I know that the product $AB$ of a Hermitian and positive definite matrix $A$ and a Hermitian matrix $B$ is itself Hermitian.
From simulations (where $A$ is diagonal, but I don't think this matters), I have a suspicion that the signature (number of positive/negative eigenvalues) of $AB$ is the same as $B$. I think Sylvester's law of inertia is applicable, but I don't know how to view $A$ as a transformation under which the signature is invariant. Also, I don't know anything about quadratic forms, so please bear with me.
I'd like to use this result in my (physics) bachelor's thesis, so any help is greatly appreciated!