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Is there any elegant proof that shows that all normal matrices are semi-simple that comes from Schur's decomposition or its corrolaries? There is a proof that normal matrices are unitary diagonizable and then that diagonizable matrices are semi-simple, but it seems a little exhaustive to combine them both. Is there any better proof?

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Consider a complex nxn matrix A. In this case, A being normal implies the Schur Decomposition gives a diagonal matrix. This says that we can diagonalize A using orthogonal vectors. And therefore the matrix A has n linearly independent eigen vectors and hence it is semi-simple.

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  • $\begingroup$ Welcome to Math.SE. Can you please edit your post and use MathJax to make the symbols look nicer. $\endgroup$ – Ertxiem - reinstate Monica Apr 24 '19 at 22:03

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