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2024-07-02 20:23:21   228人阅读

$\bf命题1:$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=(Er000)
<script id="MathJax-Element-1" type="math/tex; mode=display">PAQ = \left( {\begin{array}{*{20}{c}} {{E_r}}&0\\ 0&0 \end{array}} \right)</script>
从而可知
A=P?1(Er000)Q?1=P?1Q?1.Q(Er000)Q?1
<script id="MathJax-Element-2" type="math/tex; mode=display">\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*}</script>
取$B = {P^{ - 1}}{Q^{ - 1}}$，$C = Q\left( {
Er000
<script id="MathJax-Element-3" type="math/tex; mode=display">\begin{array}{*{20}{c}} {{E_r}}&0\\ 0&0 \end{array}</script>} \right){Q^{ - 1}}$,即证


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