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

Source abstract

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