Question

已知3阶矩阵A有3维向量A满足A^3X=3AX-A^2X,且向量组X,AX,A^2X线性无关.（1）记P=(X,AX,A

Answer 1

1) AP=A(x, Ax, A^2*x)

=(AX,A^2X,A^3X)

=(AX,A^2X,3AX-A^2X)

=(X,AX,A^2X)[ 0 0 0

1 0 3

0 1 -1 ]

=PB

B =[ 0 0 0

1 0 3

0 1 -1 ]

2) AP=PB ,X,AX,A^2X线性无关,P可逆

P逆*A*P=B ,A与B相似

|A|=|B|=0