Product of two Hermitian matrices

Question

According to Wikipedia:

The product of two Hermitian matrices $A$ and $B$ is Hermitian if and only if $AB = BA$.

So if I understood correctly, if $C=AB$, then C will be Hermitian if and only if $AB=BA$.

But... I've been able to create a matrix $S$ then did $R=SS^H$, and $R$ turned out to be Hermitian, even though $SS^H \neq S^HS$.

So I'm clearly misunderstanding that property I quoted. Could anyone help me? Thank you!

Answer 1

Note that $S^HS$ is not the adjoint of $SS^H$. The adjoint of $SS^H$ is always $SS^H$, whatever $S$ is. In your example, your $S$ is not hermitian, so the commutation of hermitian matrices does not apply.