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class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/symbol.html" aria-label="常用数学符号"><!--[--><!--[--><!--]--><!--]-->常用数学符号<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/group.html" aria-label="群"><!--[--><!--[--><!--]--><!--]-->群<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_1.html" aria-label="数论（一）"><!--[--><!--[--><!--]--><!--]-->数论（一）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_2.html" aria-label="数论（二）"><!--[--><!--[--><!--]--><!--]-->数论（二）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/math/number_theory_3.html" aria-label="数论（三）"><!--[--><!--[--><!--]--><!--]-->数论（三）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a 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active collapsible">图形学 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item active collapsible">数学基础 <span class="down arrow"></span></p><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><!--[--><!--]--><!--]-->矢量<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link route-link-active auto-link vp-sidebar-item active" href="/note/graphics/math/math_02.html" aria-label="矩阵"><!--[--><!--[--><!--]--><!--]-->矩阵<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_3-矩阵" aria-label="3. 矩阵"><!--[--><!--[--><!--]--><!--]-->3. 矩阵<!--[--><!--[--><!--]--><!--]--></a><ul style="" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="#_3-1-定义" aria-label="3.1 定义"><!--[--><!--[--><!--]--><!--]-->3.1 定义<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-2-加法" aria-label="3.2 加法"><!--[--><!--[--><!--]--><!--]-->3.2 加法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-3-乘法" aria-label="3.3 乘法"><!--[--><!--[--><!--]--><!--]-->3.3 乘法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-4-转置矩阵" aria-label="3.4 转置矩阵"><!--[--><!--[--><!--]--><!--]-->3.4 转置矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-4-3-正交矩阵" aria-label="3.4.3 正交矩阵"><!--[--><!--[--><!--]--><!--]-->3.4.3 正交矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-5-基本运算规则" aria-label="3.5 基本运算规则"><!--[--><!--[--><!--]--><!--]-->3.5 基本运算规则<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-6-单位矩阵" aria-label="3.6 单位矩阵"><!--[--><!--[--><!--]--><!--]-->3.6 单位矩阵<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-7-子式矩阵-minor-matrix" aria-label="3.7 子式矩阵(Minor Matrix)"><!--[--><!--[--><!--]--><!--]-->3.7 子式矩阵(Minor Matrix)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-8-行列式-determinant-of-a-matrix" aria-label="3.8 行列式（Determinant of a Matrix）"><!--[--><!--[--><!--]--><!--]-->3.8 行列式（Determinant of a Matrix）<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-9-伴随矩阵-adjoint-of-a-matrix" aria-label="3.9 伴随矩阵(Adjoint of a Matrix)"><!--[--><!--[--><!--]--><!--]-->3.9 伴随矩阵(Adjoint of a Matrix)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="#_3-10-逆矩阵-inverse-of-a-matrix" aria-label="3.10 逆矩阵(Inverse of a Matrix)"><!--[--><!--[--><!--]--><!--]-->3.10 逆矩阵(Inverse of a Matrix)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><!--[--><!--]--><!--]-->立体角<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_01.html" aria-label="几何变换(一)"><!--[--><!--[--><!--]--><!--]-->几何变换(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_02.html" aria-label="几何变换(二)"><!--[--><!--[--><!--]--><!--]-->几何变换(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_03.html" aria-label="法线变换"><!--[--><!--[--><!--]--><!--]-->法线变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/math/transform_04.html" aria-label="摄像机变换"><!--[--><!--[--><!--]--><!--]-->摄像机变换<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">光照模型 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/classic/illumination_model_01.html" aria-label="传统光照模型"><!--[--><!--[--><!--]--><!--]-->传统光照模型<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/luminosity/luminosity.html" aria-label="光度学"><!--[--><!--[--><!--]--><!--]-->光度学<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/BRDF/BRDF.html" aria-label="双向反射分布函数(BRDF)"><!--[--><!--[--><!--]--><!--]-->双向反射分布函数(BRDF)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_1.html" aria-label="微平面理论(一)"><!--[--><!--[--><!--]--><!--]-->微平面理论(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_2.html" aria-label="微平面理论(二)"><!--[--><!--[--><!--]--><!--]-->微平面理论(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/microfacets/microfacets_3.html" aria-label="微平面理论(三)"><!--[--><!--[--><!--]--><!--]-->微平面理论(三)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/illumination_model/render_function/render_function_1.html" aria-label="光照方程"><!--[--><!--[--><!--]--><!--]-->光照方程<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">环境光渲染 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_01.html" aria-label="环境光渲染(一)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(一)<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/graphics/image_based_lighting/ibl_02.html" aria-label="环境光渲染(二)"><!--[--><!--[--><!--]--><!--]-->环境光渲染(二)<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><li><p tabindex="0" class="vp-sidebar-item collapsible">编程语言 <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><p tabindex="0" class="vp-sidebar-item collapsible">JavaScript <span class="right arrow"></span></p><ul style="display:none;" class="vp-sidebar-children"><!--[--><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/helloworld.html" aria-label="环境搭建"><!