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Comptes Rendus Mathématique
Volume 346, n° 15-16
pages 893-896 (août 2008)
Doi : 10.1016/j.crma.2008.07.009
Received : 28 May 2008 ;  accepted : 10 July 2008
The least singular value of a random square matrix is  
La plus petite valeur singulière dʼune matrice carrée aléatoire est en  

Mark Rudelson a, 1 , Roman Vershynin b, 2
a Department of Mathematics, University of Missouri, Columbia, MO 65211, USA 
b Department of Mathematics, University of California, Davis, CA 95616, USA 


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   is of order   with high probability. The lower estimate of this type was proved recently by the authors; in this Note we establish the matching upper estimate. To cite this article: M. Rudelson, R. Vershynin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.

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   est de lʼordre de   avec grande probabilité. La minoration de   a été récemment obtenue par les auteurs ; dans cette Note, nous prouvons la majoration. Pour citer cet article : M. Rudelson, R. Vershynin, C. R. Acad. Sci. Paris, Ser. I 346 (2008).

The full text of this article is available in PDF format.
1  M.R. was supported by NSF DMS grant 0652684.
2  R.V. was supported by the Alfred P. Sloan Foundation and by NSF DMS grant 0652617.

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