Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture

Abstract

A matrix (Formula presented.) is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every (Formula presented.) is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E. B. Davies: for each (Formula presented.), every matrix (Formula presented.) is at least (Formula presented.) -close to one whose eigenvectors have condition number at worst (Formula presented.), for some (Formula presented.) depending only on n. We further show that the dependence on δ cannot be improved to (Formula presented.) for any constant (Formula presented.). 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 Śniady implies a conjecture of Sankar, Spielman, and Teng on the optimal constant for smoothed analysis of condition numbers. © 2021 Wiley Periodicals, Inc.