--[--><!--[--><!--]--><!--]-->环境搭建<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/grammar.html" aria-label="基本语法"><!--[--><!--[--><!--]--><!--]-->基本语法<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/function.html" aria-label="函数"><!--[--><!--[--><!--]--><!--]-->函数<!--[--><!--[--><!--]--><!--]--></a><!----></li><li><a class="route-link auto-link vp-sidebar-item" href="/note/language/javascript/object_and_class.html" aria-label="对象和类"><!--[--><!--[--><!--]--><!--]-->对象和类<!--[--><!--[--><!--]--><!--]--></a><!----></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul></li><!--]--></ul><!--[--><!--]--></aside><!--]--><!--[--><main class="vp-page"><!--[--><!--]--><div vp-content><!--[--><!--]--><div><h1 id="数学基础-二" tabindex="-1"><a class="header-anchor" href="#数学基础-二"><span>数学基础(二）</span></a></h1><hr><h2 id="_3-矩阵" tabindex="-1"><a class="header-anchor" href="#_3-矩阵"><span>3. 矩阵</span></a></h2><h3 id="_3-1-定义" tabindex="-1"><a class="header-anchor" href="#_3-1-定义"><span>3.1 定义</span></a></h3><p>一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>的矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container>是一个有<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></mjx-assistive-mml></mjx-container>行<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></mjx-assistive-mml></mjx-container>列的数组， <mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">A</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>表示第<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></mjx-assistive-mml></mjx-container>行第<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></mjx-assistive-mml></mjx-container>列的元素， <mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">A</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mo>∗</mo></mrow></msub></math></mjx-assistive-mml></mjx-container>表示第<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>i</mi></math></mjx-assistive-mml></mjx-container>行所有元素， <mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi mathvariant="bold-italic">A</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>表示第<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>j</mi></math></mjx-assistive-mml></mjx-container>列所有元素，</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:10.747em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:6.62em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.912em;vertical-align:-1.706em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:5.737em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2190"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2190"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2190"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2192"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.912em;vertical-align:-1.706em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.837em;vertical-align:-1.669em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:7.21em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2191"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2191"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2191"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2193"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2193"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2193"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.837em;vertical-align:-1.669em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi mathvariant="bold-italic">A</mi></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mo stretchy="false">←</mo></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>,</mo><mo>∗</mo></mrow></msub></mtd><mtd><mo stretchy="false">→</mo></mtd></mtr><mtr><mtd><mo stretchy="false">←</mo></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>2</mn><mo>,</mo><mo>∗</mo></mrow></msub></mtd><mtd><mo stretchy="false">→</mo></mtd></mtr><mtr><mtd><mo stretchy="false">←</mo></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>3</mn><mo>,</mo><mo>∗</mo></mrow></msub></mtd><mtd><mo stretchy="false">→</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mo stretchy="false">↑</mo></mtd><mtd><mo stretchy="false">↑</mo></mtd><mtd><mo stretchy="false">↑</mo></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mn>1</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mn>2</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mn>3</mn></mrow></msub></mtd></mtr><mtr><mtd><mo stretchy="false">↓</mo></mtd><mtd><mo stretchy="false">↓</mo></mtd><mtd><mo stretchy="false">↓</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h3 id="_3-2-加法" tabindex="-1"><a class="header-anchor" href="#_3-2-加法"><span>3.2 加法</span></a></h3><p>如果矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container>是一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>的矩阵，矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">B</mi></math></mjx-assistive-mml></mjx-container>是一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>的矩阵，那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="3"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mo>+</mo><mi mathvariant="bold-italic">B</mi></math></mjx-assistive-mml></mjx-container>是一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>的矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D46A TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">C</mi></math></mjx-assistive-mml></mjx-container>，并且</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.045em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>C</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><h3 id="_3-3-乘法" tabindex="-1"><a class="header-anchor" href="#_3-3-乘法"><span>3.3 乘法</span></a></h3><h4 id="_3-3-1-定义" tabindex="-1"><a class="header-anchor" href="#_3-3-1-定义"><span>3.3.1 定义</span></a></h4><p>矩阵之间的乘法，如果矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container>是一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>的矩阵，矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">B</mi></math></mjx-assistive-mml></mjx-container>是一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>×</mo><mi>p</mi></math></mjx-assistive-mml></mjx-container>的矩阵，那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">B</mi></math></mjx-assistive-mml></mjx-container>是一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mi>p</mi></math></mjx-assistive-mml></mjx-container>的矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D46A TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">C</mi></math></mjx-assistive-mml></mjx-container>，并且</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.045em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D435 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>C</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>,</mo><mo>∗</mo></mrow></msub><mo>⋅</mo><msub><mi>B</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><p>一般来说<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2260"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="4"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">B</mi><mo>≠</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container></p><h4 id="_3-3-2-举例" tabindex="-1"><a class="header-anchor" href="#_3-3-2-举例"><span>3.3.2 举例</span></a></h4><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:33.52em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:4.278em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-mrow space="2"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:5.056em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c 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class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:29.395em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c30"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c22C5"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n" space="2"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:5.556em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c39"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c37"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c35"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">B</mi></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mo>−</mo><mn>1</mn></mtd><mtd><mn>5</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>−</mo><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>1</mn></mtd><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>,</mo><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>,</mo><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>5</mn><mo>,</mo><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>0</mn><mo>,</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mtd><mtd><mo stretchy="false">(</mo><mn>3</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mo>−</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo stretchy="false">)</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>2</mn></mtd><mtd><mo>−</mo><mn>19</mn></mtd><mtd><mo>−</mo><mn>7</mn></mtd></mtr><mtr><mtd><mn>5</mn></mtd><mtd><mn>1</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h4 id="_3-3-3-矢量和矩阵相乘" tabindex="-1"><a class="header-anchor" href="#_3-3-3-矢量和矩阵相乘"><span>3.3.3 矢量和矩阵相乘</span></a></h4><p>以三维为例</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:14.567em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.105em;padding-left:0.372em;margin-bottom:-0.516em;"><mjx-mo class="mjx-n" style="width:0;margin-left:-0.25em;"><mjx-c class="mjx-c20D7 TEX-V"></mjx-c></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D496 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 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style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">u</mi><mo stretchy="false">→</mo></mover></mrow><mi mathvariant="bold-italic">A</mi></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mo stretchy="false">[</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">]</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mo stretchy="false">↑</mo></mtd><mtd><mo stretchy="false">↑</mo></mtd><mtd><mo stretchy="false">↑</mo></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mn>1</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mn>2</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mn>3</mn></mrow></msub></mtd></mtr><mtr><mtd><mo stretchy="false">↓</mo></mtd><mtd><mo stretchy="false">↓</mo></mtd><mtd><mo stretchy="false">↓</mo></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><mi>x</mi><msub><mi mathvariant="bold-italic">A</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mn>1</mn></mrow></msub><mo>+</mo><mi>y</mi><msub><mi mathvariant="bold-italic">A</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mn>2</mn></mrow></msub><mo>+</mo><mi>z</mi><msub><mi