Eigenvector Overlaps of Random Covariance Matrices and their Submatrices

TL;DR

This paper analyzes the asymptotic overlaps between singular vectors of submatrices and full matrices of Gaussian random matrices, providing explicit formulas in the macroscopic limit for various spectral regimes.

Contribution

It introduces a novel method using singular vector dynamics and resolvents to derive explicit overlap formulas for submatrix and full matrix singular vectors.

Findings

Explicit formulas for mean squared overlaps in the bulk spectrum

Results apply to general initial matrices, including null matrices

Simplified formulas in the Marchenko-Pastur case

Abstract

We consider the singular vectors of any $m \times n$ submatrix of a rectangular $M \times N$ Gaussian matrix and study their asymptotic overlaps with those of the full matrix, in the macroscopic regime where $N \,/\, M\,$, $m \,/\, M$ as well as $n \,/\, N$ converge to fixed ratios. Our method makes use of the dynamics of the singular vectors and of specific resolvents when the matrix coefficients follow Brownian trajectories. We obtain explicit forms for the limiting rescaled mean squared overlaps for right and left singular vectors in the bulk of both spectra, for any initial matrix $A\,$. When it is null, this corresponds to the Marchenko-Pastur setup for covariance matrices, and our formulas simplify into Cauchy-like functions.