Probability Theory
The least singular value of a random square matrix is O(n−1/2)
[La plus petite valeur singulière d'une matrice carrée aléatoire est en O(n−1/2)]

Présenté par : Gilles Pisier

Rudelson, Mark1 ; Vershynin, Roman2
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, Tome 346 (2008) no. 15-16, pp. 893-896.

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 sn(A) est de l'ordre de n−1/2 avec grande probabilité. La minoration de sn(A) a été récemment obtenue par les auteurs ; dans cette Note, nous prouvons la majoration.

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 sn(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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.07.009
Rudelson, Mark 1 ; Vershynin, Roman 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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Rudelson, Mark; Vershynin, Roman. The least singular value of a random square matrix is $ \mathrm{O}({n}^{-1/2})$. Comptes Rendus. Mathématique, Tome 346 (2008) no. 15-16, pp. 893-896. doi : 10.1016/j.crma.2008.07.009. https://www.numdam.org/articles/10.1016/j.crma.2008.07.009/

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

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

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