Prove that all normal matrices are semi-simple using Schur's decomposition

Question

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?

Answer 1

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.