Comptes Rendus
Probability Theory
The least singular value of a random square matrix is O(n1/2)
[La plus petite valeur singulière d'une matrice carrée aléatoire est en O(n1/2)]
Comptes Rendus. Mathématique, Volume 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 n1/2 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 n1/2 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

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
@article{CRMATH_2008__346_15-16_893_0,
     author = {Mark Rudelson and Roman Vershynin},
     title = {The least singular value of a random square matrix is $ \mathrm{O}({n}^{-1/2})$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {893--896},
     publisher = {Elsevier},
     volume = {346},
     number = {15-16},
     year = {2008},
     doi = {10.1016/j.crma.2008.07.009},
     language = {en},
}
TY  - JOUR
AU  - Mark Rudelson
AU  - Roman Vershynin
TI  - The least singular value of a random square matrix is $ \mathrm{O}({n}^{-1/2})$
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 893
EP  - 896
VL  - 346
IS  - 15-16
PB  - Elsevier
DO  - 10.1016/j.crma.2008.07.009
LA  - en
ID  - CRMATH_2008__346_15-16_893_0
ER  - 
%0 Journal Article
%A Mark Rudelson
%A Roman Vershynin
%T The least singular value of a random square matrix is $ \mathrm{O}({n}^{-1/2})$
%J Comptes Rendus. Mathématique
%D 2008
%P 893-896
%V 346
%N 15-16
%I Elsevier
%R 10.1016/j.crma.2008.07.009
%G en
%F CRMATH_2008__346_15-16_893_0
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/

[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

  • Martin Vogel Pseudospectra and eigenvalue asymptotics for disordered non-selfadjoint operators in the semiclassical limit, Frontiers in Applied Mathematics and Statistics, Volume 10 (2024) | DOI:10.3389/fams.2024.1508973
  • Peiliang Xu; Yun Shi Unidentifiability of errors-in-variables models with rank deficiency from measurements, Measurement, Volume 192 (2022), p. 110853 | DOI:10.1016/j.measurement.2022.110853
  • Alexander E. Litvak; Konstantin Tikhomirov; Nicole Tomczak-Jaegermann Small Ball Probability for the Condition Number of Random Matrices, Geometric Aspects of Functional Analysis, Volume 2266 (2020), p. 125 | DOI:10.1007/978-3-030-46762-3_5
  • Anna Lytova; Konstantin Tikhomirov On delocalization of eigenvectors of random non-Hermitian matrices, Probability Theory and Related Fields, Volume 177 (2020) no. 1-2, p. 465 | DOI:10.1007/s00440-019-00956-8
  • Phil Kopel; Sean O’Rourke; Van Vu Random matrix products: Universality and least singular values, The Annals of Probability, Volume 48 (2020) no. 3 | DOI:10.1214/19-aop1396
  • Igor Tsukerman; Shampy Mansha; Y.D. Chong; Vadim A. Markel Trefftz approximations in complex media: Accuracy and applications, Computers Mathematics with Applications, Volume 77 (2019) no. 6, p. 1770 | DOI:10.1016/j.camwa.2018.08.065
  • Pierre Baldi; Roman Vershynin Polynomial Threshold Functions, Hyperplane Arrangements, and Random Tensors, SIAM Journal on Mathematics of Data Science, Volume 1 (2019) no. 4, p. 699 | DOI:10.1137/19m1257792
  • Nicholas Cook Lower bounds for the smallest singular value of structured random matrices, The Annals of Probability, Volume 46 (2018) no. 6 | DOI:10.1214/17-aop1251
  • Anirban Basak; Mark Rudelson Invertibility of sparse non-Hermitian matrices, Advances in Mathematics, Volume 310 (2017), p. 426 | DOI:10.1016/j.aim.2017.02.009
  • Feng Wei Upper bound for intermediate singular values of random matrices, Journal of Mathematical Analysis and Applications, Volume 445 (2017) no. 2, p. 1530 | DOI:10.1016/j.jmaa.2016.08.007
  • Mark Rudelson; Roman Vershynin No-gaps delocalization for general random matrices, Geometric and Functional Analysis, Volume 26 (2016) no. 6, p. 1716 | DOI:10.1007/s00039-016-0389-0
  • Konstantin Tikhomirov The limit of the smallest singular value of random matrices with i.i.d. entries, Advances in Mathematics, Volume 284 (2015), p. 1 | DOI:10.1016/j.aim.2015.07.020
  • Robert Qiu; Michael Wicks Non-asymptotic, Local Theory of Random Matrices, Cognitive Networked Sensing and Big Data (2014), p. 271 | DOI:10.1007/978-1-4614-4544-9_5
  • Carsten Schuett; Stiene Riemer On the expectation of the norm of random matrices with non-identically distributed entries, Electronic Journal of Probability, Volume 18 (2013) no. none | DOI:10.1214/ejp.v18-2103
  • Olivier Bordellès Prime Numbers, Arithmetic Tales (2012), p. 57 | DOI:10.1007/978-1-4471-4096-2_3
  • Roman Vershynin Spectral norm of products of random and deterministic matrices, Probability Theory and Related Fields, Volume 150 (2011) no. 3-4, p. 471 | DOI:10.1007/s00440-010-0281-z
  • Terence Tao; Van Vu Random Matrices: the Distribution of the Smallest Singular Values, Geometric and Functional Analysis, Volume 20 (2010) no. 1, p. 260 | DOI:10.1007/s00039-010-0057-8
  • Terence Tao; Van Vu; Manjunath Krishnapur Random matrices: Universality of ESDs and the circular law, The Annals of Probability, Volume 38 (2010) no. 5 | DOI:10.1214/10-aop534
  • Roman Vershynin, 2009 3rd IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP) (2009), p. 189 | DOI:10.1109/camsap.2009.5413304

Cité par 19 documents. Sources : Crossref

Commentaires - Politique