Statistical properties of eigenvectors in non-Hermitian Gaussian random matrix ensembles

TL;DR

This paper investigates the statistical properties and correlations of eigenvectors in non-Hermitian Gaussian random matrices, revealing their impact on eigenvalue sensitivity and transient dynamics in physical systems.

Contribution

It provides exact and approximate methods to analyze eigenvector correlations in non-Hermitian ensembles, highlighting their physical significance.

Findings

Eigenvalues are more sensitive to perturbations than in Hermitian ensembles.

Eigenvector correlations influence transient behaviors in time-evolution problems.

Exact calculations are performed for Ginibre's ensemble, with approximate methods for Girko's ensemble.

Abstract

Statistical properties of eigenvectors in non-Hermitian random matrix ensembles are discussed, with an emphasis on correlations between left and right eigenvectors. Two approaches are described. One is an exact calculation for Ginibre's ensemble, in which each matrix element is an independent, identically distributed Gaussian complex random variable. The other is a simpler calculation using $N^{-1}$ as an expansion parameter, where $N$ is the rank of the random matrix: this is applied to Girko's ensemble. Consequences of eigenvector correlations which may be of physical importance in applications are also discussed. It is shown that eigenvalues are much more sensitive to perturbations than in the corresponding Hermitian random matrix ensembles. It is also shown that, in problems with time-evolution governed by a non- Hermitian random matrix, transients are controlled by eigenvector correlations.