Question

已知A的特征值1,2,所对应的特征向量,求A,A^2,A^100

Answer 1

由已知A为2阶方阵,且有两个线性无关的特征向量a1,a2

故令 P=(a1,a2),有 P^-1AP=diag(1,2).

所以 A = Pdiag(1,2)P^-1

=

1 0 1 0 1 0

1 1 0 2 -1 1

=

1 0

-1 2

A^2 =

1 0

-3 4

A^100 = Pdiag(1,2)^100P^-1 = Pdiag(1,2^100)P^-1

=

1 0 1 0 1 0

1 1 0 2^100 -1 1

=

1 0

1-2^100 2^100