Structure of trajectories of complex matrix eigenvalues in the Hermitian-non-Hermitian transition

TL;DR

This paper investigates how eigenvalue trajectories of Gaussian complex matrices evolve as they transition from Hermitian to non-Hermitian, revealing how real eigenvalue ordering influences their paths and final distributions.

Contribution

It provides a detailed analysis of eigenvalue trajectory structures during the Hermitian-non-Hermitian transition in Gaussian matrices, linking real eigenvalue orderings to complex plane distributions.

Findings

Eigenvalue trajectories reflect real eigenvalue ordering

Final eigenvalue distribution depends on initial real eigenvalue arrangement

Trajectory structures change systematically during the transition

Abstract

The statistical properties of trajectories of eigenvalues of Gaussian complex matrices whose Hermitian condition is progressively broken are investigated. It is shown how the ordering on the real axis of the real eigenvalues is reflected in the structure of the trajectories and also in the final distribution of the eigenvalues in the complex plane.