Gaussian Regularization of the Pseudospectrum and Davies' Conjecture

06/27/2019

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by   Jess Banks, et al.

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A matrix A∈C^n× n is diagonalizable if it has a basis of linearly independent eigenvectors. Since the set of nondiagonalizable matrices has measure zero, every A∈C^n× n is the limit of diagonalizable matrices. We prove a quantitative version of this fact conjectured by E.B. Davies: for each δ∈ (0,1), every matrix A∈C^n× n is at least δA-close to one whose eigenvectors have condition number at worst c_n/δ, 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.