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How to prove the following fact or where can I find its proof?

Let $A \in M(\mathbb{R}^d)$ and assume that $\rho(A)>0$ (where $\rho(A)$ stands for the spectral radius of $A$). Then, for each matrix norm $\|\cdot\|$ the sequence $(n^{-1} \ln (\|A^n\|)_{n\in\mathbb{N}}$ converges to $\ln(\rho(A))$.

I know that in order to prove it we should use the Jordan decomposition but I have no idea how to cope with this...

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up vote 3 down vote accepted

This is immediate consequence of spectral radius formula which holds for arbitrary bounded operator on arbitrary Banach space.

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Yes, you are right, a was looking everywhere but not in wikipedia :) –  dawid Dec 23 '12 at 21:41
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