Home > Higher Algebra, Olympiad > Characterization of a normal matrix

## Characterization of a normal matrix

A $n\times n,\ (n \geq 2)$ matrix $A$ with complex entries is called normal if $AA^*=A^*A$, where $A^*$ is the conjugate transpose of $A$. Prove that $A$ is normal if and only if $\text{rank}\; (AA^*-A^*A) \leq 1$.

Hint: Prove that if $C$ is a matrix of rank $1$ then $C^2=\text{Tr}(C)\cdot C$.

What happens when rank $(AA^* - A^*A) = 1$?
When rank of $AA^*-A^*A$ is $1$ use the hint, since the trace of this matrix is zero and therefore $(AA^*-A^*A)^2=0$. Then if you denote $X=AA^*-A^*A$ and note that $X^*=X$ you get $XX^*=X^2=0$, and therefore $X=0$.