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z%^+M>l6U)Vc2<|nbpyHjDHZ&{vdaOe_S*1pIncnzJ|_e=3o2CY)OMQ zEg)mjd>E9~AXL(Rl=C_YBkNvelIg2$afun{l$&Zw9}|ppgsk)J59Zt=mZcvDD=v}i zvQKMuPd{2J_)po}!^fFjBNh~I_Q40N_B$fR9L?4;c23BW2jLHeH t`zX?TKxpq{(>{P%)v)0H_08DX>)6^m+Zq@`F)}bSGC`4$2uq1T{U26;y959L diff --git a/linear-algebra/knowledge/2-matrix/matrix.tex b/linear-algebra/knowledge/2-matrix/matrix.tex index ad1365c..7bf499e 100644 --- a/linear-algebra/knowledge/2-matrix/matrix.tex +++ b/linear-algebra/knowledge/2-matrix/matrix.tex @@ -2,6 +2,7 @@ % UTF8编码,ctexart现实中文 \usepackage{color} % 使用颜色 +\definecolor{orange}{RGB}{255,127,0} \usepackage{geometry} \setcounter{tocdepth}{4} \setcounter{secnumdepth}{4} @@ -71,13 +72,313 @@ $m\times n$矩阵是由$m\times n$个数$a_{ij}$(元素)排成的$m$行$n$ $\varLambda=\left( \begin{array}{cccc} \lambda_1 & 0 & \cdots & 0 \\ + 0 & \lambda_2 & \cdots & 0 \\ + \vdots & \vdots & \vdots & \vdots \\ + 0 & 0 & \cdots & \lambda_n \end{array} \right)$ - 单位矩阵或单位阵:$\lambda_1=\lambda_2=\cdots=\lambda_n=1$的对角矩阵,记为$E$。 - + 单位矩阵或单位阵:$\lambda_1=\lambda_2=\cdots=\lambda_n=1$的对角矩阵,记为$E$。这种线性变换叫做恒等变换,$AE=A$。 + $E=\left( + \begin{array}{cccc} + 1 & 0 & \cdots & 0 \\ + 0 & 1 & \cdots & 0 \\ + \vdots & \vdots & \vdots & \vdots \\ + 0 & 0 & \cdots & 1 + \end{array} + \right)$ \end{multicols} +\section{矩阵运算} + +\subsection{矩阵加法减法} + +设与两个矩阵都是同型矩阵$m\times n$$A=(a_{ij})$和$B=(b_{ij})$,则其加法就是$A+B$。 + +$$A+B=\left( + \begin{array}{cccc} + a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ + a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+b_{2n} \\ + \vdots & \vdots & \vdots & \vdots \\ + a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{m+n}+b_{m+n} + \end{array} +\right)$$ + +\begin{itemize} + \item $A+B=B+A$。 + \item $(A+B)+C=A+(B+C)$。 +\end{itemize} + +若$-A=(-a_{ij})$,则$-A$是$A$的负矩阵,$A+(-A)=O$。 + +从而矩阵的减法为$A-B=A+(-B)$。 + +\subsection{数乘矩阵} + +数$\lambda$与矩阵$A$的乘积记为$\lambda A$或$A\lambda$,规定: + +$$\lambda A=A\lambda=\left( + \begin{array}{cccc} + \lambda a_{11} & \lambda a_{12} & \cdots & \lambda a_{1n} \\ + \lambda a_{21} & \lambda a_{22} & \cdots & \lambda a_{2n} \\ + \vdots & \vdots & \ddots & \vdots \\ + \lambda a_{m1} & \lambda a_{m2} & \cdots & \lambda a_{mn} + \end{array} +\right)$$ + +假设$A$、$B$都是$m\times n$的矩阵,$\lambda$、$\mu$为数: + +\begin{itemize} + \item $(\lambda\mu)A=\lambda(\mu A)$。 + \item $(\lambda+\mu)A=\lambda A+\mu A$。 + \item $\lambda(A+B)=\lambda A+\lambda B$。 +\end{itemize} + +矩阵加法与数乘矩阵都是矩阵的线性运算。 + +\subsection{矩阵相乘} + +设$A=(a_{ij})$是一个$m\times s$的矩阵,$B=(b_{ij})$是一个$s\times n$的矩阵,那么$A\times B=AB=C_{m\times n}=(c_{ij})$。即: + +$$c_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{is}b_{sj}=\sum\limits_{k=1}^sa_{ik}b_{kj}\,\text{(}i=1,2,\cdots,m;j=1,2,\cdots,n\text{)}$$ + +所以按此定义一个$1\times s$行矩阵与$s\times 1$列矩阵的乘积就是一个1阶方针即一个数: + +$(a_{i1},a_{i2},\cdots,a_{is})\left( + \begin{array}{c} + b_{1j} \\ + b_{2j} \\ + \cdots \\ + b_{sj} + \end{array} +\right)=a_{i1}b_{1j}+a_{i2}b_{2j}+\cdots+a_{is}b_{sj}=\sum\limits_{k=1}^sa_{ik}b_{kj}=c_{ij}$。 + +从而$AB=C$的$c_{ij}$就是$A$的第$i$行与$B$的$j$列的乘积。 + +\textcolor{orange}{注意:}只有左矩阵的列数等于右矩阵的行数才能相乘。 + +只有$AB$都是方阵的时候才能$AB$与$BA$。 + +矩阵的左乘与右乘不一定相等,即$AB\neq BA$。 + +若方阵$AB$乘积满足$AB=BA$,则表示其是可交换的。 + +$A\neq O$,$B\neq O$,但是不能推出$AB\neq O$或$BA\neq O$。 + +$AB=O$不能推出$A=O$或$B=O$。 + +$A(X-Y)=O$当$A\neq O$也不能推出$X=Y$。 + +\begin{itemize} + \item $(AB)C=A(BC)$。 + \item $\lambda(AB)=(\lambda A)B=A(\lambda B)$。 + \item $A(B+C)=AB+AC$。 + \item $(B+C)A=BA+CA$。 + \item $EA=AE=A$。 +\end{itemize} + +$\lambda E$称为纯量阵,$(\lambda E_n)A_n=\lambda A_n=A_n(\lambda E_n)$。 + +若$A_{m\times s}$,$B_{s\times n}=(\beta_1,\cdots,\beta_s)$,其中$\beta$为$n$行的列矩阵,则: + +$AB=A(\beta_1,\cdots,\beta_s)=(A\beta_1,\cdots,A\beta_n)$。 + +\subsection{矩阵幂} + +只有方阵才能连乘,从而只有方阵才有幂。 + +若$A$是$n$阶方阵,所以: + +$$A^1=A\text{,}A^2=A^1A^1\text{,}\cdots\text{,}A^{k+1}=A^kA^1$$ + +\begin{itemize} + \item $A^kA^l=A^{k+l}$。 + \item $(A^k)^l=A^{kl}$。 +\end{itemize} + +因为矩阵乘法一般不满足交换率,所以$(AB)^k\neq A^kB^k$。只有$AB$可交换时才相等。 + +若$A\neq 0$不能推出$A^k\neq 0$,如: + +$A=\left( + \begin{array}{cc} + 0 & 2 \\ + 0 & 0 + \end{array} +\right)\neq 0$。$A^2=\left( + \begin{array}{cc} + 0 & 2 \\ + 0 & 0 + \end{array} +\right)\left( + \begin{array}{cc} + 0 & 2 \\ + 0 & 0 + \end{array} +\right)=\left( + \begin{array}{cc} + 0 & 0 \\ + 0 & 0 + \end{array} +\right)=O$。 + +$A=\left( + \begin{array}{ccc} + 0 & 1 & 1 \\ + 0 & 0 & 1 \\ + 0 & 0 & 0 + \end{array} +\right)$,$A^3=O$。 + +矩阵幂可以同普通多项式进行处理。 + +如$f(x)=a_nx^n+\cdots+a_1x+n$,对于$A$就是$f(A)=a_nA^n+\cdots+a_1A+a_nE$。 + +$f(A)=A^2-A-6E=(A+2E)(A-3E)$。 + +\subsection{矩阵转置} + +把矩阵$A$的行换成同序数的列就得到一个新矩阵,就是$A$的转置矩阵$A^T$。若$A$为$m\times n$,则$A^T$为$n\times m$。 + +\begin{itemize} + \item $(A^T)^T=A$。 + \item $(A+B)^T=A^T+B^T$。 + \item $(\lambda A)^T=\lambda A^T$。 + \item $(AB)^T=B^TA^T$。 +\end{itemize} + +对称矩阵或对称阵:元素以对角线为对称轴对应相等,$A=A^T$。 + +\subsection{方阵行列式} + +由$n$阶方阵$A$的元素所构成的行列式称为矩阵$A$的行列式,记为$\textrm{det}\,A$或$\vert A\vert$。 + +\subsection{线性方程组与矩阵} + +\begin{itemize} + \item $\vert A^T\vert=\vert A\vert$。 + \item $\vert\lambda A\vert=\lambda^n\vert A\vert$。 + \item $\vert AB\vert=\vert A\vert\cdot\vert B\vert=\vert BA\vert$。 +\end{itemize} + +伴随矩阵或伴随阵:行列式$\vert A\vert$各个元素的代数余子式$A_{ij}$构成的矩阵。 + +$$A^*=\left( + \begin{array}{cccc} + A_{11} & A_{21} & \cdots & A_{n1} \\ + A_{12} & A_{22} & \cdots & A_{n2} \\ + \vdots & \vdots & \ddots & \vdots \\ + A_{1n} & A_{2n} & \cdots & A_{nn} + \end{array} +\right)$$ + +其中$AA^*=A^*A=\vert A\vert E$。 + +\section{线性方程组} + +矩阵是根据线性方程组得到。 + +\subsection{线性方程组与矩阵} + +\begin{multicols}{2} + + $\begin{cases} + a_{11}x_1+\cdots+a_{1n}x_n=0 \\ + \cdots \\ + a_{m1}x_1+\cdots+a_{mn}x_n=0 + \end{cases}$ \medskip + + $n$元齐次线性方程组。 + + $\begin{cases} + a_{11}x_1+\cdots+a_{1n}x_n=b_1 \\ + \cdots \\ + a_{m1}x_1+\cdots+a_{mn}x_n=b_n + \end{cases}$ \medskip + + $n$元非齐次线性方程组。 + +\end{multicols} + +对于齐次方程,$x_1=\cdots=x_n=0$一定是其解,称为其零解,若有一组不全为零的解,则称为其非零解。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。 + +令系数矩阵$A_{m\times n}=\left( + \begin{array}{ccc} + a_{11} & \cdots & a_{1n} \\ + \cdots \\ + a_{m1} & \cdots & a_{mn} + \end{array} +\right)$,未知数矩阵$X_{n\times 1}=\left( + \begin{array}{c} + x_1 \\ + \cdots \\ + x_n + \end{array} +\right)$,常数项矩阵$b_{m\times 1}=\left( + \begin{array}{c} + b_1 \\ + \cdots \\ + b_m + \end{array} +\right)$,增广矩阵$B_{m\times(n+1)}=\left( + \begin{array}{cccc} + a_{11} & \cdots & a_{1n} & b_1\\ + \cdots \\ + a_{m1} & \cdots & a_{mn} & b_n + \end{array} +\right)$。 + + + +所以$AX=\left( + \begin{array}{c} + a_11x_1+\cdots+a_{1n}x_n \\ + \cdots \\ + a_{m1}x_1+\cdots+a_{mn}x_n + \end{array} +\right)$。 + +从而$AX=b$等价于$\begin{cases} + a_{11}x_1+\cdots+a_{1n}x_n=b_1 \\ + \cdots \\ + a_{m1}x_1+\cdots+a_{mn}x_n=b_n +\end{cases}$,当$b=O$就是齐次线性方程。 + +从而矩阵可以简单表示线性方程。 + +\subsection{线性方程组的解} + +对于一元一次线性方程:$ax=b$: + +\begin{itemize} + \item 当$a\neq 0$时,可以解得$x=\dfrac{b}{a}$。 + \item 当$a=0$时,若$b\neq 0$时,无解,若$b=0$时,无数解。 +\end{itemize} + +当推广到多元一次线性方程组:$AX=b$,如何求出$X$这一系列的$x$的解? + +从数学逻辑上看,已知多元一次方程,有$m$个约束方程,有$n$个未知数,假定$m\leqslant n$。 + +当$m