INVERTIBILITY OF RANDOM MATRICES: UNITARY AND ORTHOGONAL PERTURBATIONS
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INVERTIBILITY OF RANDOM MATRICES: UNITARY AND ORTHOGONAL PERTURBATIONS

  • Author(s): Rudelson, Mark
  • Vershynin, Roman
  • et al.
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.

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