖는다. 즉, 0이 아닌 두 행렬의 곱은 0일 수 있으며, <a href="/wiki/%EC%86%8C%EA%B1%B0_%EB%B2%95%EC%B9%99" title="소거 법칙">소거 법칙</a>이 일반적으로 성립하지 않는다. 예를 들어, 실수 행렬에서 </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{pmatrix}2&-1\\-2&1\end{pmatrix}}{\begin{pmatrix}1&3\\2&6\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}}"> <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>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>6</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> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}2&-1\\-2&1\end{pmatrix}}{\begin{pmatrix}1&3\\2&6\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e87a15773b719c4c00f56db84477a7469863353" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:33.175ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}2&-1\\-2&1\end{pmatrix}}{\begin{pmatrix}1&3\\2&6\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\end{pmatrix}}}"></span></dd></dl> <p>이다. </p> <div class="mw-heading mw-heading4"><h4 id="역행렬"><span id=".EC.97.AD.ED.96.89.EB.A0.AC"></span>역행렬</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=7" title="부분 편집: 역행렬"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>행렬환 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e9f24a35d629b7bc279a9565e553b1893523d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.2ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mat} (n;R)}"></span>의 <a href="/wiki/%EA%B0%80%EC%97%AD%EC%9B%90" title="가역원">가역원</a>은 <b><a href="/wiki/%EA%B0%80%EC%97%AD_%ED%96%89%EB%A0%AC" class="mw-redirect" title="가역 행렬">가역 행렬</a></b>이라고 하며, 그 곱셈 <a href="/wiki/%EC%97%AD%EC%9B%90" title="역원">역원</a>은 <b><a href="/wiki/%EC%97%AD%ED%96%89%EB%A0%AC" class="mw-redirect" title="역행렬">역행렬</a></b>이라고 한다. 일반적으로 행렬환은 (<a href="/wiki/%EC%B2%B4_(%EC%88%98%ED%95%99)" title="체 (수학)">체</a> 위에서도) 0이 아닌 비<a href="/wiki/%EA%B0%80%EC%97%AD_%ED%96%89%EB%A0%AC" class="mw-redirect" title="가역 행렬">가역 행렬</a>을 갖는다. 예를 들어, 실수 2×2 정사각 행렬 </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{pmatrix}0&5\\0&3\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}0&5\\0&3\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34e7cfb05ef56f1a3579b77430c85b3089d91eec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.82ex; height:6.176ex;" alt="{\displaystyle {\begin{pmatrix}0&5\\0&3\end{pmatrix}}}"></span></dd></dl> <p>은 <a href="/wiki/%EA%B0%80%EC%97%AD_%ED%96%89%EB%A0%AC" class="mw-redirect" title="가역 행렬">가역 행렬</a>이 아니다. </p><p>만약 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>가 <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a>일 경우, 가역 행렬은 <a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a>이 환의 가역원인 것과 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이며, 특히 <a href="/wiki/%EC%B2%B4_(%EC%88%98%ED%95%99)" title="체 (수학)">체</a>의 경우 <a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a>이 0이 아닌 것과 <a href="/wiki/%EB%8F%99%EC%B9%98" title="동치">동치</a>이다. 또한, <a href="/wiki/%EA%B0%80%EC%97%AD_%ED%96%89%EB%A0%AC" class="mw-redirect" title="가역 행렬">가역 행렬</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\in \operatorname {Unit} (\operatorname {Mat} (n;R))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Unit</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Unit} (\operatorname {Mat} (n;R))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a89ab6373565b763d7251c214e014f6180f0b445" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.18ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Unit} (\operatorname {Mat} (n;R))}"></span>의 <a href="/wiki/%EC%97%AD%ED%96%89%EB%A0%AC" class="mw-redirect" title="역행렬">역행렬</a>은 <a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a>과 <a href="/wiki/%EC%88%98%EB%B0%98_%ED%96%89%EB%A0%AC" class="mw-redirect" title="수반 행렬">수반 행렬</a>을 통하여 다음과 같이 나타낼 수 있다. </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^{-1}={\frac {1}{\det A}}\operatorname {adj} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> </mrow> </mfrac> </mrow> <mi>adj</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{-1}={\frac {1}{\det A}}\operatorname {adj} A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fae7545a9026be9d0e9fb4f59ed8946a5cd2c1c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.054ex; height:5.343ex;" alt="{\displaystyle A^{-1}={\frac {1}{\det A}}\operatorname {adj} A}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="전치_행렬"><span id=".EC.A0.84.EC.B9.98_.ED.96.89.EB.A0.AC"></span>전치 행렬</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=8" title="부분 편집: 전치 행렬"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EC%A0%84%EC%B9%98_%ED%96%89%EB%A0%AC" title="전치 행렬">전치 행렬</a>입니다.</div> <p><a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> 행렬 <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\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0a7b74fc96ebe6b858c776506c1065f6932bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.858ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (m,n;R)}"></span>의 <b><a href="/wiki/%EC%A0%84%EC%B9%98_%ED%96%89%EB%A0%AC" title="전치 행렬">전치 행렬</a></b> <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^{\top }\in \operatorname {Mat} (n,m;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\top }\in \operatorname {Mat} (n,m;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7dc62dd101edca58b928122c7906d20d59fd74f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.369ex; height:3.176ex;" alt="{\displaystyle A^{\top }\in \operatorname {Mat} (n,m;R)}"></span>는 행과 열을 교환한 <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d82325a2a02ad79bc7c347ba9702ad46eb0de824" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle n\times m}"></span> 행렬이다. 