# Followup question in Brian Hall's Lie Groups and Algebras.

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.

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