你知道哪些矩阵运算满足交换律吗？

一、题目

已知 $\mathit{A}$ 是 $n$ 阶矩阵, ${\mathit{A}}^{\mathrm{T}}$ 是 $\mathit{A}$ 的转置矩阵, ${\mathit{A}}^{\ast }$ 是 $\mathit{A}$ 的伴随矩阵, $\mathit{E}$ 是 $n$ 阶单位矩阵, ${\mathbf{\Lambda }}_1$, ${\mathrm{\Lambda }}_2$ 都是 $n$ 阶对角矩阵, 在下列运算中, 交换律一定成立的是哪个或者哪些？

(1) $A{A}^{\ast }=A^{\ast }A$

(2) ${\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_2={\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_1$

(3) $A^m{A}^t=A^t{A}^m$

(4) $\mathit{A}{\mathit{A}}^{\mathrm{\top }}={\mathit{A}}^{\mathrm{\top }}\mathit{A}$

(5) $\mathit{A}{\mathbf{\Lambda }}_1={\mathbf{\Lambda }}_1\mathit{A}$

(6) $(\mathit{A}+\mathit{E})(\mathit{A}-\mathit{E})=(\mathit{A}-\mathit{E})(\mathit{A}+\mathit{E})$

难度评级：

二、解析

(1)

$$A{A}^{\ast }=A^{\ast }A=|A|E$$

(2)

$${\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_2={\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_1\Rightarrow$$

$$\left[\begin{array}{ccc}{a}_1& 0& 0\\ 0& a_2& 0\\ 0& 0& a_3\end{array}\right]\left[\begin{array}{ccc}{b}_1& 0& 0\\ 0& b_2& 0\\ 0& 0& b_3\end{array}\right]=$$

$$\left[\begin{array}{ccc}{a}_1{b}_1& 0& 0\\ 0& a_2{b}_2& 0\\ 0& 0& a_3{b}_3\end{array}\right]=$$

$$\left[\begin{array}{ccc}{b}_1& 0& 0\\ 0& b_2& 0\\ 0& 0& b_3\end{array}\right]\left[\begin{array}{ccc}{a}_1& 0& 0\\ 0& a_2& 0\\ 0& 0& a_3\end{array}\right]$$

(3)

$$A^m{A}^t=A^{m+t}\iff A^t{A}^m=A^{t+m}$$

(4)

$$A{A}^{\mathrm{\top }}\ne A^{\mathrm{\top }}A\Rightarrow$$

$$(1,2,3)\left(\begin{array}{l}1\\ 2\\ 3\end{array}\right)=14$$

$$\left(\begin{array}{l}1\\ 2\\ 3\end{array}\right)(1,2,3)=\left[\begin{array}{lll}1& 2& 3\\ 2& 4& 6\\ 3& 6& 9\end{array}\right]$$

(5)

$$A{\mathrm{\Lambda }}_1\ne {\mathrm{\Lambda }}_{1}A\Rightarrow$$

$$\left[\begin{array}{ll}a& 0\\ 0& b\end{array}\right]\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right]=\left[\begin{array}{ll}a& 2a\\ 3b& 4b\end{array}\right]$$

$$\left[\begin{array}{ll}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{ll}a& 0\\ 0& b\end{array}\right]=\left[\begin{array}{ll}a& 2b\\ 3a& 4b\end{array}\right]$$

(6)

$$(A+E)(A-E)=A^2-A+A-E=A^2-E$$

$$(A-E)(A+E)=A^2+A-A-Z=A^2-E$$

于是：

$$(A+E)(A-E)=(A-E)(A+E)$$

综上可知，一定满足交换律的有：

(1) $A{A}^{\ast }=A^{\ast }A$

(2) ${\mathrm{\Lambda }}_1{\mathrm{\Lambda }}_2={\mathrm{\Lambda }}_2{\mathrm{\Lambda }}_1$

(3) $A^m{A}^t=A^t{A}^m$

(6) $(\mathit{A}+\mathit{E})(\mathit{A}-\mathit{E})=(\mathit{A}-\mathit{E})(\mathit{A}+\mathit{E})$

可能不满足交换律的有：

(4) $\mathit{A}{\mathit{A}}^{\mathrm{\top }}={\mathit{A}}^{\mathrm{\top }}\mathit{A}$

(5) $\mathit{A}{\mathbf{\Lambda }}_1={\mathbf{\Lambda }}_1\mathit{A}$

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