989

$\bf命题2:$任意方阵$A$均可分解为可逆阵$B$与对称阵$C$之积

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