즉, 각 <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\in \{1,\dotsc ,n\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i\in \{1,\dotsc ,n\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3b2faa5249b1490e9d5387bdc70d127181e8d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.703ex; height:2.843ex;" alt="{\displaystyle i\in \{1,\dotsc ,n\}}"></span> 및 <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 j\in \{1,\dotsc ,m\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>∈</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j\in \{1,\dotsc ,m\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ffacb98d56d973c09e0e83dd939fddcf955a79f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:14.531ex; height:2.843ex;" alt="{\displaystyle j\in \{1,\dotsc ,m\}}"></span>에 대하여, </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^{\top })_{ij}=A_{ji}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A^{\top })_{ij}=A_{ji}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5734f52b2961568124a7fb57c0ee16c1fd53a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.859ex; height:3.343ex;" alt="{\displaystyle (A^{\top })_{ij}=A_{ji}}"></span></dd></dl> <p>이다.<sup id="cite_ref-Kharab_2-5" class="reference"><a href="#cite_note-Kharab-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:99</sup></span> </p><p>다음은 실수 행렬의 예다. </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{pmatrix}9&8&7\\-1&3&4\end{pmatrix}}^{\top }={\begin{pmatrix}9&-1\\8&3\\7&4\end{pmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>9</mn> </mtd> <mtd> <mn>8</mn> </mtd> <mtd> <mn>7</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>9</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>8</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>7</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}9&8&7\\-1&3&4\end{pmatrix}}^{\top }={\begin{pmatrix}9&-1\\8&3\\7&4\end{pmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/486f4ccdacab364b8191e66e1bb4a3c7d9a5e6cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:29.997ex; height:9.176ex;" alt="{\displaystyle {\begin{pmatrix}9&8&7\\-1&3&4\end{pmatrix}}^{\top }={\begin{pmatrix}9&-1\\8&3\\7&4\end{pmatrix}}}"></span></dd></dl> <p>이다. </p><p>전치 행렬은 함수 </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 ^{\top }\colon \operatorname {Mat} (m,n;R)\to \operatorname {Mat} (n,m;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>:</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>m</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{\top }\colon \operatorname {Mat} (m,n;R)\to \operatorname {Mat} (n,m;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9910a26b26114f9278dac9468f90c0817e88d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.708ex; height:3.176ex;" alt="{\displaystyle ^{\top }\colon \operatorname {Mat} (m,n;R)\to \operatorname {Mat} (n,m;R)}"></span></dd></dl> <p>로서 <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,R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R,R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c541d71440092290621d4785f95797a1859fce7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.371ex; height:2.843ex;" alt="{\displaystyle (R,R)}"></span>-<a href="/wiki/%EC%8C%8D%EA%B0%80%EA%B5%B0" title="쌍가군">쌍가군</a> <a href="/wiki/%EB%8F%99%ED%98%95" class="mw-redirect" title="동형">동형</a>을 이루며, 그 <a href="/wiki/%EC%97%AD%ED%95%A8%EC%88%98" title="역함수">역함수</a> 또한 (<a href="/wiki/%EC%A0%95%EC%9D%98%EC%97%AD" title="정의역">정의역</a>과 <a href="/wiki/%EA%B3%B5%EC%97%AD" title="공역">공역</a>이 뒤바뀐) 전치 행렬이다. </p><p>또한, 임의의 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> 행렬 <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\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0a7b74fc96ebe6b858c776506c1065f6932bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.858ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (m,n;R)}"></span> 및 <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43ad58cdd60e9b0ab2bec828151c740accf92028" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.405ex; height:2.009ex;" alt="{\displaystyle n\times p}"></span> 행렬 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B\in \operatorname {Mat} (n,p;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>p</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B\in \operatorname {Mat} (n,p;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4160920bf95c7dbdedc36b83f6c78bc84ed0b779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.008ex; height:2.843ex;" alt="{\displaystyle B\in \operatorname {Mat} (n,p;R)}"></span>에 대하여, </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 (AB)^{\top }=B^{\top }A^{\top }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (AB)^{\top }=B^{\top }A^{\top }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33917942a0ad427cca82ae479f8c0b9aeb58ed89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.454ex; height:3.176ex;" alt="{\displaystyle (AB)^{\top }=B^{\top }A^{\top }}"></span></dd></dl> <p>이다. </p><p>특히, <a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 정사각 행렬의 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-<a href="/wiki/%EA%B2%B0%ED%95%A9_%EB%8C%80%EC%88%98" title="결합 대수">결합 대수</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 \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e9f24a35d629b7bc279a9565e553b1893523d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.2ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mat} (n;R)}"></span> 위에서, <a href="/wiki/%EC%A0%84%EC%B9%98_%ED%96%89%EB%A0%AC" title="전치 행렬">전치 행렬</a>은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e9f24a35d629b7bc279a9565e553b1893523d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.2ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mat} (n;R)}"></span>와 그 <a href="/wiki/%EB%B0%98%EB%8C%80%ED%99%98" class="mw-redirect" title="반대환">반대환</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 \operatorname {Mat} (n;R)^{\operatorname {op} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>op</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} (n;R)^{\operatorname {op} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82f9cf95f0d348dd38dffc1832a2ae34c11f1928" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.168ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mat} (n;R)^{\operatorname {op} }}"></span> 사이의 <a href="/wiki/%EB%8C%80%ED%95%A9_(%EC%88%98%ED%95%99)" title="대합 (수학)">대합</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-<a href="/wiki/%EA%B2%B0%ED%95%A9_%EB%8C%80%EC%88%98" title="결합 대수">결합 대수</a> <a href="/wiki/%EB%8F%99%ED%98%95" class="mw-redirect" title="동형">동형</a>이며, 만약 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>가 <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a>일 경우 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91e9f24a35d629b7bc279a9565e553b1893523d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.2ex; height:2.843ex;" alt="{\displaystyle \operatorname {Mat} (n;R)}"></span>는 전치 행렬에 따라 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-<a href="/wiki/%EB%8C%80%ED%95%A9_%EB%8C%80%EC%88%98" title="대합 대수">대합 대수</a>를 이룬다. </p> <div class="mw-heading mw-heading3"><h3 id="대각합"><span id=".EB.8C.80.EA.B0.81.ED.95.A9"></span>대각합</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=9" title="부분 편집: 대각합"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EB%8C%80%EA%B0%81%ED%95%A9" title="대각합">대각합</a>입니다.</div> <p><a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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> 정사각 행렬 <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\in \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bae03fe1013ece5fefa7e970c4bd28c3493109e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.784ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (n;R)}"></span>의 <b><a href="/wiki/%EB%8C%80%EA%B0%81%ED%95%A9" title="대각합">대각합</a></b>은 모든 대각 성분들의 합이다. </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 \operatorname {tr} A=\sum _{i=1}^{n}A_{ii}=A_{11}+A_{22}+\cdots +A_{nn}\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mi>A</mi> <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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} A=\sum _{i=1}^{n}A_{ii}=A_{11}+A_{22}+\cdots +A_{nn}\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c8c7d07df3bf0bb21b60126f004f0e83199c72e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:44.031ex; height:6.843ex;" alt="{\displaystyle \operatorname {tr} A=\sum _{i=1}^{n}A_{ii}=A_{11}+A_{22}+\cdots +A_{nn}\in R}"></span></dd></dl> <p>대각합 </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 \operatorname {tr} \colon \operatorname {Mat} (n;R)\to R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>:</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} \colon \operatorname {Mat} (n;R)\to R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fedcd08b706adbaf800dc756f03b212b34ae6d92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.428ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} \colon \operatorname {Mat} (n;R)\to R}"></span></dd></dl> <p>는 <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,R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (R,R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c541d71440092290621d4785f95797a1859fce7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.371ex; height:2.843ex;" alt="{\displaystyle (R,R)}"></span>-<a href="/wiki/%EC%84%A0%ED%98%95_%EB%B3%80%ED%99%98" title="선형 변환">선형 변환</a>을 이룬다. 또한, 임의의 <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\in \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bae03fe1013ece5fefa7e970c4bd28c3493109e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.784ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (n;R)}"></span>에 대하여, 그 대각합은 그 <a href="/wiki/%EC%A0%84%EC%B9%98_%ED%96%89%EB%A0%AC" title="전치 행렬">전치 행렬</a>의 대각합과 같다. </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 \operatorname {tr} (A^{\top })=\operatorname {tr} A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (A^{\top })=\operatorname {tr} A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f55ee01817cb3bf7398d169a7f9fb1f76d8cb1c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.924ex; height:3.176ex;" alt="{\displaystyle \operatorname {tr} (A^{\top })=\operatorname {tr} A}"></span></dd></dl> <p>만약 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>가 <a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a>일 경우, 임의의 두 행렬 <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,B\in \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B\in \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5eb6096c69733a6be70b9b2d8574d2341db504a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.582ex; height:2.843ex;" alt="{\displaystyle A,B\in \operatorname {Mat} (n;R)}"></span>에 대하여, 두 행렬의 곱의 대각합은 곱하는 순서와 무관하게 같다. </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 \operatorname {tr} (AB)=\operatorname {tr} (BA)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e3cc6cb1fb477706acf1c248d6f66882df7176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.364ex; height:2.843ex;" alt="{\displaystyle \operatorname {tr} (AB)=\operatorname {tr} (BA)}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="행렬식"><span id=".ED.96.89.EB.A0.AC.EC.8B.9D"></span>행렬식</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=10" title="부분 편집: 행렬식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a>입니다.