Spectra of nearly Hermitian random matrices

TL;DR

This paper investigates the eigenvalues and eigenvectors of nearly Hermitian random matrices, revealing that eigenvalues can be complex, most lie close to the real line, and it provides new bounds on outliers for certain perturbations.

Contribution

It introduces new results on the spectral properties of matrices formed by adding low-rank deterministic perturbations to Wigner and sample covariance matrices, including bounds on eigenvalue locations.

Findings

Eigenvalues of M + P can be complex, even with rank-one P.

Most eigenvalues of M + P are within 1/n of the real line.

New bounds on outlier eigenvalues for multiplicative perturbations S(I + P).

Abstract

We consider the eigenvalues and eigenvectors of matrices of the form M + P, where M is an n by n Wigner random matrix and P is an arbitrary n by n deterministic matrix with low rank. In general, we show that none of the eigenvalues of M + P need be real, even when P has rank one. We also show that, except for a few outlier eigenvalues, most of the eigenvalues of M + P are within 1/n of the real line, up to small order corrections. We also prove a new result quantifying the outlier eigenvalues for multiplicative perturbations of the form S ( I + P ), where S is a sample covariance matrix and I is the identity matrix. We extend our result showing all eigenvalues except the outliers are close to the real line to this case as well. As an application, we study the critical points of the characteristic polynomials of nearly Hermitian random matrices.