2.5 矩阵行列式

矩阵行列式(http://zh.wikipedia.org/wiki/行列式)是一个特殊的函数,它可以将一个正方矩阵映射为一个实数,正方矩阵A的行列式通常用符号detA表示。行列式描述的是一个线性变换对“体积”所造成的影响。此外,当线性方程组对应的行列式不为零时,由克莱姆法则(http://zh.wikipedia.org/wiki/克萊姆法則),可以直接以行列式的形式写出方程组的解。但是,我们使用行列式的主要目的是为了用它得到逆矩阵(2.7节的主题)。此外,还可以证明:当且仅当正方矩阵A的行列式detA≠0时,它才是可逆的。这个结论非常有用,因为它提供了一个判断矩阵是否可逆的计算工具。在对行列式下定义之前,我们首先介绍余子式的概念。

2.5.1余子式

给定一个n×n矩阵A,余子式\({\overline {\bf{A}} _{ij}}\)是指删除了第i行和第j列后的(n − 1)×(n − 1)矩阵。

例2.8

找到下列矩阵的余子式\({\overline {\bf{A}} _{11}}\)、\({\overline {\bf{A}} _{22}}\)和\({\overline {\bf{A}} _{13}}\):

\[{\bf{A}} = \left[ {\begin{array}{*{20}{c}}{{A_{11}}}&{{A_{12}}}&{{A_{13}}}\\{{A_{21}}}&{{A_{22}}}&{{A_{23}}}\\{{A_{31}}}&{{A_{32}}}&{{A_{33}}}\end{array}} \right]\]

删除第1行和第1列可得:

\[{\overline {\bf{A}} _{11}} = \left[ {\begin{array}{*{20}{c}}{{A_{22}}}&{{A_{23}}}\\{{A_{32}}}&{{A_{33}}}\end{array}} \right]\]

删除第2行和第2列可得:

\[{\overline {\bf{A}} _{22}} = \left[ {\begin{array}{*{20}{c}}{{A_{11}}}&{{A_{13}}}\\{{A_{31}}}&{{A_{33}}}\end{array}} \right]\]

删除第1行和第3列可得:

\[{\overline {\bf{A}} _{13}} = \left[ {\begin{array}{*{20}{c}}{{A_{21}}}&{{A_{22}}}\\{{A_{31}}}&{{A_{32}}}\end{array}} \right]\]

2.5.2 定义

行列式是递归定义的;例如,4×4矩阵的行列式是以3×3矩阵的形式定义的,3×3矩阵的定义式是以2×2矩阵的形式定义的,2×2矩阵的定义式是以1×1矩阵的形式定义的(1×1矩阵A=[A11]可简单地表示为det[A11] = A11)。若A为一个n×n矩阵,在n>1时我们可以定义:

\(\det {\bf{A}} = \sum\limits_{j = 1}^n {{A_{1j}}{{( - 1)}^{1 + j}}\det {{\overline {\bf{A}} }_{1j}}} \)(公式2.4)

回忆一下2×2矩阵的余子式\({\overline {\bf{A}} _{ij}}\)的定义,可以得到以下式子:

\[\det \left[ {\begin{array}{*{20}{c}}{{A_{11}}}&{{A_{12}}}\\{{A_{21}}}&{{A_{22}}}\end{array}} \right] = {A_{11}}\det \left[ {{A_{22}}} \right] - {A_{12}}\det \left[ {{A_{21}}} \right] = {A_{11}}{A_{22}} - {A_{12}}{A_{21}}\]

若是3×3矩阵,则公式如下:

\[\det \left[ {\begin{array}{*{20}{c}}{{A_{11}}}&{{A_{12}}}&{{A_{13}}}\\{{A_{21}}}&{{A_{22}}}&{{A_{23}}}\\{{A_{31}}}&{{A_{32}}}&{{A_{33}}}\end{array}} \right] = {A_{11}}\det \left[ {\begin{array}{*{20}{c}}{{A_{22}}}&{{A_{23}}}\\{{A_{32}}}&{{A_{33}}}\end{array}} \right] - {A_{12}}\det \left[ {\begin{array}{*{20}{c}}{{A_{21}}}&{{A_{23}}}\\{{A_{31}}}&{{A_{33}}}\end{array}} \right] + {A_{13}}\det \left[ {\begin{array}{*{20}{c}}{{A_{21}}}&{{A_{22}}}\\{{A_{31}}}&{{A_{32}}}\end{array}} \right]\]

