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

In ex 9, page 60, he writes down that in order to prove that each invertible matrix $A$ can be written as $A=e^X$, where $X\in M_{n\times n}$, one need to use the fact that if $A$ is unipotent then $\exp (\log A) = A$ and the fact that $A$ is similar to a block diagonal matrix where in the diagonal there are matrices of the form $\lambda I+ N_{\lambda}$, where $N_{\lambda}$ is a nilpotent matrix.

Obviously after I'll prove that $A$ is unipotent then I only need to take $X=\log A$.

I am not sure how to show this, I need to show that $A-I$ is nilpotent. I thought of taking the maximum integer, $\max\{k_1,...,k_n\}$, where $N_{\lambda_j}^{k_j} =0$, and $k_j$ is the minimal that satisfies this.

Don't know how to show this?

Any hints? Thanks.

share|cite|improve this question
    
put it in jordan canonical form,en.wikipedia.org/wiki/Jordan_normal_form – mike Oct 30 '12 at 10:24

Your Answer

 
discard

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

Browse other questions tagged or ask your own question.