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Spectral norm of products of random and deterministic matrices
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  • Published: 19 March 2010

Spectral norm of products of random and deterministic matrices

  • Roman Vershynin1 

Probability Theory and Related Fields volume 150, pages 471–509 (2011)Cite this article

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Abstract

We study the spectral norm of matrices W that can be factored as W = BA, where A is a random matrix with independent mean zero entries and B is a fixed matrix. Under the (4 + ε)th moment assumption on the entries of A, we show that the spectral norm of such an m × n matrix W is bounded by \({\sqrt{m} + \sqrt{n}}\), which is sharp. In other words, in regard to the spectral norm, products of random and deterministic matrices behave similarly to random matrices with independent entries. This result along with the previous work of Rudelson and the author implies that the smallest singular value of a random m × n matrix with i.i.d. mean zero entries and bounded (4 + ε)th moment is bounded below by \({\sqrt{m} - \sqrt{n-1}}\) with high probability.

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Authors and Affiliations

  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109, USA

    Roman Vershynin

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  1. Roman Vershynin
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Correspondence to Roman Vershynin.

Additional information

Partially supported by NSF grant DMS FRG 0652617, 0918623 and Alfred P. Sloan Research Fellowship.

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Vershynin, R. Spectral norm of products of random and deterministic matrices. Probab. Theory Relat. Fields 150, 471–509 (2011). https://doi.org/10.1007/s00440-010-0281-z

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  • Received: 10 March 2009

  • Revised: 16 February 2010

  • Published: 19 March 2010

  • Issue Date: August 2011

  • DOI: https://doi.org/10.1007/s00440-010-0281-z

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Mathematics Subject Classification (2000)

  • 60B20
  • 60E15
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