On the real Davies' conjecture

TL;DR

This paper demonstrates that any real matrix can be approximated by a nearby matrix with well-conditioned eigenvectors, confirming a conjecture for a broad class of random perturbations and providing new eigenvalue separation estimates.

Contribution

It proves that every real matrix is close to one with well-conditioned eigenvectors using sub-Gaussian perturbations, extending previous results from complex Gaussian matrices.

Findings

Any real matrix can be approximated by a matrix with eigenvectors of bounded condition number.

High probability results for sub-Gaussian perturbations achieving the approximation.

New estimates on the minimal distance between eigenvalues of matrices with arbitrary mean entries.

Abstract

We show that every matrix $A \in \mathbb{R}^{n\times n}$ is at least $\delta$$\|A\|$-close to a real matrix $A+E \in \mathbb{R}^{n\times n}$ whose eigenvectors have condition number at most $\tilde{O}_{n}(\delta^{-1})$. In fact, we prove that, with high probability, taking $E$ to be a sufficiently small multiple of an i.i.d. real sub-Gaussian matrix of bounded density suffices. This essentially confirms a speculation of Davies, and of Banks, Kulkarni, Mukherjee, and Srivastava, who recently proved such a result for i.i.d. complex Gaussian matrices. Along the way, we also prove non-asymptotic estimates on the minimum possible distance between any two eigenvalues of a random matrix whose entries have arbitrary means; this part of our paper may be of independent interest.