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As part of an exercise I needed to decompose the Jordan form of a complex matrix into nilpotent and diagonalizable parts, and also explain why they commute. The decomposition is trivial since strictly triangular matrices are nilpotent, but I don't see - why do these matrices commute?

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marked as duplicate by Andrew D. Hwang, Jack's wasted life, M. Vinay, user228113, hardmath Jul 4 '16 at 18:29

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    $\begingroup$ math.stackexchange.com/questions/1274980/… $\endgroup$ – Bowditch Jul 4 '16 at 13:31
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    $\begingroup$ In the context of Jordan form decomposition, it is often a question of block diagonal/nilpotent matrices. For every Jordan block, commuting of a nilpotent and a matrix of the form $\lambda I$ is not a problem. $\endgroup$ – A.Γ. Jul 4 '16 at 14:34