We know that not all matrices can be diagonalized, but all matrices can be block diagonalized (with just one block) How can we find a similarity transformation leading to block diagonalization with the greatest possible number of blocks?
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$\begingroup$Every matrix with elements in $\mathbb C$ has a Jordan Normal Form. The transform in the canonical basis will have blocks of sizes equal to the sizes of the generalized eigenspaces of the matrix.
The Jordan blocks have a very particular structure:
$$\left[\begin{array}{ccc}\lambda&1&0&\cdots&0\\0&\lambda&1&0&0\\0&\ddots&\ddots&\ddots&0\\0&0&0&\lambda&1\\0&0&0&0&\lambda\end{array}\right]$$
where the $\lambda$ is an eigenvalue for the matrix. It should be possible to prove that the block above can not be further reduced (although I have no proof ready in my magic pockets right now).
In the case you want real elements everywhere you can take a look here
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