Statistical properties of eigenvalues of an ensemble of pseudo-Hermitian Gaussian matrices

TL;DR

This paper studies the eigenvalue distributions of pseudo-Hermitian Gaussian matrices, revealing how real and complex eigenvalues exhibit different spectral properties and repulsion behaviors.

Contribution

It provides a detailed analysis of the statistical properties and eigenvalue repulsion in pseudo-Hermitian Gaussian matrices, highlighting differences between real and complex eigenvalues.

Findings

Real eigenvalues form an intermediate thinned spectrum.

Complex eigenvalues exhibit cubic or higher order repulsion.

Eigenvalue behavior varies with matrix type (real, complex, quaternion).

Abstract

We investigate the statistical properties of eigenvalues of pseudo-Hermitian random matrices whose eigenvalues are real or complex conjugate. It is shown that when the spectrum splits into separated sets of real and complex conjugate eigenvalues, the real ones show characteristics of an intermediate incomplete spectrum, that is, of a so-called thinned ensemble. On the other hand, the complex ones show repulsion compatible with cubic-order repulsion of non normal matrices for the real matrices, but higher order repulsion for the complex and quaternion matrices.