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Let $B$ a matrix over $\mathbb{C}$ and $J$ be a jordan matrix for $B$. Then we have $B=P^{-1}JP$. How to prove that $B^k=P^{-1}J^kP$?

Thanks.

Pedro
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1 Answers1

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Use induction. It's obviously true for $k = 1$ (by definition). Assume it's also true for some $k$, i.e. $$ B^k = P^{-1} J^k P $$ Now, consider $$ B^{k+1} = B^k B = P^{-1}J^k P B $$ Now, use the expression that's Jordan decomposition, then $$ B^{k+1} = P^{-1}J^k P P^{-1} J P = P^{-1}J^{k+1} P $$ so it's also true for $k+1$, i.e. according to the principle of mathematical induction, it's true for all $k \in \mathbb N$.

Kaster
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