Invertibility of random matrices: Unitary and orthogonal perturbations
Published Web Locationhttps://doi.org/10.1090/s0894-0347-2013-00771-7
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.