Which Metrics Are Consistent with a Given Pseudo-Hermitian Matrix?

TL;DR

This paper characterizes all metrics under which a given diagonalizable matrix with complex conjugate and real eigenvalues is pseudo-hermitian, providing explicit parametrizations and topological structure of the solution space.

Contribution

It explicitly determines the metrics compatible with a pseudo-hermitian matrix and describes the topology of the space of all such metrics.

Findings

Compatible metrics must have p pairs of opposite eigenvalues.

The space of admissible metrics is topologically a p-dimensional torus tensor with a power of Z_2.

Provides explicit parametrization of all such metrics.

Abstract

Given a diagonalizable $N\times N$ 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 parametrization 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 $Z_2$.