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设P是n阶可逆矩阵．如果B=P^-1AP，证明B^m=P^-1A^mP，这里m为任意正整数．

高老师2年前 (2024-03-26)线性代数（经管类）(04184)7

设P是n阶可逆矩阵．如果B=P^-1AP，证明B^m=P^-1A^mP，这里m为任意正整数．

证明：当m=1时，B=P^-1AP成立设当m=k(k≥1)时，B=P^-1A^kP成立则当m=k+1时，B=P^-1A^kP•P^-1AP =P^-1A^k+1P 所以当m=k+1时，等式成立．综上所述知B^m=P^-1A^mP

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