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So I've worked through a couple examples which were straight forward.

Matrices like $A = \left[\begin{array}{cc} 4 & 1\\ 0 & e\end{array}\right]$, were easy because $A$ is diagonalizable.

The only non-diagonalizable example we covered in class were of the form

$\lambda I + N$, where $N^r = 0$ for some positive integer $r$, then we used the formula

$\log(\lambda I + N) = \log(\lambda I) + \sum\limits_{n=1}^{r-1}\frac{(-N)^{n}}{j\lambda^{j}}$.

How can you calculate the log if it doesn't fall under one of these two forms?

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The Jordan normal form of a complex matrix can be written as a "block diagonal" matrix, where each block on the diagonal is of the form $\lambda I + N$ in your form. So you can compute the logarithm of each block. See: http://en.wikipedia.org/wiki/Jordan_normal_form#Complex_matrices and http://en.wikipedia.org/wiki/Logarithm_of_a_matrix#The_logarithm_of_a_non-diagonalizable_matrix.

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That's where the Jordan decomposition comes in. Remember that there is always a similarity transformation for a matrix that turns it into a block-diagonal matrix whose diagonal blocks are either scalars or Jordan blocks. Once you have the Jordan decomposition, apply the logarithm formulae you know on the scalar and/or Jordan blocks, and then undo the similarity transformation.

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    You'll want to see [this](http://books.google.com/books?id=PlYQN0ypTwEC&pg=PA119).2011-10-23