Eigenvector statistics in non-Hermitian random matrix ensembles

TL;DR

This paper investigates the statistical properties and correlations of eigenvectors in non-Hermitian random matrices, specifically within Ginibre's complex Gaussian ensemble, revealing significant eigenvector correlations near close eigenvalues.

Contribution

It provides a detailed analysis of eigenvector correlations in non-Hermitian matrices, highlighting their behavior in the large N limit and implications for physical systems.

Findings

Eigenvectors with close eigenvalues are highly correlated.

Eigenvector correlations become significant in the large N limit.

Implications for physical applications of non-Hermitian matrices.

Abstract

We study statistical properties of the eigenvectors of non-Hermitian random matrices, concentrating on Ginibre's complex Gaussian ensemble, in which the real and imaginary parts of each element of an N x N matrix, J, are independent random variables. Calculating ensemble averages based on the quantity $< L_\alpha | L_\beta > < R_\beta | R_\alpha >$, where $< L_\alpha |$ and $| R_\beta >$ are left and right eigenvectors of J, we show for large N that eigenvectors associated with a pair of eigenvalues are highly correlated if the two eigenvalues lie close in the complex plane. We examine consequences of these correlations that are likely to be important in physical applications.