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.

46 pages. A more general result on orthogonal perturbations of complex matrices added. It rectified an inaccuracy in application to Single Ring Theorem for orthogonal matrices

References in corpus (3)

• Local semicircle law and complete delocalization for Wigner random matrices
• Invertibility of symmetric random matrices
• Support convergence in the single ring theorem

Cited by in corpus (2)

• Limiting Spectral Distributions of Sums of Products of Non-Hermitian Random Matrices
• Smallest Singular Value for Perturbations of Random Permutation Matrices