Smallest singular value of a random rectangular matrix

Communications on Pure and Applied Mathematics

Volumn 62, Issue 12, 2009, Pages 1707-1739

  Rudelson, Mark ^a   Vershynin, Roman ^b  

^a UNIVERSITY OF MISSOURI   (United States)

^b UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 76749128899     PISSN: 00103640     EISSN: 00103640     Source Type: Journal    
DOI: 10.1002/cpa.20294     Document Type: Article

References (24)

1. N. Israel J. Math. 156 (2006), 141-155.
2. Bai, Z. D.; Yin, Y. Q. Limit of the smallest eigenvalue of a large-dimensional sample covariance matrix. Ann. Probab. 21 (1993), no. 3, 1275-1294.
3. p. 1< p<2. Israel J. Math. 26 (1977), no. 2, 178-187.
4. Davidson, K.; Szarek, S. J. Local operator theory, random matrices and Banach spaces. Handbook of the geometry of Banach spaces, vol. 1, 317-366. North-Holland, Amsterdam, 2001.
5. Edelman, A. Eigenvalues and condition numbers of random matrices. SIAM J. Matrix Anal. Appl. 9 (1988), no. 4, 543-560.
6. Edelman, A.; Sutton, B. D. Tails of condition number distributions. SIAM J. Matrix Anal. Appl. 27 (2005), no. 2, 547-560 (electronic).
7. Esseen, C. G. On the Kolmogorov-Rogozin inequality for the concentration function. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 5 (1966), 210-216.
8. Friedland, O.; Sodin, S. Bounds on the concentration function in terms of Diophantine approx- imation. C. R. Math. Acad. Sci. Paris 345 (2007), no. 9, 513-518.
9. Friedland, O., Sodin, S. Private communication.
10. Gordon, Y. Some inequalities for Gaussian processes and applications. Israel J. Math. 50 (1985), no. 4, 265-289.
11. Halász, G. Estimates for the concentration function of combinatorial number theory and probability. Period. Math. Hungar. 8 (1977), no. 3-4, 197-211.
12. Kolmogorov, A. Sur les propriétés des fonctions de concentrations de M. P. Lévy. Ann. Inst. H. Poincaré 16 (1958), 27-34.
13. Ledoux, M. Deviation inequalities on largest eigenvalues. Geometric aspects of functional analysis, 167-219. Lecture Notes in Mathematics, 1910. Springer, Berlin, 2007.
14. Lévy, P. Théorie de l'addition des variables aléatoires. Gauthier-Villars, Paris, 1937.
15. Li, W. V.; Shao, Q.-M. Gaussian processes: inequalities, small ball probabilities and applications. Stochastic processes: theory and methods, 533-597. Handbook of Statistics, 19. North-Holland, Amsterdam, 2001.
16. Litvak, A. E.; Pajor, A.; Rudelson, M.; Tomczak-Jaegermann, N. Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195 (2005), no. 2, 491-523.
17. Milman, V. D.; Schechtman, G. Asymptotic theory of finite-dimensional normed spaces. Lecture Notes in Mathematics, 1200. Springer, Berlin, 1986.
18. Rudelson, M. Lower estimates for the singular values of random matrices. C. R. Math. Acad. Sci. Paris 342 (2006), no. 4, 247-252.
19. Rudelson, M. Invertibility of random matrices: norm of the inverse. Ann. of Math. (2) 168 (2008), no. 2, 575-600.
20. Rudelson, M.; Vershynin, R. The Littlewood-Offord problem and invertibility of random ma- trices. Adv. Math. 218 (2008), no. 2, 600-633.
21. Silverstein, J. W. The smallest eigenvalue of a large-dimensional Wishart matrix. Ann. Probab. 13 (1985), no. 4, 1364-1368.
22. Szarek, S. J. Condition numbers of random matrices. J. Complexity 7 (1991), no. 2, 131-149.
23. Tao, T.; Vu, V. Additive combinatorics. Cambridge Studies in Advanced Mathematics, 105. Cambridge University Press, Cambridge, 2006.
24. Vershynin, R. Some problems in asymptotic convex geometry and random matrices motivated by numerical algorithms. Banach spaces and their applications in analysis, 209-218. de Gruyter, Berlin, 2007.