已知一个方阵为A={n10;0n1;00n}求该方阵A的K次幂,希望能够解答,

Question

已知一个方阵为 A={n 1 0; 0 n 1; 0 0 n} 求该方阵A的K次幂,希望能够解答,

Answer 1

A=
n 1 0
0 n 1
0 0 n
= nE + B
B =
0 1 0
0 0 1
0 0 0
由于 nE 与 B 可交换,且 B^2=
0 0 1
0 0 0
0 0 0
B^3 = 0
所以 A^k = (nE)^k + k(nE)^(k-1)B + [k(k-1)/2] (nE)^(k-2) B^2
= n^kE + kn^(k-1)B + [k(k-1)/2] n^(k-2) B^2
=
n^k kn^(k-1) [k(k-1)/2] n^(k-2)
0 n^k kn^(k-1)
0 0 n^k