Gaussian Regularization of the Pseudospectrum and Davies' Conjecture

TL;DR

This paper proves a quantitative version of Davies' conjecture, showing that any matrix can be approximated by a diagonalizable matrix with controlled eigenvector condition number using Gaussian regularization.

Contribution

It provides a new probabilistic approach to regularize the pseudospectrum of matrices and confirms a conjecture on smoothed condition number analysis.

Findings

Matrices can be approximated by well-conditioned diagonalizable matrices.

Gaussian perturbations effectively regularize the pseudospectrum.

Confirmed a conjecture on optimal smoothed analysis constants.

Abstract

A matrix $A\in\mathbb{C}^{n\times n}$ is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every $A\in \mathbb{C}^{n\times n}$ is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E.B. Davies: for each $\delta\in (0,1)$, every matrix $A\in \mathbb{C}^{n\times n}$ is at least $\delta\|A\|$-close to one whose eigenvectors have condition number at worst $c_n/\delta$, for some constants $c_n$ dependent only on $n$. Our proof uses tools from random matrix theory to show that the pseudospectrum of $A$ can be regularized with the addition of a complex Gaussian perturbation. Along the way, we explain how a variant of a theorem of \'Sniady implies a conjecture of Sankar, Spielman and Teng on the optimal constant for smoothed analysis of condition numbers.