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

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 $Z_2$.

4 pages, latex; version 2: one affiliation updated, one reference updated, no other changes

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Cited by in corpus (4)

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• Entropy and entanglement in a bipartite quasi-Hermitian system and its Hermitian counterparts
• Pseudo-hermitian Chebyshev differential matrix and non-Hermitian Liouville quantum mechanics