</div> <p><a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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> 정사각 행렬 <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\in \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bae03fe1013ece5fefa7e970c4bd28c3493109e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.784ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (n;R)}"></span>의 <b><a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a></b>은 다음과 같다. </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 \det A=\sum _{\sigma \in \operatorname {Sym} (n)}\operatorname {sgn} \sigma \prod _{i=1}^{n}A_{i,\sigma (i)}\in R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>σ</mi> <mo>∈</mo> <mi>Sym</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mi>sgn</mi> <mo>⁡</mo> <mi>σ</mi> <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>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>σ</mi> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msub> <mo>∈</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det A=\sum _{\sigma \in \operatorname {Sym} (n)}\operatorname {sgn} \sigma \prod _{i=1}^{n}A_{i,\sigma (i)}\in R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/883f400d7aa54fe49136c66fa04f3e90774c5a52" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:35.522ex; height:7.343ex;" alt="{\displaystyle \det A=\sum _{\sigma \in \operatorname {Sym} (n)}\operatorname {sgn} \sigma \prod _{i=1}^{n}A_{i,\sigma (i)}\in R}"></span></dd></dl> <p>여기서 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Sym} (n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Sym</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {Sym} (n)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b53f7a393ab1e376574e61af57e6cd8591b0d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.66ex; height:2.843ex;" alt="{\displaystyle \operatorname {Sym} (n)}"></span>은 <a href="/wiki/%EB%8C%80%EC%B9%AD%EA%B5%B0" class="mw-disambig" title="대칭군">대칭군</a>이며, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>sgn</mi> <mo>⁡</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4231b394565cb9eaba08224d8ef8a2a151824f91" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.088ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} \sigma }"></span>는 <a href="/wiki/%EC%88%9C%EC%97%B4%EC%9D%98_%EB%B6%80%ED%98%B8" class="mw-redirect" title="순열의 부호">순열의 부호</a>이다. 행렬 <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>의 행렬식은 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f2d8fe180a2f848cf11e82a535b193cfe718742" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.36ex; height:2.176ex;" alt="{\displaystyle \det A}"></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 |A|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </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/648fce92f29d925f04d39244ccfe435320dfc6de" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\displaystyle |A|}"></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 \operatorname {D} (A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">D</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {D} (A)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8030d64cbd9cfa7e113da01f10ee5196ac9a918" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.328ex; height:2.843ex;" alt="{\displaystyle \operatorname {D} (A)}"></span> 등으로 표기한다. 특히, 2×2 행렬 <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\in \operatorname {Mat} (2;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (2;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c34abae8e3d199ca7126a3ca618892d3064c8b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.552ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (2;R)}"></span>의 행렬식은 다음과 같다. </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 \det A={\begin{vmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{vmatrix}}=A_{11}A_{22}-A_{12}A_{21}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det A={\begin{vmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{vmatrix}}=A_{11}A_{22}-A_{12}A_{21}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d25ccb01586cc09cd571d7fdfef220aeba9d828f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.481ex; height:6.176ex;" alt="{\displaystyle \det A={\begin{vmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{vmatrix}}=A_{11}A_{22}-A_{12}A_{21}}"></span></dd></dl> <p>행렬식은 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>개의 행벡터(또는 열벡터)의 함수 </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 \det \colon \operatorname {Mat} (n;R)=\underbrace {\operatorname {Mat} (1,n;R)\oplus \cdots \oplus \operatorname {Mat} (1,n;R)} _{n}\to R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo>:</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munder> <mo stretchy="false">→</mo> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det \colon \operatorname {Mat} (n;R)=\underbrace {\operatorname {Mat} (1,n;R)\oplus \cdots \oplus \operatorname {Mat} (1,n;R)} _{n}\to R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5d474fb5eed809fe77f2f64eacf71c1dd59760" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:56.137ex; height:6.176ex;" alt="{\displaystyle \det \colon \operatorname {Mat} (n;R)=\underbrace {\operatorname {Mat} (1,n;R)\oplus \cdots \oplus \operatorname {Mat} (1,n;R)} _{n}\to R}"></span></dd></dl> <p>로서, <a href="/wiki/%EB%8B%A8%EC%9C%84_%ED%96%89%EB%A0%AC" class="mw-redirect" title="단위 행렬">단위 행렬</a>의 <a href="/wiki/%EC%83%81_(%EC%88%98%ED%95%99)" title="상 (수학)">상</a>이 1인 유일한 <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>-<a href="/w/index.php?title=%EA%B5%90%EB%8C%80_%EB%8B%A4%EC%A4%91_%EC%84%A0%ED%98%95_%ED%98%95%EC%8B%9D&action=edit&redlink=1" class="new" title="교대 다중 선형 형식 (없는 문서)">교대 다중 선형 형식</a>이다. 또한, 행렬식은 두 환의 곱셈 <a href="/wiki/%EB%AA%A8%EB%85%B8%EC%9D%B4%EB%93%9C" title="모노이드">모노이드</a> 사이의 준동형이며, <a href="/wiki/%EC%A0%84%EC%B9%98_%ED%96%89%EB%A0%AC" title="전치 행렬">전치 행렬</a>에 대하여 불변이다. 