Gaussian pseudo-Orthogonal Ensemble of Real Random Matrices

TL;DR

This paper introduces Gaussian pseudo-orthogonal ensembles of real pseudo-symmetric matrices, analyzing their eigenvalue distributions and spectral properties, with implications for PT-symmetric quantum systems.

Contribution

The paper constructs and studies new Gaussian pseudo-orthogonal ensembles of pseudo-symmetric matrices, revealing their spectral behaviors and group properties, connecting to PT-symmetric quantum physics.

Findings

Eigenvalues are real or complex conjugate pairs depending on parameters.

Spectral distributions follow Wigner's statistics when eigenvalues are real.

Diagonalizing matrices are pseudo-orthogonal under a constant metric.

Abstract

Here, using two real non-zero parameters $\lambda$ and $\mu$, we construct Gaussian pseudo-orthogonal ensembles of a large number $N$ of $n \times n$ ($n$ even and large) real pseudo-symmetric matrices under the metric $\eta$ using $ \altmathcal {N}=n(n+1)/2$ elements independently drawn from a Gaussian random population and investigate the statistical properties of the eigenvalues. When $\lambda \mu >0$, we show that the pseudo-symmetric matrix is similar to a real symmetric matrix, consequently, all the eigenvalues are real and so the spectral distributions satisfy Wigner's statistics. But when $\lambda \mu <0$ the eigenvalues are either real or complex conjugate pairs. We find that these real eigenvalues exhibit intermediate statistics. We show that the diagonalizing matrices ${ \cal D}$ of these pseudo-symmetric matrices are pseudo-orthogonal under a constant metric $\zeta$ as $ \altmathcal{D}^t \zeta \altmathcal{D}= \zeta$, and hence they belong to a pseudo-orthogonal group. These pseudo-symmetric matrices serve to represent the parity-time (PT)-symmetric quantum systems having exact (un-broken) or broken PT-symmetry.