关于矩阵的特征多项式与最小多项式的专题讨论
$\bf命题:$设$g\left( \lambda \right)$为任意多项式,方阵$A$的最小多项式为$m\left( \lambda \right)$,则$g(A)$可逆的充要条件是$\left( {g\left( \lambda \right),m\left( \lambda \right)} \right) = 1$
$\bf命题:$设$A,B$分别为$m$阶与$n$阶矩阵,则矩阵方程$AX=XB$只有零解的充要条件是$A,B$无公共特征值
$\bf命题:$设$A$为$n$阶方阵,则$A$可逆当且仅当存在常数项不为零的多项式$f(\lambda )$,使得$f(A)=0$
$\bf命题:$设$A \in {M_n}\left( F \right)$,$m\left( \lambda \right),f\left( \lambda \right)$分别为$A$的最小多项式与特征多项式,则存在正整数$t$,使得$f\left( \lambda \right)|{m^t}\left( \lambda \right)$
$\bf命题:$$\bf(04浙大二)$设$A \in {P^{n \times n}},f\left( x \right) \in P\left[ x \right],f\left( A \right)$,可逆,证明:存在$g\left( x \right) \in P\left[ x \right]$,使得${\left( {f\left( A \right)} \right)^{ - 1}} = g\left( A \right)$
$\bf命题:$$\bf(06江苏九)$设$n$阶矩阵$A$的特征多项式为$f(\lambda)$,且\[\left( {f\left( \lambda \right),f'\left( \lambda \right)} \right) = d\left( \lambda \right),h\left( \lambda \right) = f\left( \lambda \right)/d\left( \lambda \right)\]证明:$A$相似于对角阵的充要条件是$h(A)=0$
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$\bf命题:$