즉, 임의의 <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,B\in \operatorname {Mat} (n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A,B\in \operatorname {Mat} (n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5eb6096c69733a6be70b9b2d8574d2341db504a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.582ex; height:2.843ex;" alt="{\displaystyle A,B\in \operatorname {Mat} (n;R)}"></span>에 대하여, </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 \det(AB)=\det A\det B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> <mo movablelimits="true" form="prefix">det</mo> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(AB)=\det A\det B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83244264d25b1b2cd8c0424fa1e01878a22b68e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.772ex; height:2.843ex;" alt="{\displaystyle \det(AB)=\det A\det B}"></span></dd> <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 \det A^{\top }=\det A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det A^{\top }=\det A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0076f34f2b45d94bfb67c4a0be7c22eced3ab3fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.329ex; height:2.676ex;" alt="{\displaystyle \det A^{\top }=\det A}"></span></dd></dl> <p>이다. </p><p>행렬식은 <a href="/wiki/%ED%81%AC%EB%9D%BC%EB%A9%94%EB%A5%B4_%EA%B3%B5%EC%8B%9D" class="mw-redirect" title="크라메르 공식">크라메르 공식</a>에서 사용된다. </p> <div class="mw-heading mw-heading3"><h3 id="부분_행렬과_소행렬식"><span id=".EB.B6.80.EB.B6.84_.ED.96.89.EB.A0.AC.EA.B3.BC_.EC.86.8C.ED.96.89.EB.A0.AC.EC.8B.9D"></span>부분 행렬과 소행렬식</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=11" title="부분 편집: 부분 행렬과 소행렬식"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r34311305"><div role="note" class="hatnote navigation-not-searchable"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/18px-Icons8_flat_search.svg.png" decoding="async" width="18" height="18" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/27px-Icons8_flat_search.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/52/Icons8_flat_search.svg/36px-Icons8_flat_search.svg.png 2x" data-file-width="512" data-file-height="512" /></span></span> 이 부분의 본문은 <a href="/wiki/%EB%B6%80%EB%B6%84_%ED%96%89%EB%A0%AC" title="부분 행렬">부분 행렬</a>입니다.</div> <p><a href="/wiki/%ED%99%98_(%EC%88%98%ED%95%99)" title="환 (수학)">환</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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span> 위의 <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 m\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}"></span> 행렬 <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\in \operatorname {Mat} (m,n;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\in \operatorname {Mat} (m,n;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0a7b74fc96ebe6b858c776506c1065f6932bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.858ex; height:2.843ex;" alt="{\displaystyle A\in \operatorname {Mat} (m,n;R)}"></span>의, 행과 열의 집합 </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 I=\{i_{1},i_{2},\dotsc ,i_{|I|}\}\subseteq \{1,\dotsc ,m\}\qquad (i_{1}<i_{2}<\cdots <i_{|I|})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <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> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo fence="false" stretchy="false">}</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <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> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I=\{i_{1},i_{2},\dotsc ,i_{|I|}\}\subseteq \{1,\dotsc ,m\}\qquad (i_{1}<i_{2}<\cdots <i_{|I|})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9524030511441fcc3c8e8797628e1157ca36c51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:58.068ex; height:3.176ex;" alt="{\displaystyle I=\{i_{1},i_{2},\dotsc ,i_{|I|}\}\subseteq \{1,\dotsc ,m\}\qquad (i_{1}<i_{2}<\cdots <i_{|I|})}"></span></dd> <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 J=\{j_{1},j_{2},\dotsc ,j_{|J|}\}\subseteq \{1,\dotsc ,n\}\qquad (j_{1}<j_{2}<\cdots <j_{|J|})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>⊆</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> <mspace width="2em" /> <mo stretchy="false">(</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo><</mo> <mo>⋯</mo> <mo><</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=\{j_{1},j_{2},\dotsc ,j_{|J|}\}\subseteq \{1,\dotsc ,n\}\qquad (j_{1}<j_{2}<\cdots <j_{|J|})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c56b6727c0f310c70b120ab71c16cf5ec5ea92d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:59.079ex; height:3.176ex;" alt="{\displaystyle J=\{j_{1},j_{2},\dotsc ,j_{|J|}\}\subseteq \{1,\dotsc ,n\}\qquad (j_{1}<j_{2}<\cdots <j_{|J|})}"></span></dd></dl> <p>에 속하는 행과 열을 취한 <b><a href="/wiki/%EB%B6%80%EB%B6%84_%ED%96%89%EB%A0%AC" title="부분 행렬">부분 행렬</a></b>은 다음과 같다. </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_{I,J}={\begin{pmatrix}A_{i_{1},j_{1}}&A_{i_{1},j_{2}}&\cdots &A_{i_{1},j_{|J|}}\\A_{i_{2},j_{1}}&A_{i_{2},j_{2}}&\cdots &A_{i_{2},j_{|J|}}\\\vdots &\vdots &\ddots &\vdots \\A_{i_{|I|},j_{1}}&A_{i_{|I|},j_{2}}&\cdots &A_{i_{|I|},j_{|J|}}\end{pmatrix}}\in \operatorname {Mat} (|I|,|J|;R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>,</mo> <mi>J</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mrow> </msub> </mrow> </msub> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>∈</mo> <mi>Mat</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>;</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{I,J}={\begin{pmatrix}A_{i_{1},j_{1}}&A_{i_{1},j_{2}}&\cdots &A_{i_{1},j_{|J|}}\\A_{i_{2},j_{1}}&A_{i_{2},j_{2}}&\cdots &A_{i_{2},j_{|J|}}\\\vdots &\vdots &\ddots &\vdots \\A_{i_{|I|},j_{1}}&A_{i_{|I|},j_{2}}&\cdots &A_{i_{|I|},j_{|J|}}\end{pmatrix}}\in \operatorname {Mat} (|I|,|J|;R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67a108318e5a16df08fe42f50a5f2dcf567aa016" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; width:59.262ex; height:16.176ex;" alt="{\displaystyle A_{I,J}={\begin{pmatrix}A_{i_{1},j_{1}}&A_{i_{1},j_{2}}&\cdots &A_{i_{1},j_{|J|}}\\A_{i_{2},j_{1}}&A_{i_{2},j_{2}}&\cdots &A_{i_{2},j_{|J|}}\\\vdots &\vdots &\ddots &\vdots \\A_{i_{|I|},j_{1}}&A_{i_{|I|},j_{2}}&\cdots &A_{i_{|I|},j_{|J|}}\end{pmatrix}}\in \operatorname {Mat} (|I|,|J|;R)}"></span></dd></dl> <p>특히, </p> <ul><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>의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>에 대한 <b><a href="/wiki/%EC%A3%BC%EB%B6%80%EB%B6%84_%ED%96%89%EB%A0%AC" class="mw-redirect" title="주부분 행렬">주부분 행렬</a></b>은 부분 행렬 <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,I}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>I</mi> <mo>,</mo> <mi>I</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{I,I}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/244d57898e3062d5e0c359d8a9db88be11494d82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.09ex; height:2.843ex;" alt="{\displaystyle A_{I,I}}"></span>를 뜻한다.