## 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.## References

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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. - © 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

Dedicated: In memory of Joram Lindenstrauss