Which metrics are consistent with a given pseudo-hermitian matrix?

Given a diagonalizable N × matrix H, whose non-degenerate spectrum consists of p pairs complex conjugate eigenvalues and additional − 2p real eigenvalues, we determine all metrics M, possible signatures, with respect to which H is pseudo-hermitian. In particular, show that any compatible M must have opposite in its so the minimal number both positive negative M. We provide explicit parameterization space admissible it topologically p-dimensional torus tensored an appropriate power group Z2.