$f命题1:$任意方阵$A$均可分解为可逆阵$B$与幂等阵$C$之积
证明：设$rleft( A
ight) = r$，则存在可逆阵$P,Q$，使得
[PAQ = left( {egin{array}{*{20}{c}}
{{E_r}}&0\
0&0
end{array}}
ight)]
从而可知egin{align*}
A& = {P^{ - 1}}left( {egin{array}{*{20}{c}}
{{E_r}}&0\
0&0
end{array}}
ight){Q^{ - 1}}\&
{
m{ = }}{P^{ - 1}}{Q^{ - 1}}.Qleft( {egin{array}{*{20}{c}}
{{E_r}}&0\
0&0
end{array}}
ight){Q^{ - 1}}
end{align*}
取$B = {P^{ - 1}}{Q^{ - 1}}$，$C = Qleft( {egin{array}{*{20}{c}}
{{E_r}}&0\
0&0
end{array}}
ight){Q^{ - 1}}$,即证