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Small Ball Probability for the Condition Number of Random Matrices

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Geometric Aspects of Functional Analysis

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2266))

Abstract

Let A be an n × n random matrix with i.i.d. entries of zero mean, unit variance and a bounded sub-Gaussian moment. We show that the condition number \(s_{\max }(A)/s_{\min }(A)\) satisfies the small ball probability estimate

$$\displaystyle {\mathbb P}\big \{s_{\max }(A)/s_{\min }(A)\leq n/t\big \}\leq 2\exp (-c t^2),\quad t\geq 1, $$

where c > 0 may only depend on the sub-Gaussian moment. Although the estimate can be obtained as a combination of known results and techniques, it was not noticed in the literature before. As a key step of the proof, we apply estimates for the singular values of A, \({\mathbb P}\big \{s_{n-k+1}(A)\leq ck/\sqrt {n}\big \}\leq 2 \exp (-c k^2), \quad 1\leq k\leq n,\) obtained (under some additional assumptions) by Nguyen.

AMS 2010 Classification 60B20, 15B52, 46B06, 15A18

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Acknowledgements

The authors are grateful to the anonymous referee for careful reading and valuable suggestions that have helped to improve the presentation. The second named author would like to thank the Department of Mathematical and Statistical Sciences, University of Alberta for ideal working conditions.

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Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, Canada

    Alexander E. Litvak & Nicole Tomczak-Jaegermann

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, USA

    Konstantin Tikhomirov

Authors
  1. Alexander E. Litvak
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  2. Konstantin Tikhomirov
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  3. Nicole Tomczak-Jaegermann
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Editors and Affiliations

  1. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel; Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel

    Bo'az Klartag

  2. Department of Mathematics, Technion – Israel Institute of Technology, Haifa, Israel

    Emanuel Milman

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Litvak, A.E., Tikhomirov, K., Tomczak-Jaegermann, N. (2020). Small Ball Probability for the Condition Number of Random Matrices. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2266. Springer, Cham. https://doi.org/10.1007/978-3-030-46762-3_5

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