mathvariant="bold-italic">A</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo><mo>,</mo><mn>3</mn></mrow></msub></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h3 id="_3-4-转置矩阵" tabindex="-1"><a class="header-anchor" href="#_3-4-转置矩阵"><span>3.4 转置矩阵</span></a></h3><h4 id="_3-4-1-定义" tabindex="-1"><a class="header-anchor" href="#_3-4-1-定义"><span>3.4.1 定义</span></a></h4><p>将一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>的矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container>的行和列颠倒，变成一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>×</mo><mi>m</mi></math></mjx-assistive-mml></mjx-container>的矩阵，成为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container>的转置矩阵，记为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mi>T</mi></msup></math></mjx-assistive-mml></mjx-container></p><h4 id="_3-4-2-举例" tabindex="-1"><a class="header-anchor" href="#_3-4-2-举例"><span>3.4.2 举例</span></a></h4><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:8.895em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:5.056em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:3.556em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c36"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mn class="mjx-n"><mjx-c class="mjx-c38"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c34"></mjx-c></mjx-mn><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi mathvariant="bold-italic">A</mi></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>2</mn></mtd><mtd><mo>−</mo><mn>1</mn></mtd><mtd><mn>8</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>6</mn></mtd><mtd><mo>−</mo><mn>4</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd><msup><mi mathvariant="bold-italic">A</mi><mi>T</mi></msup></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mo>−</mo><mn>1</mn></mtd><mtd><mn>6</mn></mtd></mtr><mtr><mtd><mn>8</mn></mtd><mtd><mo>−</mo><mn>4</mn></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h3 id="_3-4-3-正交矩阵" tabindex="-1"><a class="header-anchor" href="#_3-4-3-正交矩阵"><span>3.4.3 正交矩阵</span></a></h3><p>如果一个矩阵满足<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="4"><mjx-c class="mjx-c1D470 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><msup><mi mathvariant="bold-italic">A</mi><mi>T</mi></msup><mo>=</mo><mi mathvariant="bold-italic">I</mi></math></mjx-assistive-mml></mjx-container>，则称<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container>为正交矩阵(<a href="https://en.wikipedia.org/wiki/Orthogonal_matrix" target="_blank" rel="noopener noreferrer">Orthogonal matrix</a>)。 正交矩阵满足如下特性：</p><ul><li>必须是一个方阵，即n行n列；</li><li>矩阵中的每一列若视作向量，则这些向量均两两相互垂直；</li><li>矩阵中的每一列若视作向量，则这些向量的长度均为1；</li></ul><h3 id="_3-5-基本运算规则" tabindex="-1"><a class="header-anchor" href="#_3-5-基本运算规则"><span>3.5 基本运算规则</span></a></h3><ul><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="3"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="4"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="3"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mo>+</mo><mi mathvariant="bold-italic">B</mi><mo>=</mo><mi mathvariant="bold-italic">B</mi><mo>+</mo><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="3"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="3"><mjx-c class="mjx-c1D46A TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="4"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="3"><mjx-c class="mjx-c1D46A TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mo>+</mo><mi mathvariant="bold-italic">B</mi><mo stretchy="false">)</mo><mo>+</mo><mi mathvariant="bold-italic">C</mi><mo>=</mo><mi mathvariant="bold-italic">A</mi><mo>+</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">B</mi><mo>+</mo><mi mathvariant="bold-italic">C</mi><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45F TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-b 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mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mi mathvariant="bold-italic">A</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi mathvariant="bold-italic">A</mi><mi>T</mi></msup><msup><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup></math></mjx-assistive-mml></mjx-container> 对于矩阵,交换律不生效，大部分情况下 <mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c2260"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="4"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">B</mi><mo>≠</mo><mi mathvariant="bold-italic">B</mi><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container></li></ul><h3 id="_3-6-单位矩阵" tabindex="-1"><a class="header-anchor" href="#_3-6-单位矩阵"><span>3.6 单位矩阵</span></a></h3><h4 id="_3-6-1-定义" tabindex="-1"><a class="header-anchor" href="#_3-6-1-定义"><span>3.6.1 定义</span></a></h4><p>一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>的矩阵，从左上角到右下角都是1，其他元素都是0，称为单位矩阵，用<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D470 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">I</mi></math></mjx-assistive-mml></mjx-container>表示，如果<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container>是一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45A TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>矩阵，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">B</mi></math></mjx-assistive-mml></mjx-container>是一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45D TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>×</mo><mi>p</mi></math></mjx-assistive-mml></mjx-container>矩阵，那么</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D470 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="4"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="2"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="4"><mjx-c class="mjx-c1D469 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="bold-italic">A</mi><mi mathvariant="bold-italic">I</mi><mo>=</mo><mi mathvariant="bold-italic">A</mi><mo>,</mo><mi mathvariant="bold-italic">B</mi><mo>=</mo><mi mathvariant="bold-italic">B</mi></math></mjx-assistive-mml></mjx-container><h3 id="_3-7-子式矩阵-minor-matrix" tabindex="-1"><a class="header-anchor" href="#_3-7-子式矩阵-minor-matrix"><span>3.7 子式矩阵(Minor Matrix)</span></a></h3><h4 id="_3-7-1-定义" tabindex="-1"><a class="header-anchor" href="#_3-7-1-定义"><span>3.7.1 定义</span></a></h4><p>对于一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>的矩阵，把第i行和第j列去掉之后的子矩阵，称为Minor Matrix(中文翻译比较乱），记为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mover><mjx-over style="padding-bottom:0.18em;margin-bottom:-0.248em;"><mjx-mo class="mjx-n"><mjx-stretchy-h style="width:0.869em;" class="mjx-c2013"><mjx-ext><mjx-c></mjx-c></mjx-ext></mjx-stretchy-h></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mover><mi mathvariant="bold-italic">A</mi><mo accent="true">―</mo></mover><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container>，比如对于一个3x3的矩阵</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5B"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo><mjx-mtable style="min-width:6.62em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding:0.2em 0.5em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-n"><mjx-stretchy-v style="height:3.8em;vertical-align:-1.65em;" class="mjx-c5D"><mjx-beg><mjx-c></mjx-c></mjx-beg><mjx-ext><mjx-c></mjx-c></mjx-ext><mjx-end><mjx-c></mjx-c></mjx-end><mjx-mark></mjx-mark></mjx-stretchy-v></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi mathvariant="bold-italic">A</mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mover><mjx-over style="padding-bottom:0.18em;margin-bottom:-0.248em;"><mjx-mo class="mjx-n"><mjx-stretchy-h style="width:0.869em;" class="mjx-c2013"><mjx-ext><mjx-c></mjx-c></mjx-ext></mjx-stretchy-h></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:4.08em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c2C"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mover><mjx-over style="padding-bottom:0.18em;margin-bottom:-0.248em;"><mjx-mo class="mjx-n"><mjx-stretchy-h style="width:0.869em;" class="mjx-c2013"><mjx-ext><mjx-c></mjx-c></mjx-ext></mjx-stretchy-h></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:4.08em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mover><mi mathvariant="bold-italic">A</mi><mo accent="true">―</mo></mover><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>,</mo><msub><mover><mi mathvariant="bold-italic">A</mi><mo accent="true">―</mo></mover><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></math></mjx-assistive-mml></mjx-container><h3 id="_3-8-行列式-determinant-of-a-matrix" tabindex="-1"><a class="header-anchor" href="#_3-8-行列式-determinant-of-a-matrix"><span>3.8 行列式（Determinant of a Matrix）</span></a></h3><h4 id="_3-8-1-定义" tabindex="-1"><a class="header-anchor" href="#_3-8-1-定义"><span>3.8.1 定义</span></a></h4><p>一个矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container>的行列式是一个数值，记为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="OP" movablelimits="true">det</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">A</mi></mrow></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-munderover space="4"><mjx-over style="padding-bottom:0.192em;padding-left:0.51em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-over><mjx-box><mjx-munder><mjx-row><mjx-base><mjx-mo class="mjx-lop"><mjx-c class="mjx-c2211 TEX-S2"></mjx-c></mjx-mo></mjx-base></mjx-row><mjx-row><mjx-under style="padding-top:0.167em;padding-left:0.124em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-under></mjx-row></mjx-munder></mjx-box></mjx-munderover><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-TeXAtom texclass="ORD"><mjx-mover><mjx-over style="padding-bottom:0.18em;margin-bottom:-0.248em;"><mjx-mo class="mjx-n"><mjx-stretchy-h style="width:0.869em;" class="mjx-c2013"><mjx-ext><mjx-c></mjx-c></mjx-ext></mjx-stretchy-h></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover></mjx-TeXAtom><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo data-mjx-texclass="OP" movablelimits="true">det</mo><mrow data-mjx-texclass="ORD"><mi mathvariant="bold-italic">A</mi></mrow><mo>=</mo><munderover><mo data-mjx-texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><mo stretchy="false">(</mo><msub><mi mathvariant="bold-italic">A</mi><mrow data-mjx-texclass="ORD"><mn>1</mn><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mn>1</mn><mo>+</mo><mi>j</mi></mrow></msup><mo data-mjx-texclass="OP" movablelimits="true">det</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi mathvariant="bold-italic">A</mi><mo accent="true">―</mo></mover></mrow><mrow data-mjx-texclass="ORD"><mn>1</mn><mi>j</mi></mrow></msub><mo stretchy="false">)</mo></math></mjx-assistive-mml></mjx-container><p>当矩阵是一个1x1的矩阵时，<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c5B"></mjx-c></mjx-mo><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n"><mjx-c