关于等价标准形的专题讨论

$\bf命题:$任意方阵$A$均可分解为可逆阵$B$与幂等阵$C$之积

1

$\bf命题:$任意方阵$A$均可分解为可逆阵$B$与对称阵$C$之积

1

$\bf命题:$设$A,B \in {P^{n \times n}}$，且$r\left( A \right) + r\left( B \right) \le n$，则存在$n$阶可逆阵$M$，使得$AMB = 0$

1

$\bf命题:$设$A$为$n$阶方阵，则存在$n$阶方阵$B$，使得$A=ABA,B=BAB$

1

$\bf命题:$设$A$是秩为$r$的$m \times r$矩阵$\left( {m > r} \right)$，$B$为$r \times s$矩阵，则存在可逆阵$P$，使得$PA$的后$m-r$行全为零

1

$\bf命题:$设$T \in L\left( {V,n,F} \right)$，则存在$S \in L\left( {V,n,F} \right)$，使得$TST = T$

1

$\bf命题:$设$A \in {M_{m \times n}}\left( F \right),B \in {M_{n \times m}}\left( F \right),m \ge n,\lambda \ne 0$，则

\[{\rm{ }}\left| {\lambda {E_m} - AB} \right| = {\lambda ^{m - n}}\left| {\lambda {E_n} - BA} \right|\]

1

$\bf命题:$设$A,B,C$为$n$阶矩阵，且$AC=CB$，$r\left( C \right) = r$，则$A$与$B$至少有$r$个相同特征值

1

$\bf命题:$设$A,B$为$n$阶矩阵，且$BA=A$，$r\left( A \right) =r\left( B \right) $，则${B^2} = B$

1

$\bf命题:$设${\alpha _1},{\alpha _2}, \cdots ,{\alpha _n}$为${V_n}\left( F \right)$的一个基，$A \in {M_{n \times s}}\left( F \right)$，且\[\left( {{\beta _1},{\beta _2}, \cdots ,{\beta _s}} \right) = \left( {{\alpha _1},{\alpha _2}, \cdots ,{\alpha _n}} \right)A\]证明：$\dim L\left( {{\beta _1},{\beta _2}, \cdots ,{\beta _s}} \right) = r\left( A \right)$

1

$\bf命题:$设$A,B$为$n$阶矩阵，若$r\left( {AB} \right) = r\left( {BA} \right)$对任意的$B$成立，则$A = 0$或$A$可逆

1

$\bf命题:$设$P \in {F^{r \times m}},Q \in {F^{n \times s}}$，若对任意的$A \in {F^{m \times n}}$，都有$PAQ=0$，证明：$P=0或Q=0$

1

$\bf命题:$设$A \in {M_m}\left( F \right),C \in {M_n}\left( F \right)$，若对于$B \in {M_{mn}}\left( F \right)$，有$r\left( {\begin{array}{*{20}{c}}A&B \\ 0&C \end{array}} \right) = r\left( A \right) + r\left( C \right)$，证明：$A或C$可逆

1

$\bf命题:$若$矩阵{A_{m \times n}}{B_{n \times p}}{C_{p \times q}}$的秩对一切秩$1$的矩阵$B$总为$1$，则$A$为列满秩，且$C$为行满秩

1

$\bf命题:$设$A \in {M_n}\left( F \right),r\left( A \right) = r\left( {{A^2}} \right)$，则存在可逆阵$P$，使得$A = P\left( {\begin{array}{*{20}{c}}D&0 \\ 0&0 \end{array}} \right){P^{ - 1}}$

1

$(04浙大七)$设$V = {P^{n \times n}}$看成数域$P$上的线性空间，取定$A,B,C,D \in {P^{n \times n}}$，对任意$X \in {P^{n \times n}}$，令\[\sigma \left( X \right) = AXB + CX + XD\]

证明：$(1)$$\sigma $是$V$上的线性变换  $(2)$当$C = D = 0$时，$\sigma $可逆的充要条件是$\left| {AB} \right| \ne 0$