Gaussian Regularization of the Pseudospectrum and Davies' Conjecture

A matrix A is an element of Double-struck capital Cnxn is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every A is an element of Double-struck capital Cnxn is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E. B. Davies: for each delta is an element of("0,1,) every matrix A is an element of Double-struck capital Cnxn is at least delta parallel to A parallel to-close to one whose eigenvectors have condition number at worst cn/delta, for some cn depending only on n. We further show that the dependence on delta cannot be improved to 1/delta p for any constant p<1. 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. (c) 2021 Wiley Periodicals, Inc.