Invertibility of random matrices: unitary and orthogonal perturbations

Mark Rudelson, Roman Vershynin

Published 2012-06-22, updated 2013-01-30Version 2

We show that a perturbation of any fixed square matrix D by a random unitary matrix is well invertible with high probability. A similar result holds for perturbations by random orthogonal matrices; the only notable exception is when D is close to orthogonal. As an application, these results completely eliminate a hard-to-check condition from the Single Ring Theorem by Guionnet, Krishnapur and Zeitouni.