class="mjx-c5D"></mjx-c></mjx-mo></mjx-TeXAtom><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msub space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo data-mjx-texclass="OP" movablelimits="true">det</mo><mrow data-mjx-texclass="ORD"><mo stretchy="false">[</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub><mo stretchy="false">]</mo></mrow><mo>=</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></math></mjx-assistive-mml></mjx-container></p><h4 id="_3-8-2-举例" tabindex="-1"><a class="header-anchor" href="#_3-8-2-举例"><span>3.8.2 举例</span></a></h4><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:21.656em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mo><mjx-TeXAtom space="2" texclass="ORD"><mjx-mrow><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:4.08em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow></mjx-TeXAtom><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd 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class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-msub space="3"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c><mjx-c class="mjx-c33"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c32"></mjx-c><mjx-c class="mjx-c32"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mn class="mjx-n"><mjx-c class="mjx-c33"></mjx-c><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msub><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd></mtd><mtd><mi></mi><mo data-mjx-texclass="OP" movablelimits="true">det</mo><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub><mo data-mjx-texclass="OP" movablelimits="true">det</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>−</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub><mo data-mjx-texclass="OP" movablelimits="true">det</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow><mo>+</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub><mo data-mjx-texclass="OP" movablelimits="true">det</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd><mtd><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mi></mi><mo>=</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub><mo>−</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>11</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub><mo>−</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>33</mn></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>12</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>23</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub><mo>+</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>21</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>32</mn></mrow></msub><mo>−</mo><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>13</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>22</mn></mrow></msub><msub><mi>A</mi><mrow data-mjx-texclass="ORD"><mn>31</mn></mrow></msub></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h3 id="_3-9-伴随矩阵-adjoint-of-a-matrix" tabindex="-1"><a class="header-anchor" href="#_3-9-伴随矩阵-adjoint-of-a-matrix"><span>3.9 伴随矩阵(Adjoint of a Matrix)</span></a></h3><h4 id="_3-9-1-定义" tabindex="-1"><a class="header-anchor" href="#_3-9-1-定义"><span>3.9.1 定义</span></a></h4><p>对于<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>的矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container>，定义另一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D46A TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">C</mi></math></mjx-assistive-mml></mjx-container>，使得</p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msub><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D436 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.15em;margin-left:-0.045em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.413em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2B"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="2"><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mo><mjx-msub space="2"><mjx-mover><mjx-over style="padding-bottom:0.18em;margin-bottom:-0.248em;"><mjx-mo class="mjx-n"><mjx-stretchy-h style="width:0.869em;" class="mjx-c2013"><mjx-ext><mjx-c></mjx-c></mjx-ext></mjx-stretchy-h></mjx-mo></mjx-over><mjx-base><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-base></mjx-mover><mjx-script style="vertical-align:-0.15em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D456 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D457 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msub></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msub><mi>C</mi><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow data-mjx-texclass="ORD"><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><mo data-mjx-texclass="OP" movablelimits="true">det</mo><msub><mover><mi mathvariant="bold-italic">A</mi><mo accent="true">―</mo></mover><mrow data-mjx-texclass="ORD"><mi>i</mi><mi>j</mi></mrow></msub></math></mjx-assistive-mml></mjx-container><p>那么<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D46A TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">C</mi></math></mjx-assistive-mml></mjx-container>的转置矩阵称为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi></math></mjx-assistive-mml></mjx-container>的伴随矩阵，记为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-n" size="s"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">A</mi><mo>∗</mo></msup></math></mjx-assistive-mml></mjx-container></p><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mo