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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Invertibility of random matrices: Unitary and orthogonal perturbations
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by Mark Rudelson and Roman Vershynin PDF
J. Amer. Math. Soc. 27 (2014), 293-338 Request permission

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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Additional Information
  • Mark Rudelson
  • Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
  • MR Author ID: 243851
  • Email: rudelson@umich.edu
  • Roman Vershynin
  • Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
  • MR Author ID: 636015
  • Email: romanv@umich.edu
  • Received by editor(s): June 22, 2012
  • Received by editor(s) in revised form: January 30, 2013
  • Published electronically: May 8, 2013
  • Additional Notes: The first author was partially supported by NSF grant DMS 1161372
    The second author was partially supported by NSF grant DMS 1001829.

  • Dedicated: In memory of Joram Lindenstrauss
  • © Copyright 2013 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 27 (2014), 293-338
  • MSC (2010): Primary 60B20
  • DOI: https://doi.org/10.1090/S0894-0347-2013-00771-7
  • MathSciNet review: 3164983