计算矩阵的n次幂

Question

如题所述

Answer 1

用试乘的方法计算
a^2,a^3,
找出一般规律,
然后用归纳法证明.
1.
这是对角矩阵,
其n次方仍是对角矩阵,
且主对角线上元素为原元素的n次方
a
=
diag(a1,a2,...,as),
则
a^n
=
diag(a1^n,a2^n,...,as^n)
2.
试乘
a^2
=
2
0
2
0
4
0
2
0
2
a^3
=
4
0
4
0
8
0
4
0
4
归纳假设
a^k
=
2^(k-1)
0
2^(k-1)
0
2^k
0
2^(k-1)
0
2^(k-1)
则
a^k
=
aa^(k-1)
=
2^k
0
2^k
0
2^(k+1)
0
2^k
0
2^k
故
a^n
=
2^(n-1)
0
2^(n-1)
0
2^n
0
2^(n-1)
0
2^(n-1)