Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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...

share|cite|improve this question
up vote 3 down vote accepted

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

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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.