Can this condition infer that the matrix is Hermite?

Question

$\boldsymbol{A^H A=AA^H}$ does this imply that $\boldsymbol{A}$ is Hermite matrix? Why?

$\boldsymbol{A^H}$ is the conjugate transpose of $\boldsymbol{A}$

Answer 1

No, your condition means that the matrix is normal. Not all normal matrices are hermitian (for example, pick your favorite unitary matrix).

Answer 2

Obviously counterexample: Skew-Hermitian matrices and unitary matrices.