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Hi I was wondering if this is true. Let $M_{n}[\mathbb{C}]$ be matrices with complex values. Then any normal matrix is the sum of a hermitian matrix and another hermitian matrix multiplied by $i$. and these two matrices commute. But I do not know a proof.

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    $\begingroup$ Start with the conclusion. $A = X + iY$ then $A^* = X^* - i Y^* = X - iY$. Now solve for $X$ and $Y$. $\endgroup$ – steven gregory May 5 at 11:52
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Hint: normal matrices are unitarily diagonalisable.

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