The least singular value of a random square matrix is $ \mathrm{O}({n}^{-1/2})$

Mark Rudelson; Roman Vershynin

doi:10.1016/j.crma.2008.07.009

Probability Theory

The least singular value of a random square matrix is

O (n^{- 1 / 2})

Presented by: Gilles Pisier

Mark Rudelson ¹; Roman Vershynin ²

¹ Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
² Department of Mathematics, University of California, Davis, CA 95616, USA

Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 893-896.

Abstract
Résumé

Let A be a matrix whose entries are real i.i.d. centered random variables with unit variance and suitable moment assumptions. Then the smallest singular value $s_{n} (A)$ is of order $n^{- 1 / 2}$ with high probability. The lower estimate of this type was proved recently by the authors; in this Note we establish the matching upper estimate.

Soit A une matrice dont les entrées sont des variables aléatoires centrées réelles i.i.d. de variance 1 vérifiant une hypothèse adéquate de moment. Alors la plus petite valeur singulière $s_{n} (A)$ est de l'ordre de $n^{- 1 / 2}$ avec grande probabilité. La minoration de $s_{n} (A)$ a été récemment obtenue par les auteurs ; dans cette Note, nous prouvons la majoration.

Received: 2008-05-28
Accepted: 2008-07-10
Published online: 2008-08-01

DOI: 10.1016/j.crma.2008.07.009

Author's affiliations:

Mark Rudelson ¹; Roman Vershynin ²

¹ Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
² Department of Mathematics, University of California, Davis, CA 95616, USA

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Mark Rudelson; Roman Vershynin. The least singular value of a random square matrix is $ \mathrm{O}({n}^{-1/2})$. Comptes Rendus. Mathématique, Volume 346 (2008) no. 15-16, pp. 893-896. doi : 10.1016/j.crma.2008.07.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.07.009/

References
Cited by

[1] A. Edelman Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl., Volume 9 (1988), pp. 543-560

[2] M. Rudelson; R. Vershynin The Littlewood–Offord problem and invertibility of random matrices, Adv. Math., Volume 218 (2008), pp. 600-633

[3] M. Rudelson, R. Vershynin, The smallest singular value of a random rectangular matrix, submitted for publication

[4] J. von Neumann Collected Works, vol. V: Design of Computers, Theory of Automata and Numerical Analysis (A.H. Taub, ed.), A Pergamon Press Book The Macmillan Co., New York, 1963

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