<sup id="cite_ref-Golub_3-0" class="reference"><a href="#cite_note-Golub-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:24, §1.3.3</sup></span></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>의 <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 k\times k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>×</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\times k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77bcf9346bcb189917b6b49c4331b4483f4a4a2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.263ex; height:2.176ex;" alt="{\displaystyle k\times k}"></span> <b><a href="/wiki/%EC%84%A0%ED%96%89_%EC%A3%BC%EB%B6%80%EB%B6%84_%ED%96%89%EB%A0%AC" class="mw-redirect" title="선행 주부분 행렬">선행 주부분 행렬</a></b>은 부분 행렬 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\{1,\dotsc ,k\},\{1,\dotsc ,k\}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\{1,\dotsc ,k\},\{1,\dotsc ,k\}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/438336e8ed05649a83193fb36e406268ff097f92" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.758ex; height:3.009ex;" alt="{\displaystyle A_{\{1,\dotsc ,k\},\{1,\dotsc ,k\}}}"></span>를 뜻한다.<sup id="cite_ref-Golub_3-1" class="reference"><a href="#cite_note-Golub-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><span class="reference" style="white-space: nowrap;"><sup>:24, §1.3.3</sup></span></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>의 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>번째 <b><a href="/wiki/%ED%96%89%EB%B2%A1%ED%84%B0" class="mw-redirect" title="행벡터">행벡터</a></b>는 <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,\dotsc ,n\}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo fence="false" stretchy="false">}</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i,\{1,\dotsc ,n\}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba04db2bb47d369c951878da308eb1896475df9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.293ex; height:3.009ex;" alt="{\displaystyle A_{i,\{1,\dotsc ,n\}}}"></span>이다.</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>의 <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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>번째 <b><a href="/wiki/%EC%97%B4%EB%B2%A1%ED%84%B0" class="mw-redirect" title="열벡터">열벡터</a></b>는 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A_{\{1,\dotsc ,m\},j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">{</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{\{1,\dotsc ,m\},j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/179108db61ed97e970cb9dac9247b7dfb65473c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.859ex; height:3.009ex;" alt="{\displaystyle A_{\{1,\dotsc ,m\},j}}"></span>이다.</li></ul> <p><a href="/wiki/%EA%B0%80%ED%99%98%ED%99%98" title="가환환">가환환</a> 위의 행렬의 부분 <a href="/wiki/%EC%A0%95%EC%82%AC%EA%B0%81_%ED%96%89%EB%A0%AC" class="mw-redirect" title="정사각 행렬">정사각 행렬</a>의 <a href="/wiki/%ED%96%89%EB%A0%AC%EC%8B%9D" title="행렬식">행렬식</a>을 <b><a href="/wiki/%EC%86%8C%ED%96%89%EB%A0%AC%EC%8B%9D" class="mw-redirect" title="소행렬식">소행렬식</a></b>이라고 한다. </p> <div class="mw-heading mw-heading2"><h2 id="예"><span id=".EC.98.88"></span>예</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=12" title="부분 편집: 예"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>몇몇 특수한 행렬들은 다음이 있다. </p> <ul><li><a href="/wiki/%EC%98%81%ED%96%89%EB%A0%AC" title="영행렬">영행렬</a></li> <li><a href="/wiki/%EB%8B%A8%EC%9C%84_%ED%96%89%EB%A0%AC" class="mw-redirect" title="단위 행렬">단위 행렬</a></li> <li><a href="/wiki/%EC%8A%A4%EC%B9%BC%EB%9D%BC_%ED%96%89%EB%A0%AC" title="스칼라 행렬">스칼라 행렬</a></li> <li><a href="/wiki/%EB%8C%80%EA%B0%81_%ED%96%89%EB%A0%AC" title="대각 행렬">대각 행렬</a></li> <li><a href="/wiki/%EC%82%BC%EA%B0%81_%ED%96%89%EB%A0%AC" class="mw-redirect" title="삼각 행렬">삼각 행렬</a></li> <li><a href="/wiki/%EB%8C%80%EC%B9%AD_%ED%96%89%EB%A0%AC" class="mw-redirect" title="대칭 행렬">대칭 행렬</a></li> <li><a href="/wiki/%EB%B0%98%EB%8C%80%EC%B9%AD_%ED%96%89%EB%A0%AC" title="반대칭 행렬">반대칭 행렬</a></li> <li><a href="/wiki/%EC%A7%81%EA%B5%90_%ED%96%89%EB%A0%AC" class="mw-redirect" title="직교 행렬">직교 행렬</a></li> <li><a href="/wiki/%EC%97%90%EB%A5%B4%EB%AF%B8%ED%8A%B8_%ED%96%89%EB%A0%AC" title="에르미트 행렬">에르미트 행렬</a></li> <li><a href="/w/index.php?title=%EB%B0%98%EC%97%90%EB%A5%B4%EB%AF%B8%ED%8A%B8_%ED%96%89%EB%A0%AC&action=edit&redlink=1" class="new" title="반에르미트 행렬 (없는 문서)">반에르미트 행렬</a></li> <li><a href="/wiki/%EC%9C%A0%EB%8B%88%ED%84%B0%EB%A6%AC_%ED%96%89%EB%A0%AC" title="유니터리 행렬">유니터리 행렬</a></li> <li><a href="/wiki/%EC%A0%95%EA%B7%9C_%ED%96%89%EB%A0%AC" title="정규 행렬">정규 행렬</a></li> <li><a href="/wiki/%EC%A0%95%EB%B6%80%ED%98%B8_%ED%96%89%EB%A0%AC" title="정부호 행렬">정부호 행렬</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="역사와_어원"><span id=".EC.97.AD.EC.82.AC.EC.99.80_.EC.96.B4.EC.9B.90"></span>역사와 어원</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=13" title="부분 편집: 역사와 어원"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>1848년 수학에 처음으로 <a href="/wiki/%EC%A0%9C%EC%9E%84%EC%8A%A4_%EC%A1%B0%EC%A7%80%ED%94%84_%EC%8B%A4%EB%B2%A0%EC%8A%A4%ED%84%B0" title="제임스 조지프 실베스터">실베스터</a>가 사용한 <b>행렬(matrix)</b>이라는 단어의 <a href="/wiki/%EC%96%B4%EC%9B%90" class="mw-redirect" title="어원">어원</a>은 해부학에서 <a href="/wiki/%EC%9E%90%EA%B6%81" title="자궁">자궁</a>(子宮,모체母體)을 뜻한다. 행렬식에 대해서 행렬의 의미를 표현한 것으로 전해진다.