Given $C$ a hermitian complex non-singular square matrix, and $\Delta$ real and diagonal with distinct, non-zero diagonal elements, I need to find an orthogonal matrix $H$ such that $$C=H\Delta H^H.$$
Is there a way to do this? One thought was to find the eigendecomposition of $C$: $$C=GDG^H.$$ Since $D$ is also real and diagonal, and $G$ is orthogonal, it has the same structure as the first equation and I thought I might be able to calculate $H$ from $G$ and $D$. However, I haven't been able to make much progress.
Is there a way to determine (a possible, if not unique) $H$ from the assumptions?
(I've found similar questions but none exactly the same; for instance in Matrix equation with a constraint the matrix $H$ is not constrained to be orthogonal.)