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Singularity of Random Integer Matrices with Large Entries

Authors Sankeerth Rao Karingula , Shachar Lovett

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Author Details

Sankeerth Rao Karingula
  • Department of Computer Science, University of California San Diego, CA, USA
Shachar Lovett
  • Department of Computer Science, University of California San Diego, CA, USA


We would like to thank Roman Vershynin and Konstantin Tikhomirov for helpful discussions.

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Sankeerth Rao Karingula and Shachar Lovett. Singularity of Random Integer Matrices with Large Entries. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 33:1-33:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


We study the singularity probability of random integer matrices. Concretely, the probability that a random n × n matrix, with integer entries chosen uniformly from {-m,…,m}, is singular. This problem has been well studied in two regimes: large n and constant m; or large m and constant n. In this paper, we extend previous techniques to handle the regime where both n,m are large. We show that the probability that such a matrix is singular is m^{-cn} for some absolute constant c > 0. We also provide some connections of our result to coding theory.

Subject Classification

ACM Subject Classification
  • Theory of computation → Randomness, geometry and discrete structures
  • Coding Theory
  • Random matrix theory
  • Singularity probability MDS codes
  • Error correction codes
  • Littlewood Offord
  • Fourier Analysis


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  1. Simeon Ball. On sets of vectors of a finite vector space in which every subset of basis size is a basis. Journal of the European Mathematical Society, 14(3):733-748, 2012. Google Scholar
  2. Jean Bourgain, Van H Vu, and Philip Matchett Wood. On the singularity probability of discrete random matrices. Journal of Functional Analysis, 258(2):559-603, 2010. Google Scholar
  3. Anthony Carbery and James Wright. Distributional and L^q norm inequalities for polynomials over convex bodies in ℝⁿ. Mathematical research letters, 8(3):233-248, 2001. Google Scholar
  4. CG Esseen. On the Kolmogorov-Rogozin inequality for the concentration function. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 5(3):210-216, 1966. Google Scholar
  5. Jeff Kahn, János Komlós, and Endre Szemerédi. On the probability that a random ± 1-matrix is singular. Journal of the American Mathematical Society, 8(1):223-240, 1995. Google Scholar
  6. Yonathan Katznelson. Singular matrices and a uniform bound for congruence groups of SL_n(ℤ). Duke Mathematical Journal, 69(1):121-136, 1993. Google Scholar
  7. János Komlós. On determinant of (0, 1) matrices. Studia Science Mathematics Hungarica, 2:7-21, 1967. Google Scholar
  8. L Leindler. On a certain converse of Hölder’s inequality. In Proceedings of the 1971 Oberwolfach Conference, BirkhHuser Verlag. Basel-Stuttgart, 1972. Google Scholar
  9. Vitali D Milman and Gideon Schechtman. Asymptotic theory of finite dimensional normed spaces: Isoperimetric inequalities in riemannian manifolds, volume 1200. Springer, 2009. Google Scholar
  10. András Prékopa. On logarithmic concave measures and functions. Acta Scientiarum Mathematicarum, 34:335-343, 1973. Google Scholar
  11. Mark Rudelson. Lecture notes on non-asymptotic theory of random matrices, 2013. Google Scholar
  12. Mark Rudelson and Roman Vershynin. The Littlewood-Offord problem and invertibility of random matrices. Advances in Mathematics, 218(2):600-633, 2008. Google Scholar
  13. Mark Rudelson and Roman Vershynin. Non-asymptotic theory of random matrices: extreme singular values. In Proceedings of the International Congress of Mathematicians 2010 (ICM 2010) (In 4 Volumes) Vol. I: Plenary Lectures and Ceremonies Vols. II-IV: Invited Lectures, pages 1576-1602. World Scientific, 2010. Google Scholar
  14. Jacob T Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM (JACM), 27(4):701-717, 1980. Google Scholar
  15. Beniamino Segre. Curve razionali normali ek-archi negli spazi finiti. Annali di Matematica Pura ed Applicata, 39(1):357-379, 1955. Google Scholar
  16. Richard Singleton. Maximum distance q-nary codes. IEEE Transactions on Information Theory, 10(2):116-118, 1964. Google Scholar
  17. Terence Tao and Van Vu. On random ± 1 matrices: singularity and determinant. Random Structures & Algorithms, 28(1):1-23, 2006. Google Scholar
  18. Terence Tao and Van Vu. On the singularity probability of random Bernoulli matrices. Journal of the American Mathematical Society, 20(3):603-628, 2007. Google Scholar
  19. Konstantin Tikhomirov. Singularity of random Bernoulli matrices. Annals of Mathematics, 191(2):593-634, 2020. Google Scholar
  20. Santosh S Vempala, Ruosong Wang, and David P Woodruff. The communication complexity of optimization. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1733-1752. SIAM, 2020. Google Scholar
  21. Richard Zippel. Probabilistic algorithms for sparse polynomials. In International symposium on symbolic and algebraic manipulation, pages 216-226. Springer, 1979. Google Scholar
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