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="같이_보기"><span id=".EA.B0.99.EC.9D.B4_.EB.B3.B4.EA.B8.B0"></span>같이 보기</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=14" title="부분 편집: 같이 보기"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%ED%85%90%EC%84%9C" title="텐서">텐서</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="참고_문헌"><span id=".EC.B0.B8.EA.B3.A0_.EB.AC.B8.ED.97.8C"></span>참고 문헌</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=15" title="부분 편집: 참고 문헌"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r35556958">.mw-parser-output .reflist{font-size:90%;margin-bottom:0.5em;list-style-type:decimal}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Lang-1"><span class="mw-cite-backlink"><a href="#cite_ref-Lang_1-0">↑</a></span> <span class="reference-text"><cite class="citation book"><a href="/wiki/%EC%84%9C%EC%A7%80_%EB%9E%AD" title="서지 랭">Lang, Serge</a> (2002). 《Algebra》. Graduate Texts in Mathematics (영어) <b>211</b> 개정 3판. New York, NY: Springer. <a href="/wiki/%EB%94%94%EC%A7%80%ED%84%B8_%EA%B0%9D%EC%B2%B4_%EC%8B%9D%EB%B3%84%EC%9E%90" title="디지털 객체 식별자">doi</a>:<a rel="nofollow" class="external text" href="https://dx.doi.org/10.1007%2F978-1-4613-0041-0">10.1007/978-1-4613-0041-0</a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-4612-6551-1" title="특수:책찾기/978-1-4612-6551-1"><bdi>978-1-4612-6551-1</bdi></a>. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EC%9D%BC%EB%A0%A8_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 일련 번호">ISSN</a> <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/0072-5285">0072-5285</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1878556">1878556</a>. <a href="/wiki/%EC%B2%B8%ED%8A%B8%EB%9E%84%EB%B8%94%EB%9D%BC%ED%8A%B8_%EB%A7%88%ED%8A%B8" title="첸트랄블라트 마트">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:0984.00001">0984.00001</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.place=New+York%2C+NY&rft.series=Graduate+Texts+in+Mathematics&rft.edition=%EA%B0%9C%EC%A0%95+3&rft.pub=Springer&rft.date=2002&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A0984.00001&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1878556&rft_id=info%3Adoi%2F10.1007%2F978-1-4613-0041-0&rft.issn=0072-5285&rft.isbn=978-1-4612-6551-1&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%96%89%EB%A0%AC" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Kharab-2"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Kharab_2-0">가</a></sup> <sup><a href="#cite_ref-Kharab_2-1">나</a></sup> <sup><a href="#cite_ref-Kharab_2-2">다</a></sup> <sup><a href="#cite_ref-Kharab_2-3">라</a></sup> <sup><a href="#cite_ref-Kharab_2-4">마</a></sup> <sup><a href="#cite_ref-Kharab_2-5">바</a></sup></span> <span class="reference-text"><cite class="citation book">Abdelwahab Kharab; Ronald B. Guenther (2013). 《An Introduction to Numerical Methods A MATLAB Approach》 [이공학도를 위한 수치해석]. 학산미디어. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-89-966211-8-8" title="특수:책찾기/978-89-966211-8-8"><bdi>978-89-966211-8-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+Numerical+Methods+A+MATLAB+Approach&rft.pub=%ED%95%99%EC%82%B0%EB%AF%B8%EB%94%94%EC%96%B4&rft.date=2013&rft.isbn=978-89-966211-8-8&rft.au=Abdelwahab+Kharab&rft.au=Ronald+B.+Guenther&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%96%89%EB%A0%AC" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-Golub-3"><span class="mw-cite-backlink">↑ <sup><a href="#cite_ref-Golub_3-0">가</a></sup> <sup><a href="#cite_ref-Golub_3-1">나</a></sup></span> <span class="reference-text"><cite class="citation book">Golub, Gene H.; Van Loan, Charles F. (2013). <a rel="nofollow" class="external text" href="https://archive.org/details/matrixcomputatio0004golu">《Matrix computations》</a>. Johns Hopkins Studies in the Mathematical Sciences (영어) 4판. Baltimore: The Johns Hopkins University Press. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-4214-0794-4" title="특수:책찾기/978-1-4214-0794-4"><bdi>978-1-4214-0794-4</bdi></a>. <a href="/wiki/%EB%AF%B8%EA%B5%AD_%EC%9D%98%ED%9A%8C%EB%8F%84%EC%84%9C%EA%B4%80_%EC%A0%9C%EC%96%B4_%EB%B2%88%ED%98%B8" title="미국 의회도서관 제어 번호">LCCN</a> <a rel="nofollow" class="external text" href="//lccn.loc.gov/2012943449">2012943449</a>. <a href="/wiki/%EC%88%98%ED%95%99_%EB%A6%AC%EB%B7%B0" title="수학 리뷰">MR</a> <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=3024913">3024913</a>. <a href="/wiki/%EC%B2%B8%ED%8A%B8%EB%9E%84%EB%B8%94%EB%9D%BC%ED%8A%B8_%EB%A7%88%ED%8A%B8" title="첸트랄블라트 마트">Zbl</a> <a rel="nofollow" class="external text" href="//zbmath.org/?format=complete&q=an:1268.65037">1268.65037</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+computations&rft.place=Baltimore&rft.series=Johns+Hopkins+Studies+in+the+Mathematical+Sciences&rft.edition=4&rft.pub=The+Johns+Hopkins+University+Press&rft.date=2013&rft_id=%2F%2Fzbmath.org%2F%3Fformat%3Dcomplete%26q%3Dan%3A1268.65037&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D3024913&rft_id=info%3Alccn%2F2012943449&rft.isbn=978-1-4214-0794-4&rft.aulast=Golub&rft.aufirst=Gene+H.&rft.au=Van+Loan%2C+Charles+F.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmatrixcomputatio0004golu&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%96%89%EB%A0%AC" class="Z3988"><span style="display:none;"> </span></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">고등기하와 벡터, 성지출판 (<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 {I} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">I</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {I} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a458c8aeb096ce732abf346ae8edf3e4f53a126" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.014ex; height:2.176ex;" alt="{\displaystyle \mathbf {I} }"></span> 일차변환과 행렬) 수학이야기-행렬과 행렬식30p</span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="외부_링크"><span id=".EC.99.B8.EB.B6.80_.EB.A7.81.ED.81.AC"></span>외부 링크</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%ED%96%89%EB%A0%AC&action=edit&section=16" title="부분 편집: 외부 링크"><span>편집</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/12px-Commons-logo.svg.png" decoding="async" width="12" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/18px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/24px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span> 위키미디어 공용에 <span class="plainlinks"><a class="external text" href="https://commons.wikimedia.org/wiki/Category:Matrices?uselang=ko">행렬</a></span> 관련 미디어 분류가 있습니다.