Suppose we are given a Hermitian matrix $E \in \mathbb{C}^{n\times n}$.
I want to find sufficient conditions on the entries of a real symmetric matrix $M$ (depending on the entries of the given matrix $E$) such that the matrix $EM$ is again Hermitian.
Is there a systematic way to determine the set of real symmetric matrices $M$ that fulfill this condition?
A necessary condition that I am already aware of is that the product of two Hermitian matrices $A,B$ is hermitian iff $AB=BA$.
I would prefer an explicit description of this set, but I am also open to suitable optimization formulation.