class="mjx-n" size="s"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msubsup space="4"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D46A TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:-0.273em;margin-left:-0.038em;"><mjx-mi class="mjx-i" size="s" style="margin-left:0.127em;"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi><mjx-spacer style="margin-top:0.18em;"></mjx-spacer><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D434 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msubsup></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi mathvariant="bold-italic">A</mi><mo>∗</mo></msup><mo>=</mo><msubsup><mi mathvariant="bold-italic">C</mi><mi>A</mi><mi>T</mi></msubsup></math></mjx-assistive-mml></mjx-container><h4 id="_3-9-2-举例" tabindex="-1"><a class="header-anchor" href="#_3-9-2-举例"><span>3.9.2 举例</span></a></h4><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-mtable style="min-width:7.113em;"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-bottom:0.15em;"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-bottom:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:2.049em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="text-align:right;padding-right:0;padding-top:0.15em;"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;"><mjx-mo class="mjx-n" size="s"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="text-align:left;padding-left:0;padding-top:0.15em;"><mjx-mi class="mjx-n"></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mrow space="4"><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5B TEX-S3"></mjx-c></mjx-mo><mjx-mtable style="min-width:3.418em;" justify="left"><mjx-table><mjx-itable><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-bottom:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D451 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-bottom:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44F TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr><mjx-mtr><mjx-mtd style="padding-right:0.5em;padding-top:0.2em;"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D450 TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd><mjx-mtd style="padding-left:0.5em;padding-top:0.2em;"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D44E TEX-I"></mjx-c></mjx-mi><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable><mjx-mo class="mjx-s3"><mjx-c class="mjx-c5D TEX-S3"></mjx-c></mjx-mo></mjx-mrow><mjx-tstrut></mjx-tstrut></mjx-mtd></mjx-mtr></mjx-itable></mjx-table></mjx-mtable></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mtable displaystyle="true" columnalign="right left" columnspacing="0em" rowspacing="3pt"><mtr><mtd><mi mathvariant="bold-italic">A</mi></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>a</mi></mtd><mtd><mi>b</mi></mtd></mtr><mtr><mtd><mi>c</mi></mtd><mtd><mi>d</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr><mtr><mtd><msup><mi mathvariant="bold-italic">A</mi><mo>∗</mo></msup></mtd><mtd><mi></mi><mo>=</mo><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">[</mo><mtable columnalign="center" columnspacing="1em" rowspacing="4pt"><mtr><mtd><mi>d</mi></mtd><mtd><mo>−</mo><mi>b</mi></mtd></mtr><mtr><mtd><mo>−</mo><mi>c</mi></mtd><mtd><mi>a</mi></mtd></mtr></mtable><mo data-mjx-texclass="CLOSE">]</mo></mrow></mtd></mtr></mtable></math></mjx-assistive-mml></mjx-container><h4 id="_3-9-3-基本运算" tabindex="-1"><a class="header-anchor" href="#_3-9-3-基本运算"><span>3.9.3 基本运算</span></a></h4><ul><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-n" size="s"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mi class="mjx-i" size="s"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-script></mjx-msup><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-n" size="s"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><msup><mi mathvariant="bold-italic">A</mi><mo>∗</mo></msup><msup><mo stretchy="false">)</mo><mi>T</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi mathvariant="bold-italic">A</mi><mi>T</mi></msup><msup><mo stretchy="false">)</mo><mo>∗</mo></msup></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-msup><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-n" size="s"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D458 TEX-I"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-n" size="s"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>k</mi><mi mathvariant="bold-italic">A</mi><msup><mo stretchy="false">)</mo><mo>∗</mo></msup><mo>=</mo><msup><mi>k</mi><mrow data-mjx-texclass="ORD"><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><msup><mi mathvariant="bold-italic">A</mi><mo>∗</mo></msup></math></mjx-assistive-mml></mjx-container></li><li><mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-n" size="s"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;"><mjx-mo class="mjx-n" size="s"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-script></mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D470 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">A</mi><msup><mi mathvariant="bold-italic">A</mi><mo>∗</mo></msup><mo>=</mo><msup><mi mathvariant="bold-italic">A</mi><mo>∗</mo></msup><mi mathvariant="bold-italic">A</mi><mo>=</mo><mo