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@@ -76,6 +76,14 @@ $\therefore A^n=\left\{\begin{array}{lcl} 4^kA, & & n=2k+1 \end{array}\right.$。 +\subsection{行列结合} + +将一个矩阵拆成$\alpha\beta^T$的形式,其中都是列向量,从而进行幂运算可以进行结合$\beta^T\alpha$为一个常数。 + +\textbf{例题:}设$\alpha=(1,3,-2)^T$,$\beta=(2,0,0)^T$,$A=\alpha\beta^T$,求$A^3$。 + +解:$\because\beta^T\alpha=[2,0,0][1,3,-2]^T=2$,$\therefore A^3=(\alpha\beta^T)(\alpha\beta^T)(\alpha\beta^T)=\alpha(\beta^T\alpha)$\\$(\beta^T\alpha)\beta^T=4\alpha\beta^T=4A$。 + \subsection{拆分矩阵} 将$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 @@ -144,6 +152,16 @@ $=\left(\begin{array}{ccc} 0 & 0 & 0 \end{array}\right)$ +\subsection{分块矩阵} + +$\left[\begin{array}{cc} + A & O \\ + O & B +\end{array}\right]^n=\left[\begin{array}{cc} + A^n & O \\ + O & B^n +\end{array}\right]$。 + \section{初等变换} \subsection{可逆矩阵} @@ -228,7 +246,7 @@ $AB-A=B$,所以提出$A(B-E)=B$,即$A(B-E)=B-E+E$,$(A-E)(B-E)=E$,所以$ \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$A^{-1}=C^{-1}B^{-1}$。 +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$A^{-1}=C^{-1}B^{-1}$。同理$(ABC)^{-1}=C^{-1}B^{-1}A^{-1}$。 \textbf{例题:}设$A$,$B$为同阶可逆方阵,且$A^{-1}+B^{-1}$可逆,求$(A+B)^{-1}$。 @@ -352,6 +370,31 @@ $A=\left(\begin{array}{ccc} A_1^{-1} \end{array}\right)$。 +\textbf{例题:}已知矩阵$A$的伴随矩阵$A^*=\left[\begin{array}{cccc} + 4 & -2 & 0 & 0 \\ + -3 & 1 & 0 & 0 \\ + 0 & 0 & -4 & 0 \\ + 0 & 0 & 0 & -1 +\end{array}\right]$,求$A$。 + +解:由于$A^{-1}=\dfrac{A^*}{\vert A\vert}$,所以$A=\vert A\vert(A^*)^{-1}$。已知$A^*$可知$(A^*)^{-1}$,所以重点就是求$\vert A\vert$。 + +又$\vert A^*\vert=\vert A\vert^{n-1}$,$\vert A^*\vert=-8$,$\vert A\vert=-2$。 + +所以根据分块矩阵的逆运算,可以得到$(A^*)^{-1}=\left[\begin{array}{cccc} + -\dfrac{1}{2} & -1 & 0 & 0 \\ + -\dfrac{3}{2} & -2 & 0 & 0 \\ + 0 & 0 & -\dfrac{1}{4} & 0 \\ + 0 & 0 & 0 & -1 +\end{array}\right]$。 + +所以$A=\left[\begin{array}{cccc} + 1 & 2 & 0 & 0 \\ + 3 & 4 & 0 & 0 \\ + 0 & 0 & \dfrac{1}{2} & 0 \\ + 0 & 0 & 0 & 2 +\end{array}\right]$。 + \section{方阵行列式} \subsection{两项积商} @@ -361,6 +404,7 @@ $A=\left(\begin{array}{ccc} \item $\vert A^{-1}\vert=\dfrac{1}{\vert A\vert}$。 \item $\vert\lambda A\vert=\lambda^n\vert A\vert$。 \item $\vert AB\vert=\vert A\vert\cdot\vert B\vert=\vert BA\vert$。 + \item $\vert A^*\vert=\vert A\vert^{n-1}$。 \end{itemize} 因为两项积商比较简单,所以基本上会变换$A$和$B$,让其变为转置或逆矩阵。 @@ -379,6 +423,8 @@ $A=\left(\begin{array}{ccc} 若$A$、$B$可逆,且可以分别得到$X=A^{-1}B$,$X=BA^{-1}$,$X=A^{-1}CB^{-1}$。 +\subsection{直接化简} + \textbf{例题:}设3阶方阵$A$,$B$满足$A^{-1}BA=6A+BA$,且$A=\left(\begin{array}{ccc} \dfrac{1}{3} & 0 & 0 \\ 0 & \dfrac{1}{4} & 0 \\ @@ -389,4 +435,32 @@ $A=\left(\begin{array}{ccc} $\therefore B=6(A^{-1}-E)^{-1}$。 +\subsection{凑目标式} + +有时候直接化简非常麻烦,因为所求的式子很复杂,甚至出现结果不能得到的情况。 + +\textbf{例题:}已知$AB=A+B$,其中$B=\left[\begin{array}{ccc} + 1 & 1 & 0 \\ + 1 & 1 & 0 \\ + 0 & 0 & 2 +\end{array}\right]$,求$(A-E)^{-1}$。 + +解:已知$AB=A+B$,求$A-E$,则向目标计算。 + +$AB-B=A$,即$(A-E)B=A$,$(A-E)^{-1}=BA^{-1}$。因为$A$未知,所以要消去$A$。 + +根据$AB=A+B$,得到$AB-A=B$,即$A(B-E)=B$,$A^{-1}=(B-E)B^{-1}$。 + +$(A-E)^{-1}=BA^{-1}=B(B-E)B^{-1}$,然后就不知道接下来怎么办了。 + +我们很希望$BB^{-1}$在一起消掉,但是无论如何操作都无法完成。但是也可以通过此得到解题的启示,按$(A-E)(B-E)$去凑。 + +回到$(A-E)B=A$,去凑$B-E$,先尝试两边减去$E$,得到$(A-E)B-E=A-E$,正好左移右项$(A-E)(B-E)=E$,解得$(A-E)^{-1}=B-E$。 + +即$=\left[\begin{array}{ccc} + 0 & 1 & 0 \\ + 1 & 0 & 0 \\ + 0 & 0 & 1 +\end{array}\right]$。 + \end{document}