Invertibility of random matrices: Unitary and orthogonal perturbations

Abstract

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.