Quantitative Diagonalizability

Abstract: A diagonalizable matrix has linearly independent eigenvectors. Since the set of non diagonalizable matrices has measure zero, every matrix is a limit of diagonalizable matrices. We prove a quantitative version of this fact: every n x n complex matrix is within distance delta in the operator norm of a matrix whose eigenvectors have condition number poly(n)/delta, confirming a conjecture of E. B. Davies. The proof is based adding a complex Gaussian perturbation to the matrix and studying its pseudospectrum. Joint work with J. Banks, A. Kulkarni, S. Mukherjee

operator algebras

Audience: researchers in the topic

Conference on operator algebras and related topics in Istanbul, 2021