 换成4×4矩阵,公式变为:

\[\begin{array}{l}\det \left[ {\begin{array}{*{20}{c}}{{A_{11}}}&{{A_{12}}}&{{A_{13}}}&{{A_{14}}}\\{{A_{21}}}&{{A_{22}}}&{{A_{23}}}&{{A_{24}}}\\{{A_{31}}}&{{A_{32}}}&{{A_{33}}}&{{A_{34}}}\\{{A_{41}}}&{{A_{42}}}&{{A_{43}}}&{{A_{44}}}\end{array}} \right]\\ = {A_{11}}\det \left[ {\begin{array}{*{20}{c}}{{A_{22}}}&{{A_{23}}}&{{A_{24}}}\\{{A_{31}}}&{{A_{33}}}&{{A_{34}}}\\{{A_{42}}}&{{A_{43}}}&{{A_{44}}}\end{array}} \right] - {A_{12}}\det \left[ {\begin{array}{*{20}{c}}{{A_{21}}}&{{A_{23}}}&{{A_{24}}}\\{{A_{31}}}&{{A_{33}}}&{{A_{34}}}\\{{A_{41}}}&{{A_{43}}}&{{A_{44}}}\end{array}} \right]\\ + {A_{13}}\det \left[ {\begin{array}{*{20}{c}}{{A_{21}}}&{{A_{22}}}&{{A_{24}}}\\{{A_{31}}}&{{A_{32}}}&{{A_{34}}}\\{{A_{41}}}&{{A_{42}}}&{{A_{44}}}\end{array}} \right] - {A_{14}}\det \left[ {\begin{array}{*{20}{c}}{{A_{21}}}&{{A_{22}}}&{{A_{23}}}\\{{A_{31}}}&{{A_{32}}}&{{A_{33}}}\\{{A_{41}}}&{{A_{42}}}&{{A_{43}}}\end{array}} \right]\end{array}\]

在3D图形中,我们主要使用4×4矩阵,所以就不再讨论n>4时的公式了。

例2.9

求下面矩阵的行列式:

\[{\bf{A}} = \left[ {\begin{array}{*{20}{c}}2&{ - 5}&3\\1&3&4\\{ - 2}&3&7\end{array}} \right]\]

我们可以得到:

\[\begin{array}{l}\det {\bf{A}} = {A_{11}}\det \left[ {\begin{array}{*{20}{c}}{{A_{22}}}&{{A_{23}}}\\{{A_{32}}}&{{A_{33}}}\end{array}} \right] - {A_{12}}\det \left[ {\begin{array}{*{20}{c}}{{A_{21}}}&{{A_{23}}}\\{{A_{31}}}&{{A_{33}}}\end{array}} \right] + {A_{13}}\det \left[ {\begin{array}{*{20}{c}}{{A_{21}}}&{{A_{22}}}\\{{A_{31}}}&{{A_{32}}}\end{array}} \right]\\\det {\bf{A}} = 2\det \left[ {\begin{array}{*{20}{c}}3&4\\3&7\end{array}} \right] - ( - 5)\det \left[ {\begin{array}{*{20}{c}}1&4\\{ - 2}&7\end{array}} \right] + 3\det \left[ {\begin{array}{*{20}{c}}1&3\\{ - 2}&3\end{array}} \right]\\ = 2(3 \cdot 7 - 4 \cdot 3) + 5(1 \cdot 7 - 4 \cdot ( - 2)) + 3(1 \cdot 3 - 3 \cdot ( - 2))\\ = 18 + 75 + 27\\ = 120\end{array}\]

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发布时间:2014/9/21 下午7:06:35  阅读次数:3860

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