Is there some simple transformation (or a simple way to find it) which would convert any given Hermitian matrix $A$ to a symmetric matrix $B$ with the same spectrum as that of $A$ (so I guess that transformation should be unitary)?
I know I can represent an hermitian matrix as a $2\times2$ block matrix like
$$A\to\left(\array{\Re A&-\Im A\\ \Im A&\Re A}\right),$$ but I'd like to preserve dimensions of $A$.
By simple I mean that I shouldn't have to diagonalize $A$ to achieve what I need (diagonalization is in fact what I want to do with the result, not having to take 4 times more space in complex case).
If not in general, then is there maybe some special-case way for such $A$ that $\Im A$ is a diagonal matrix?