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)