TITLE: If $AA^*=AA$, how to prove $A$ is an Hermitian?
QUESTION [5 upvotes]: If $A$ is an $n \times n$ matrix and $AA^*=AA,$ how to prove $A$ is Hermitian?

REPLY [7 votes]: An arbitrary square matrix $A$ can be written as $B + i C$ where $B = (A + A^*)/2$ and $C = (A - A^*)/(2i)$ are Hermitian.  The given condition
says $AC = 0$, i.e. $B C = -iC^2$.  But $CBC = -iC^3$, and the left side is Hermitian while the right side is only Hermitian if $C = 0$.  Therefore $C = 0$ which means $A$ is Hermitian.