On a class of H-selfadjont random matrices with one eigenvalue of nonpositive type

TL;DR

This paper studies large H-selfadjoint random matrices with one negative eigenvalue, showing the eigenvalue converges to a deterministic limit and analyzing the distribution of real eigenvalues.

Contribution

It introduces a class of H-selfadjoint matrices with a unique nonpositive eigenvalue and characterizes its asymptotic behavior and eigenvalue distribution.

Findings

The unique eigenvalue converges in probability to a deterministic limit.

The weak limit of the real eigenvalues' distribution is characterized.

The study advances understanding of spectral properties of H-selfadjoint matrices.

Abstract

Large H-selfadjoint random matrices are considered. The matrix $H$ is assumed to have one negative eigenvalue, hence the matrix in question has precisely one eigenvalue of nonpositive type. It is showed that this eigenvalue converges in probability to a deterministic limit. The weak limit of distribution of the real eigenvalues is investigated as well.