Interlacing Eigenvectors of Large Gaussian Matrices

TL;DR

This paper analyzes the asymptotic behavior of eigenvector overlaps between large Gaussian matrices and their principal minors, providing explicit formulas and revealing eigenvector interlacing phenomena.

Contribution

It introduces a method to compute limiting eigenvector overlaps for large Gaussian matrices using a Burgers-type evolution equation, extending understanding of eigenvector dynamics.

Findings

Explicit formulas for eigenvector overlaps in large Gaussian matrices.

Identification of an eigenvector analogue of eigenvalue interlacing.

Analysis of eigenvector behavior when the initial matrix has isolated eigenvalues.

Abstract

We consider the eigenvectors of the principal minor of dimension $n< N$ of the Dyson Brownian motion in $\mathbb{R}^{N}$ and investigate their asymptotic overlaps with the eigenvectors of the full matrix in the limit of large dimension. We explicitly compute the limiting rescaled mean squared overlaps in the large $n\,, N$ limit with $n\,/\,N$ tending to a fixed ratio $q\,$, for any initial symmetric matrix $A\,$. This is accomplished using a Burgers-type evolution equation for a specific resolvent. In the GOE case, our formula simplifies, and we identify an eigenvector analogue of the well-known interlacing of eigenvalues. We investigate in particular the case where $A$ has isolated eigenvalues. Our method is based on analysing the eigenvector flow under the Dyson Brownian motion.