</li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://encyclopediaofmath.org/wiki/Matrix">“Matrix”</a>. 《Encyclopedia of Mathematics》 (영어). Springer-Verlag. 2001. <a href="/wiki/%EA%B5%AD%EC%A0%9C_%ED%91%9C%EC%A4%80_%EB%8F%84%EC%84%9C_%EB%B2%88%ED%98%B8" class="mw-redirect" title="국제 표준 도서 번호">ISBN</a> <a href="/wiki/%ED%8A%B9%EC%88%98:%EC%B1%85%EC%B0%BE%EA%B8%B0/978-1-55608-010-4" title="특수:책찾기/978-1-55608-010-4"><bdi>978-1-55608-010-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Encyclopedia+of+Mathematics&rft.atitle=Matrix&rft.date=2001&rft.isbn=978-1-55608-010-4&rft_id=https%3A%2F%2Fencyclopediaofmath.org%2Fwiki%2FMatrix&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%96%89%EB%A0%AC" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Matrix.html">“Matrix”</a>. 《<a href="/wiki/%EB%A7%A4%EC%8A%A4%EC%9B%94%EB%93%9C" title="매스월드">Wolfram MathWorld</a>》 (영어). Wolfram Research.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Wolfram+MathWorld&rft.atitle=Matrix&rft.aulast=Weisstein&rft.aufirst=Eric+Wolfgang&rft_id=https%3A%2F%2Fmathworld.wolfram.com%2FMatrix.html&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%96%89%EB%A0%AC" class="Z3988"><span style="display:none;"> </span></span></li> <li><cite class="citation web"><a rel="nofollow" class="external text" href="https://ncatlab.org/nlab/show/matrix">“Matrix”</a>. 《nLab》 (영어).</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=nLab&rft.atitle=Matrix&rft_id=https%3A%2F%2Fncatlab.org%2Fnlab%2Fshow%2Fmatrix&rfr_id=info%3Asid%2Fko.wikipedia.org%3A%ED%96%89%EB%A0%AC" class="Z3988"><span style="display:none;"> </span></span></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r36480591">.mw-parser-output .hlist dl,.mw-parser-output .hlist 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title="다중선형대수학">다중선형대수학</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/%EA%B8%B0%ED%95%98%EC%A0%81_%EB%8C%80%EC%88%98%ED%95%99" title="기하적 대수학">기하적 대수학</a></li> <li><a href="/wiki/%EC%99%B8%EB%8C%80%EC%88%98" title="외대수">외대수</a></li> <li><a href="/w/index.php?title=%EC%9D%B4%EC%A4%91%EB%B2%A1%ED%84%B0&action=edit&redlink=1" class="new" title="이중벡터 (없는 문서)">이중벡터</a></li> <li><a href="/w/index.php?title=%EB%8B%A4%EC%A4%91%EB%B2%A1%ED%84%B0&action=edit&redlink=1" class="new" title="다중벡터 (없는 문서)">다중벡터</a></li> <li><a href="/wiki/%ED%85%90%EC%84%9C" title="텐서">텐서</a></li> <li><a href="/w/index.php?title=%EC%95%84%EC%9A%B0%ED%84%B0%EB%AA%A8%ED%94%BC%EC%A6%98&action=edit&redlink=1" class="new" title="아우터모피즘 (없는 문서)">아우터모피즘</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a class="mw-selflink selflink">행렬</a></th><td class="navbox-list-with-group 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href="/w/index.php?title=%EC%88%98%EC%B9%98_%EB%B6%84%EC%84%9D_%EC%86%8C%ED%94%84%ED%8A%B8%EC%9B%A8%EC%96%B4_%EB%B9%84%EA%B5%90&action=edit&redlink=1" class="new" title="수치 분석 소프트웨어 비교 (없는 문서)">수치 분석 소프트웨어 비교</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3" style="font-weight:bold;"><div> <ul><li><span typeof="mw:File"><span title="분류"><img alt="분류" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/16px-Folder_Hexagonal_Icon.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/24px-Folder_Hexagonal_Icon.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Folder_Hexagonal_Icon.svg/32px-Folder_Hexagonal_Icon.svg.png 2x" data-file-width="36" data-file-height="31" /></span></span> <a href="/wiki/%EB%B6%84%EB%A5%98:%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99" title="분류:선형대수학">분류</a></li> <li><span typeof="mw:File"><span title="목록 문서"><img alt="목록 문서" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/16px-Symbol_list_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/23px-Symbol_list_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/db/Symbol_list_class.svg/31px-Symbol_list_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/w/index.php?title=%EC%84%A0%ED%98%95%EB%8C%80%EC%88%98%ED%95%99_%EC%A3%BC%EC%A0%9C_%EB%AA%A9%EB%A1%9D&action=edit&redlink=1" class="new" title="선형대수학 주제 목록 (없는 문서)">개요</a></li> <li><span typeof="mw:File"><span title="포털"><img alt="포털" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Portal.svg/16px-Portal.svg.png" decoding="async" width="16" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Portal.svg/24px-Portal.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Portal.svg/32px-Portal.svg.png 2x" data-file-width="36" data-file-height="32" /></span></span> <a href="/wiki/%ED%8F%AC%ED%84%B8:%EC%88%98%ED%95%99" title="포털:수학">수학 포털</a></li> <li><span typeof="mw:File"><span title="위키책"><img alt="위키책" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/16px-Wikibooks-logo.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/24px-Wikibooks-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fa/Wikibooks-logo.svg/32px-Wikibooks-logo.svg.png 2x" data-file-width="300" data-file-height="300" /></span></span> <a href="https://en.wikipedia.org/wiki/wikibooks:Linear_algebra" class="extiw" title="en:wikibooks:Linear algebra">위키책</a></li> <li><span typeof="mw:File"><span title="위키배움터"><img alt="위키배움터" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/16px-Wikiversity-logo.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/24px-Wikiversity-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/91/Wikiversity-logo.svg/32px-Wikiversity-logo.svg.png 2x" data-file-width="1000" data-file-height="800" /></span></span> <a href="https://en.wikipedia.org/wiki/wikiversity:Linear_algebra" class="extiw" title="en:wikiversity:Linear algebra">위키배움터</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r36480591"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r38501015"></div><div role="navigation" class="navbox authority-control" 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cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock 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