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November 2021 On the real Davies’ conjecture
Vishesh Jain, Ashwin Sah, Mehtaab Sawhney
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Ann. Probab. 49(6): 3011-3031 (November 2021). DOI: 10.1214/21-AOP1522

Abstract

We show that every matrix ARn×n is, at least, δA-close to a real matrix A+ERn×n whose eigenvectors have condition number, at most, O˜n(δ1). In fact, we prove that, with high probability, taking E to be a sufficiently small multiple of an i.i.d. real sub-Gaussian matrix of bounded density suffices. This essentially confirms a speculation of Davies and of Banks, Kulkarni, Mukherjee and Srivastava, who recently proved such a result for i.i.d. complex Gaussian matrices.

Along the way we also prove nonasymptotic estimates on the minimum possible distance between any two eigenvalues of a random matrix whose entries have arbitrary means; this part of our paper may be of independent interest.

Acknowledgments

V.J. would like to thank Archit Kulkarni and Nikhil Srivastava for introducing him to the problem. The authors would like to thank an anonymous referee for their careful reading of the manuscript and valuable comments.

Citation

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Vishesh Jain. Ashwin Sah. Mehtaab Sawhney. "On the real Davies’ conjecture." Ann. Probab. 49 (6) 3011 - 3031, November 2021. https://doi.org/10.1214/21-AOP1522

Information

Received: 1 May 2020; Revised: 1 March 2021; Published: November 2021
First available in Project Euclid: 7 December 2021

Digital Object Identifier: 10.1214/21-AOP1522

Subjects:
Primary: 15A12
Secondary: 60B20

Keywords: Davies’ conjecture , eigenvalue gaps , eigenvector condition number , pseudospectrum

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.49 • No. 6 • November 2021
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