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$\bf命题1:$任意方阵$A$均可分解为可逆阵$B$与幂等阵$C$之积

证明：设$r\left( A \right) = r$，则存在可逆阵$P,Q$，使得
\[PAQ = \left( {\begin{array}{*{20}{c}}
{{E_r}}&0\\
0&0
\end{array}} \right)\]
从而可知\begin{align*}
A& = {P^{ - 1}}\left( {\begin{array}{*{20}{c}}
{{E_r}}&0\\
0&0
\end{array}} \right){Q^{ - 1}}\\&
{\rm{ = }}{P^{ - 1}}{Q^{ - 1}}.Q\left( {\begin{array}{*{20}{c}}
{{E_r}}&0\\
0&0
\end{array}} \right){Q^{ - 1}}
\end{align*}
取$B = {P^{ - 1}}{Q^{ - 1}}$，$C = Q\left( {\begin{array}{*{20}{c}}
{{E_r}}&0\\
0&0
\end{array}} \right){Q^{ - 1}}$,即证