Eigenvectors of non normal random matrices

TL;DR

This paper investigates the angles between eigenvectors of large non-normal random matrices, revealing their asymptotic behavior and providing precise results for the Ginibre ensemble, which enhances understanding of eigenvector correlations.

Contribution

It establishes the asymptotic distribution of eigenvector angles for a broad class of non-normal random matrices, including detailed results for the Ginibre ensemble.

Findings

Eigenvector angles are asymptotically sub-Gaussian.

Rescaled inner products between eigenvectors are well-behaved in the limit.

Results apply to matrices with convex potential functions.

Abstract

We study the angles between the eigenvectors of a random $n\times n$ complex matrix $M$ with density $\propto \mathrm{e}^{-n\operatorname{Tr}V(M^*M)}$ and $x\mapsto V(x^2)$ convex. We prove that for unit eigenvectors $\mathbf{v},\mathbf{v}'$ associated with distinct eigenvalues $\lambda,\lambda'$ that are the closest to specified points $z,z'$ in the complex plane, the rescaled inner product $$\sqrt{n}(\lambda'-\lambda)\langle\mathbf{v},\mathbf{v}'\rangle$$ is uniformly sub-Gaussian, and give a more precise statement in the case of the Ginibre ensemble.