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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Invertibility of random matrices: Unitary and orthogonal perturbations


Authors: Mark Rudelson and Roman Vershynin
Journal: J. Amer. Math. Soc. 27 (2014), 293-338
MSC (2010): Primary 60B20
Published electronically: May 8, 2013
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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
Email: rudelson@umich.edu

Roman Vershynin
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
Email: romanv@umich.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-2013-00771-7
PII: S 0894-0347(2013)00771-7
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.