Gaussian Regularization of the Pseudospectrum and Davies’ Conjecture

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Abstract
A matrix is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E. B. Davies: for each , every matrix is at least ‐close to one whose eigenvectors have condition number at worst , for some depending only on n. We further show that the dependence on δ cannot be improved to for any constant .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.

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