data-mjx-texclass="OP" movablelimits="true">det</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mo stretchy="false">)</mo><mi mathvariant="bold-italic">I</mi></math></mjx-assistive-mml></mjx-container></li></ul><h3 id="_3-10-逆矩阵-inverse-of-a-matrix" tabindex="-1"><a class="header-anchor" href="#_3-10-逆矩阵-inverse-of-a-matrix"><span>3.10 逆矩阵(Inverse of a Matrix)</span></a></h3><h4 id="_3-10-1-定义" tabindex="-1"><a class="header-anchor" href="#_3-10-1-定义"><span>3.10.1 <strong>定义</strong></span></a></h4><p>只有正方形矩阵有逆矩阵，对于一个<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="3"><mjx-c class="mjx-cD7"></mjx-c></mjx-mo><mjx-mi class="mjx-i" space="3"><mjx-c class="mjx-c1D45B TEX-I"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>×</mo><mi>n</mi></math></mjx-assistive-mml></mjx-container>的矩阵<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi></math></mjx-assistive-mml></mjx-container>，定义它的逆矩阵为<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.054em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup></math></mjx-assistive-mml></mjx-container>，满足<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.054em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.054em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i" space="4"><mjx-c class="mjx-c1D470 TEX-BI"></mjx-c></mjx-mi></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="bold-italic">M</mi><msup><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mi mathvariant="bold-italic">M</mi><mo>=</mo><mi mathvariant="bold-italic">I</mi></math></mjx-assistive-mml></mjx-container></p><h4 id="_3-10-2-计算方法" tabindex="-1"><a class="header-anchor" href="#_3-10-2-计算方法"><span>3.10.2 计算方法</span></a></h4><mjx-container class="MathJax" jax="CHTML" display="true" style="position:relative;"><mjx-math display="true" style="margin-left:0;margin-right:0;" class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.413em;margin-left:0.054em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-mfrac space="4"><mjx-frac type="d"><mjx-num><mjx-nstrut type="d"></mjx-nstrut><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.054em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2217"></mjx-c></mjx-mo></mjx-TeXAtom></mjx-script></mjx-msup></mjx-num><mjx-dbox><mjx-dtable><mjx-line type="d"></mjx-line><mjx-row><mjx-den><mjx-dstrut type="d"></mjx-dstrut><mjx-mrow><mjx-mo class="mjx-n"><mjx-c class="mjx-c64"></mjx-c><mjx-c class="mjx-c65"></mjx-c><mjx-c class="mjx-c74"></mjx-c></mjx-mo><mjx-mo class="mjx-n"><mjx-c class="mjx-c28"></mjx-c></mjx-mo><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D468 TEX-BI"></mjx-c></mjx-mi><mjx-mo class="mjx-n"><mjx-c class="mjx-c29"></mjx-c></mjx-mo></mjx-mrow></mjx-den></mjx-row></mjx-dtable></mjx-dbox></mjx-frac></mjx-mfrac></mjx-math><mjx-assistive-mml unselectable="on" display="block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><msup><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><mfrac><msup><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msup><mrow><mo data-mjx-texclass="OP" movablelimits="true">det</mo><mo stretchy="false">(</mo><mi mathvariant="bold-italic">A</mi><mo stretchy="false">)</mo></mrow></mfrac></math></mjx-assistive-mml></mjx-container><h4 id="_3-10-3-正交矩阵" tabindex="-1"><a class="header-anchor" href="#_3-10-3-正交矩阵"><span>3.10.3 正交矩阵</span></a></h4><p>对于正交矩阵，根据定义有<mjx-container class="MathJax" jax="CHTML" style="position:relative;"><mjx-math class="MJX-TEX" aria-hidden="true"><mjx-msup><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.054em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mo class="mjx-n"><mjx-c class="mjx-c2212"></mjx-c></mjx-mo><mjx-mn class="mjx-n"><mjx-c class="mjx-c31"></mjx-c></mjx-mn></mjx-TeXAtom></mjx-script></mjx-msup><mjx-mo class="mjx-n" space="4"><mjx-c class="mjx-c3D"></mjx-c></mjx-mo><mjx-msup space="4"><mjx-mi class="mjx-b mjx-i"><mjx-c class="mjx-c1D474 TEX-BI"></mjx-c></mjx-mi><mjx-script style="vertical-align:0.363em;margin-left:0.054em;"><mjx-TeXAtom size="s" texclass="ORD"><mjx-mi class="mjx-i"><mjx-c class="mjx-c1D447 TEX-I"></mjx-c></mjx-mi></mjx-TeXAtom></mjx-script></mjx-msup></mjx-math><mjx-assistive-mml unselectable="on" display="inline"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mo>−</mo><mn>1</mn></mrow></msup><mo>=</mo><msup><mi mathvariant="bold-italic">M</mi><mrow data-mjx-texclass="ORD"><mi>T</mi></mrow></msup></math></mjx-assistive-mml></mjx-container></p></div><!--[--><!--]--></div><footer class="vp-page-meta"><!----><div class="vp-meta-item git-info"><div class="vp-meta-item last-updated"><span class="meta-item-label">Last Updated: </span><!----></div><div class="vp-meta-item contributors"><span class="meta-item-label">Contributors: </span><span class="meta-item-info"><!--[--><!--[--><span class="contributor" title="email: thejinchao@gmail.com">thejinchao</span><!----><!--]--><!--]--></span></div></div></footer><nav class="vp-page-nav" aria-label="page navigation"><a class="route-link auto-link prev" href="/note/graphics/math/math_01.html" aria-label="矢量"><!--[--><div class="hint"><span class="arrow left"></span> Prev</div><div class="link"><span>矢量</span></div><!--]--></a><a class="route-link auto-link next" href="/note/graphics/math/math_03.html" aria-label="立体角"><!--[--><div class="hint">Next <span class="arrow right"></span></div><div 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