Gaussian statistics for left and right eigenvectors of complex non-Hermitian matrices

TL;DR

This paper demonstrates that for large complex non-Hermitian matrices, certain projections of eigenvectors are Gaussian and independent, revealing statistical properties of eigenvectors in such matrices.

Contribution

It establishes Gaussianity and independence of projections of eigenvectors associated with well-separated eigenvalues in large non-Hermitian matrices.

Findings

Eigenvector projections are Gaussian for large matrices.

Projections are jointly independent.

Results hold for eigenvalues with pairwise distances at least N^{-1/2+ε}.

Abstract

We consider a constant-size subset of left and right eigenvectors of an $N\times N$ i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least $N^{-\frac12+\epsilon}$. We show that arbitrary constant rank projections of these eigenvectors are Gaussian and jointly independent.