用試乘的方法計算 A^2,A^3,找出一般規(guī)律,然后用歸納法證明.
1.這是對角矩陣,其n次方仍是對角矩陣,且主對角線上元素為原元素的n次方
A = diag(a1,a2,...,as),則 A^n = diag(a1^n,a2^n,...,as^n)
2.試乘
A^2 =
2 0 2
0 4 0
2 0 2
A^3 =
4 0 4
0 8 0
4 0 4
歸納假設 A^k =
2^(k-1) 0 2^(k-1)
0 2^k 0
2^(k-1) 0 2^(k-1)
則
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)