关于矩阵的特征多项式与最小多项式的专题讨论

$f命题：$设$gleft( lambda   ight)$为任意多项式，方阵$A$的最小多项式为$mleft( lambda   ight)$，则$g(A)$可逆的充要条件是$left( {gleft( lambda   ight),mleft( lambda   ight)} ight) = 1$

1

$f命题：$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

1

$f命题：$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$f(A)=0$

1

$f命题：$设$A in {M_n}left( F ight)$，$mleft( lambda   ight),fleft( lambda   ight)$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$fleft( lambda   ight)|{m^t}left( lambda   ight)$

1

$f命题：$$f(04浙大二)$设$A in {P^{n imes n}},fleft( x ight) in Pleft[ x ight],fleft( A ight)$，可逆，证明：存在$gleft( x ight) in Pleft[ x ight]$，使得${left( {fleft( A ight)} ight)^{ - 1}} = gleft( A ight)$

$f命题：$$f(06江苏九)$设$n$阶矩阵$A$的特征多项式为$f(lambda)$，且[left( {fleft( lambda   ight),f'left( lambda   ight)} ight) = dleft( lambda   ight),hleft( lambda   ight) = fleft( lambda   ight)/dleft( lambda   ight)]证明：$A$相似于对角阵的充要条件是$h(A)=0$

1

$f命题：$