# Prove that any normal matrix is the sum of hermitian matrices

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.

• Start with the conclusion. $A = X + iY$ then $A^* = X^* - i Y^* = X - iY$. Now solve for $X$ and $Y$. – steven gregory May 5 '19 at 11:52