From 862e7223d73b8a5b3d435cb8eb4c5f0ee1cc4551 Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Sun, 4 Jul 2021 22:51:44 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E7=9F=A9=E9=98=B5?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- linear-algebra/exercise/2-matrix/matrix.pdf | Bin 0 -> 57020 bytes linear-algebra/exercise/2-matrix/matrix.tex | 189 +++++++++++++++++++ linear-algebra/knowledge/2-matrix/matrix.pdf | Bin 127531 -> 132069 bytes linear-algebra/knowledge/2-matrix/matrix.tex | 107 ++++++++--- 4 files changed, 270 insertions(+), 26 deletions(-) create mode 100644 linear-algebra/exercise/2-matrix/matrix.pdf create mode 100644 linear-algebra/exercise/2-matrix/matrix.tex diff --git a/linear-algebra/exercise/2-matrix/matrix.pdf b/linear-algebra/exercise/2-matrix/matrix.pdf new file mode 100644 index 0000000000000000000000000000000000000000..229b0d04407b08b41dbe8f571163d266e03c3012 GIT binary patch literal 57020 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+\title{矩阵} +\date{} +\begin{document} +\maketitle +\pagestyle{empty} +\thispagestyle{empty} +\tableofcontents +\thispagestyle{empty} +\newpage +\pagestyle{plain} +\setcounter{page}{1} +\section{矩阵的幂} + +\subsection{对应成比例} + +因为矩阵运算不满足交换率但是满足结合率,且一行矩阵乘一列矩阵的乘积为一个数,所以可以推出矩阵的幂的运算方法。 + +这个方法要求$r(A)=1$,即对应成比例。 + +令$A$为$n$阶方阵,将$A$拆为$A=(a_1,a_2,\cdots,a_n)^T(b_1,b_2,\cdots,b_n)=\alpha^T\beta$,所以$A^n=\alpha^T\beta\alpha^T\beta\cdots\alpha^T\beta$,利用结合率:$\alpha^T(\beta\alpha^T)(\beta\cdots\alpha^T)\beta$,中间一共$n-1$个$\beta\alpha^T$,$\beta\alpha^T$是一个数,即$A^n=(\beta\alpha^T)^{n-1}\alpha^T\beta=(\beta\alpha^T)^{n-1}A$。\medskip + +\textbf{例题:}$A=\left(\begin{array}{ccc} + 1 & 2 & 3 \\ + -2 & -4 & -6 \\ + 3 & 6 & 9 +\end{array}\right)$,求$A^n$。\medskip + +解:$A=(1,-2,3)^T(1,2,3)$,所以$A^n=((1,2,3)(1,-2,3)^T)^n(1,-2,3)^T(1,2,3)$ + +$=6^{n-1}A$。 + +若矩阵$A$的行与列都成比例,则$A^n=[tr(A)]^{n-1}A$,$[tr(A)]=\sum a_{ii}$,即矩阵迹为对角线元素值之和。 + +\subsection{试算归纳} + +对$A$进行试算,如$A^2$,若$A^k$是一个数量阵,那么计算$A^n$就只用找规律就可以了。 + +\textbf{例题:}$A=\left(\begin{array}{cccc} + 1 & -1 & -1 & -1 \\ + -1 & 1 & -1 & -1 \\ + -1 & -1 & 1 & -1 \\ + -1 & -1 & -1 & 1 \\ +\end{array}\right)$,求$A^n$($n\geqslant2$)。\medskip + +解:通过计算得知$A^2=4E$,这是一个数量阵。\medskip + +$\therefore A^n=\left\{\begin{array}{lcl} + 4^kE, & & n=2k \\ + 4^kA, & & n=2k+1 +\end{array}\right.$。 + +\subsection{拆分矩阵} + +将$A^n$拆分为两个矩阵$A^n=(B+C)^n$,其中$BC$应该是可逆的,即$BC=CB$,所以一般有一个是$E$。\medskip + +\textbf{例题:}$A=\left(\begin{array}{ccc} + 1 & 1 & 0 \\ + 0 & 1 & 1 \\ + 0 & 0 & 1 +\end{array}\right)$,求$A^n$。\medskip + +解:$A=E+B=\left(\begin{array}{ccc} + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 +\end{array}\right)+\left(\begin{array}{ccc} + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 +\end{array}\right)$。\medskip + +$\therefore A^n=(E+B)^n=C_n^0E^n+C_n^1E^{n-1}B+C_n^2E^{n-2}B^2+\cdots$。 + +又$B^2=\left(\begin{array}{ccc} + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 +\end{array}\right)\left(\begin{array}{ccc} + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 +\end{array}\right)=\left(\begin{array}{ccc} + 0 & 0 & 1 \\ + 0 & 0 & 0 \\ + 0 & 0 & 0 +\end{array}\right)$。 + +$B^3=B^2B=\left(\begin{array}{ccc} + 0 & 0 & 1 \\ + 0 & 0 & 0 \\ + 0 & 0 & 0 +\end{array}\right)\left(\begin{array}{ccc} + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 +\end{array}\right)=\left(\begin{array}{ccc} + 0 & 0 & 0 \\ + 0 & 0 & 0 \\ + 0 & 0 & 0 +\end{array}\right)=O$。 + +$\therefore B^4=B^5=\cdots=O$。 + +$\therefore A^n=(E+B)^n=C_n^0E^n+C_n^1E^{n-1}B+C_n^2E^{n-2}B^2$。\medskip + +$=\left(\begin{array}{ccc} + 1 & 0 & 0 \\ + 0 & 1 & 0 \\ + 0 & 0 & 1 +\end{array}\right)+n\left(\begin{array}{ccc} + 0 & 1 & 0 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 +\end{array}\right)+\dfrac{n(n-1)}{2}\left(\begin{array}{ccc} + 0 & 0 & 1 \\ + 0 & 0 & 0 \\ + 0 & 0 & 0 +\end{array}\right)$ + +\section{逆矩阵} + +\subsection{定义法} + +找出一个矩阵$B$,使得$AB=E$,则$A$可逆,$A^{-1}=B$。 + +\textbf{例题:}$A$,$B$均是$n$阶方阵,且$AB=A+B$,证明$A-E$可逆,并求$(A-E)^{-1}$。 + +解:要证明$A-E$,就要从$AB=A+B$中尽量凑出。 + +$AB=A+B$变为$AB-B=A$,从而提取$(A-E)B=A$,$(A-E)BA^{-1}=E$。 + +但是$A^{-1}$是未知的,所以$A-E$的逆矩阵不能用$BA^{-1}$来表示。 + +$AB-A=B$,所以提出$A(B-E)=B$,即$A(B-E)=B-E+E$,$(A-E)(B-E)=E$,所以$A-E$的逆矩阵就是$B-E$。 + +\subsection{分解乘积} + +将$A$分解为若干个可逆矩阵的乘积。若$A=BC$,$B$,$C$可逆,则$A$可逆,且$A^{-1}=C^{-1}B^{-1}$。 + +\textbf{例题:}设$A$,$B$为同阶可逆方阵,且$A^{-1}+B^{-1}$可逆,求$(A+B)^{-1}$。 + +解:已知$A^{-1}+B^{-1}$可以用来表示其他式子,需要求$A+B$的逆,则需要将$A+B$转为其逆。 + +$\because A+B=A(E+A^{-1}B)=A(B^{-1}+A^{-1})B$。 + +$\therefore (A+B)^{-1}=B^{-1}(B^{-1}+A^{-1})^{-1}A^{-1}$。 + +\subsection{分块矩阵} + +对于一些分块矩阵的逆,若$A$,$B$都可逆,则:$\left[\begin{array}{cc} + A & O \\ + O & B +\end{array}\right]^{-1}=\left[\begin{array}{cc} + A^{-1} & O \\ + O & B^{-1} +\end{array}\right]$,$\left[\begin{array}{cc} + O & A \\ + B & O +\end{array}\right]^{-1}=\left[\begin{array}{cc} + O & B^{-1} \\ + A^{-1} & O +\end{array}\right]$。 + +\end{document} diff --git a/linear-algebra/knowledge/2-matrix/matrix.pdf b/linear-algebra/knowledge/2-matrix/matrix.pdf index 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a/linear-algebra/knowledge/2-matrix/matrix.tex +++ b/linear-algebra/knowledge/2-matrix/matrix.tex @@ -123,7 +123,7 @@ $A+B=\left( \subsection{数乘矩阵} -数$\lambda$与矩阵$A$的乘积记为$\lambda A$或$A\lambda$,规定: +数$\lambda$与矩阵$A$的乘积记为$\lambda A$或$A\lambda$,规定:\medskip $\lambda A=A\lambda=\left( \begin{array}{cccc} @@ -132,7 +132,7 @@ $\lambda A=A\lambda=\left( \vdots & \vdots & \ddots & \vdots \\ \lambda a_{m1} & \lambda a_{m2} & \cdots & \lambda a_{mn} \end{array} -\right)$ +\right)$ \medskip 假设$A$、$B$都是$m\times n$的矩阵,$\lambda$、$\mu$为数: @@ -159,7 +159,7 @@ $(a_{i1},a_{i2},\cdots,a_{is})\left( \cdots \\ b_{sj} \end{array} -\right)=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{is}b_{sj}=\sum\limits_{k=1}^sa_{ik}b_{kj}=c_{ij}$。 +\right)=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{is}b_{sj}=\sum\limits_{k=1}^sa_{ik}b_{kj}=c_{ij}$。\medskip 从而$AB=C$的$c_{ij}$就是$A$的第$i$行与$B$的$j$列的乘积。 @@ -208,7 +208,7 @@ $A^1=A\text{,}A^2=A^1A^1\text{,}\cdots\text{,}A^{k+1}=A^kA^1$ 因为矩阵乘法一般不满足交换率,所以$(AB)^k\neq A^kB^k$。只有$AB$可交换时才相等。 -若$A\neq 0$不能推出$A^k\neq 0$,如: +若$A\neq 0$不能推出$A^k\neq 0$,如:\medskip $A=\left( \begin{array}{cc} @@ -230,7 +230,7 @@ $A=\left( 0 & 0 \\ 0 & 0 \end{array} -\right)=O$。 +\right)=O$。\medskip $A=\left( \begin{array}{ccc} @@ -238,7 +238,7 @@ $A=\left( 0 & 0 & 1 \\ 0 & 0 & 0 \end{array} -\right)$,$A^3=O$。 +\right)$,$A^3=O$。\medskip 矩阵幂可以同普通多项式进行处理。 @@ -258,7 +258,14 @@ $f(A)=A^2-A-6E=(A+2E)(A-3E)$。 \item 若$m=n$,$\vert A\vert=\vert A^T\vert$。 \end{itemize} -对称矩阵或对称阵\textcolor{violet}{\textbf{定义:}}元素以对角线为对称轴对应相等,$A=A^T$。 +对称矩阵或对称阵\textcolor{violet}{\textbf{定义:}}矩阵$A$是方阵,且元素以对角线为对称轴对应相等,$A=A^T$。 + +反对称矩阵\textcolor{violet}{\textbf{定义:}}矩阵$A$是方阵,且满足$-A=A^T$。即$\left\{\begin{array}{l} + a_{ij}=-a_{ji},i\neq j \\ + a_{ii}=0 +\end{array}\right.$。 + +正交矩阵\textcolor{violet}{\textbf{定义:}}矩阵$A$是方阵,且满足$A^TA=E$。 \subsection{方阵行列式} @@ -272,6 +279,8 @@ $f(A)=A^2-A-6E=(A+2E)(A-3E)$。 一般而言:$\vert A+B\vert\neq\vert A\vert+\vert B\vert$,$\vert A\vert\neq O\nRightarrow\vert A\vert\neq0$,$A\neq B\nRightarrow\vert A\vert\neq\vert B\vert$。 +\subsection{伴随矩阵} + 伴随矩阵或伴随阵\textcolor{violet}{\textbf{定义:}}行列式$\vert A\vert$各个元素的代数余子式$A_{ij}$转置构成的矩阵。 $A^*=\left( @@ -283,7 +292,30 @@ $A^*=\left( \end{array} \right)$ -其中$AA^*=A^*A=\vert A\vert E$。 +\begin{itemize} + \item 任何方阵都有伴随矩阵,其中$AA^*=A^*A=\vert A\vert E$。 + \item $A^*=\vert A\vert A^{-1}$,$A^{-1}=\dfrac{1}{\vert A\vert}A^*$,$A=\vert A\vert(A^*)^{-1}$。 + \item $\vert A^*\vert=\vert A\vert^{n-1}$,$(kA)(kA)^*=\vert kA\vert E$,$A^T(A^T)^*=\vert A^T\vert E$,$A^{-1}(A^{-1})^*=\vert A^{-1}\vert E$,$A^*(A^*)^*=\vert A^*\vert E$。 + \item $(A^T)^*=(A^*)^T$,$(A^{-1})^*=(A^*)^{-1}$,$(AB)^*=B^*A^*$,$(A^*)^*=\vert A\vert^{n-2}A$。 +\end{itemize} + +\textbf{例题:}假设$A$为$n$阶方阵,求$\vert A^*\vert$与$(A^*)^*$。 + +解:$\because AA^*=A^*A=\vert A\vert E$,$\therefore A^*(A^*)^*=\vert A^*\vert E$。 + +$(A^*)^{-1}A^*(A^*)^*=(A^*)^*=(A^*)^{-1}\vert A^*\vert E$,又$AA^*=\vert A\vert E$,$\vert AA^*\vert=\vert\vert A\vert E\vert$, + +$\therefore\vert A\vert\vert A^*\vert=\vert A\vert^n\vert E\vert$,$\therefore\vert A^*\vert=\vert A\vert^{n-1}$。 + +又$AA^*=\vert A\vert E$,$\therefore A^*=\vert A\vert A^{-1}$,$(A^*)^{-1}=(\vert A\vert A^{-1})^{-1}=\dfrac{1}{\vert A\vert}A$。 + +$\because(A^*)^*=(A^*)^{-1}\vert A^*\vert E$,$\therefore=\dfrac{1}{\vert A\vert}A\vert A^*\vert=\dfrac{1}{\vert A\vert}A\vert A\vert^{n-1}=\vert A\vert^{n-2}A$。 + +\textbf{例题:}假设$A$为$n$阶方阵,求$(kA)^*$。 + +解:根据$AA^*=\vert A\vert E$,$\therefore (kA)(kA)^*=\vert kA\vert E$,推出$(kA)^*=\vert kA\vert(kA)^{-1}$。 + +$=k^n\vert A\vert\dfrac{1}{k}A^{-1}=k^{n-1}\vert A\vert A^{-1}=k^{n-1}A^*$。 \subsection{分块矩阵} @@ -295,7 +327,7 @@ $A^*=\left( 分块矩阵的计算法则与普通矩阵计算类似。 -\textcolor{aqua}{\textbf{定理:}}若$AB$矩阵行列数相同,采用相同的分块法,则 +\textcolor{aqua}{\textbf{定理:}}若$AB$矩阵行列数相同,采用相同的分块法,则 \medskip $A=\left( \begin{array}{ccc} @@ -333,7 +365,7 @@ $A+B=\left( \end{array} \right)$。\medskip -\textcolor{aqua}{\textbf{定理:}}若$A_{m\times l}$,$B_{l\times n}$,采用相同的分块法,则 +\textcolor{aqua}{\textbf{定理:}}若$A_{m\times l}$,$B_{l\times n}$,采用相同的分块法,则 \medskip $A=\left( \begin{array}{ccc} @@ -398,7 +430,7 @@ $AB=\left( \vdots \\ a_{mj} \end{array} -\right)$,则$A$可以按列分块为$A=(a_1,a_2,\cdots,a_n)$。 +\right)$,则$A$可以按列分块为$A=(a_1,a_2,\cdots,a_n)$。\medskip 其$m$行称为$A$的$m$个行向量,若第$i$行记为$a_i^T=(a_{i1},a_{i2},\cdots,a_{in})$,则$A$可以按行分块为$A=\left(\begin{array}{c} a_1^T \\ @@ -407,7 +439,7 @@ $AB=\left( a_{m}^T \end{array}\right)$。 -若对于$A_{m\times s}$与$B_{s\times n}$的乘积矩阵$AB=C=(c_{ij})_{m\times n}$,若将$A$按行分为$m$块,$B$按列分为$n$块,则有: +若对于$A_{m\times s}$与$B_{s\times n}$的乘积矩阵$AB=C=(c_{ij})_{m\times n}$,若将$A$按行分为$m$块,$B$按列分为$n$块,则有:\medskip $AB=\left( \begin{array}{c} @@ -438,7 +470,7 @@ $=\left( 证明:$\because A=O$,$\therefore A^T=O$,$A^TA=O$。 -设$A=(a_{ij})_{m\times n}$,将$A$按列分块为$A=(a_1,a_2,\cdots,a_n)$,则 +设$A=(a_{ij})_{m\times n}$,将$A$按列分块为$A=(a_1,a_2,\cdots,a_n)$,则 \medskip $A^TA=\left( \begin{array}{c} @@ -454,11 +486,11 @@ $A^TA=\left( \vdots & \vdots & \ddots & \vdots \\ a_{m}^Ta_1 & a_{m}^Ta_2 & \cdots & a_{m}^Ta_n \end{array} -\right)\text{。}$ +\right)\text{。}$\medskip 所以$A^TA$的元为$a^T_ia_j$,又$\because A^TA=O$,$\therefore a^T_ia_j=0$($i,j=1,2,\cdots n$)。 -$\therefore a^T_ja_j=0$($j=1,2,\cdots n$),对角线元素全部为0。 +$\therefore a^T_ja_j=0$($j=1,2,\cdots n$),对角线元素全部为0。\medskip 且$a^T_ja_j=\left( \begin{array}{cccc} @@ -472,7 +504,7 @@ $\therefore a^T_ja_j=0$($j=1,2,\cdots n$),对角线元素全部为0。 a_{2j} \\ \vdots \\ a_{mj} -\end{array}\right)$ +\end{array}\right)$ \medskip $=a_{1j}^2+a_{2j}^2+\cdots+a_{mj}^2=0$,所以$a_{1j}=a_{2j}=\cdots+a_{mj}=0$。 @@ -506,7 +538,7 @@ $\therefore A=O$。 对于齐次方程,$x_1=\cdots=x_n=0$一定是其解,称为其\textbf{零解},若有一组不全为零的解,则称为其\textbf{非零解}。其一定有零解,但是不一定有非零解。 -对于非齐次方程,只有$b_1\cdots b_n$不全为零才是。 +对于非齐次方程,只有$b_1\cdots b_n$不全为零才是。\medskip 令\textbf{系数矩阵}$A_{m\times n}=\left( \begin{array}{ccc} @@ -569,9 +601,9 @@ $\begin{cases} x_1=b_{11}t_1+b_{12}t_2+\cdots+b_{1n}t_n \\ \cdots \\ x_s=b_{s1}t_1+b_{s2}t_2+\cdots+b_{sn}t_n -\end{cases}$ +\end{cases}$\medskip -原本是线性方程分别是$y$与$x$和$x$与$t$的关系式,而如果将$t$关于$x$的关系式代入$x$关于$y$的关系式中,就会得到$t$关于$y$的关系式: +原本是线性方程分别是$y$与$x$和$x$与$t$的关系式,而如果将$t$关于$x$的关系式代入$x$关于$y$的关系式中,就会得到$t$关于$y$的关系式:\medskip $\begin{cases} y_1=a_{11}(b_{11}t_1+\cdots+b_{1n}t_n)+\cdots+a_{1s}(b_{s1}t_1+b_{s2}t_2+\cdots+b_{sn}t_n) \\ @@ -583,9 +615,9 @@ $=\begin{cases} y_1=(a_{11}b_{11}+\cdots+a_{1s}b_{s1})t_1+\cdots+(a_{11}b_{1n}+\cdots+a_{1s}b_{sn})t_n \\ \cdots \\ y_m=(a_{m1}b_{11}+\cdots+a_{ms}b_{s1})t_1+\cdots+(a_{m1}b_{1n}+\cdots+a_{ms}b_{sn})t_m -\end{cases}$ +\end{cases}$ \medskip -这可以看作上面两个线性方程组相乘,也可以将线性方程组表示为矩阵,进行相乘就得到乘积,从而了解矩阵乘积与线性方程组的关系: +这可以看作上面两个线性方程组相乘,也可以将线性方程组表示为矩阵,进行相乘就得到乘积,从而了解矩阵乘积与线性方程组的关系:\medskip $\left(\begin{array}{ccc} @@ -641,7 +673,7 @@ $=\left(\begin{array}{ccc} a_{m1}x_1+\cdots+a_{mn}x_n=b_n \end{cases}$。 -按乘积表示为$A_{m\times n}x_{n\times 1}=b_{m\times 1}$,然后将$A$按列分块,$x$按行分块: +按乘积表示为$A_{m\times n}x_{n\times 1}=b_{m\times 1}$,然后将$A$按列分块,$x$按行分块:\medskip $(a_1,a_2,\cdots,a_n)\left(\begin{array}{c} x_1 \\ @@ -663,15 +695,19 @@ $(a_1,a_2,\cdots,a_n)\left(\begin{array}{c} b_2 \\ \vdots \\ b_m -\end{array}\right)\text{。}$ +\end{array}\right)\text{。}$ \medskip 这三种都是解的表示方法。 \section{逆矩阵} -\textcolor{violet}{\textbf{定义:}}逆矩阵类比倒数,若对于$n$阶矩阵$A$,有一个$n$阶矩阵$B$,使得$AB=BA=E$,则$A$可逆,$B$是$A$的逆矩阵也称为逆阵,且逆矩阵唯一,记为$B=A^{-1}$。 +\subsection{逆矩阵定义} -\textcolor{aqua}{\textbf{定理:}}若矩阵$A$可逆,则$\vert A\vert\neq 0$。 +逆矩阵就是类似矩阵的除运算。 + +\textcolor{violet}{\textbf{定义:}}逆矩阵类比倒数,若对于$n$阶方阵$A$,有一个$n$阶方阵$B$,使得$AB=BA=E$,则$A$可逆,$B$是$A$的逆矩阵也称为逆阵,且逆矩阵唯一,记为$B=A^{-1}$。 + +\textcolor{aqua}{\textbf{定理:}}若方阵$A$可逆,则$\vert A\vert\neq 0$。 证明:若$A$可逆,则$AA^{-1}=E$,所以$\vert A\vert\cdot\vert A^{-1}\vert=\vert E\vert=1$,$\vert A\vert\neq 0$。 @@ -689,14 +725,33 @@ $(a_1,a_2,\cdots,a_n)\left(\begin{array}{c} \textcolor{aqua}{\textbf{定理:}}若$AB=E$或$BA=E$,则$B=A^{-1}$。 +\subsection{逆矩阵性质} + \begin{itemize} \item 若$A$可逆,则$(A^{-1})^{-1}=A$。 \item 若$A$可逆,数$\lambda\neq0$,则$(\lambda A)^{-1}=\dfrac{1}{\lambda}A^{-1}$。 \item 若$AB$为同阶矩阵且都可逆,则$(AB)^{-1}=B^{-1}A^{-1}$。 \item 若$A$可逆,则$(A^T)^{-1}=(A^{-1})^T$。 + \item 若$A$可逆,则$\vert A^{-1}\vert=\vert A\vert^{-1}$。 \item 若$A$可逆,$\lambda\mu$为整数时,$A^\lambda A^\mu=A^{\lambda+\mu}$,$(A^\lambda)^\mu=A^{\lambda\mu}$。 \end{itemize} +若$A$、$B$可逆,则$A+B$不一定可逆,且$(A+B)^{-1}\neq A^{-1}+B^{-1}$。 + +\subsection{求逆矩阵} + +\subsubsection{伴随矩阵} + +根据伴随矩阵的定义,即求出矩阵所代表行列式的各行元素的代数余子式,然后按列进行排列。 + +只能求四阶以下的矩阵,过高阶的矩阵很难求出。 + +\begin{enumerate} + \item 求出$\vert A\vert$,判断是否为0。 + \item 写出$A^*$。 + \item 计算$A^{-1}=\dfrac{1}{\vert A\vert}A^*$。 +\end{enumerate} + \section{矩阵初等变换} 求逆矩阵可以使用伴随矩阵来求,但是只针对三阶以及以下的矩阵,若阶数过高则会十分困难。可以使用矩阵初等变换来实现求逆矩阵。且初等变换还可以用来求线性方程组的解。 @@ -778,7 +833,7 @@ $(a_1,a_2,\cdots,a_n)\left(\begin{array}{c} \subsection{初等行变换求逆} -已知$A^{-1}=\dfrac{A^*}{\vert A\vert}$,但是伴随矩阵计算非常麻烦,并且若矩阵在三阶以上计算就很难办到,所以还有一种方法,就是若该矩阵$A$是可逆矩阵,就将$AX=B$的增广矩阵$(A,B)$化为最简形矩阵,从而得到方程解。 +已知$A^{-1}=\dfrac{A^*}{\vert A\vert}$,但是伴随矩阵计算非常麻烦,并且若矩阵在三阶以上计算就很难办到,所以还有一种方法,就是若该矩阵$A$是可逆矩阵,就将$AX=B$的增广矩阵$(A,B)$化为最简形矩阵,从而得到方程解。\medskip $(A\vdots B)\overset{r}{\thicksim}(E\vdots A^{-1})$,$\left(\begin{array}{c} A \\