...可逆矩阵。如果B=P-1AP,证明:Bm=P-1AmP,这里的m为任意正整数。_百度...
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发布时间:2024-10-17 20:23
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时间:2024-10-18 03:45
B^m = B * B^(m-1)= P^(-1)AP * B^(m-1) = P^(-1)AP * P^(-1)AP * B^(m-2)
= P^(-1)A^2 P * B^(m-2)= ... = P^(-1)A^(m-1) P * B = P^(-1)A^(m-1) P * P^(-1)AP
= P^(-1)A^m P
还可以用数学归纳法来看, 推算会简单一点
B=P-1AP
B^2 =B*B = P-1AP P-1AP= P-1AAP = P^-1A^2 P
假设
B^k = P-1A^k P
B^(k+1) = B^k *B = P-1A^k P *B= P-1A^k P *P-1AP = P-1A^(k+1) P
假设成立!