A simple observation on random matrices with continuous diagonal entries

Omer Friedland (Université Pierre et Marie Curie)
Ohad Giladi (University of Alberta)


Let $T$ be an $n\times n$ random matrix, such that each diagonal entry $T_{i, i}$ is a continuous random variable, independent from all the other entries of $T$. Then for every $n\times n$ matrix $A$ and every $t\ge0$$$\mathbb{P}\Big[|\det(A+T)|^{1/n}\le t\Big]\le2bnt, $$where $b>0$ is a uniform upper bound on the densities of $T_{i, i}$.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-7

Publication Date: June 30, 2013

DOI: 10.1214/ECP.v18-2633


  • Adamczak, Radosław; Guédon, Olivier; Litvak, Alexander; Pajor, Alain; Tomczak-Jaegermann, Nicole. Smallest singular value of random matrices with independent columns. C. R. Math. Acad. Sci. Paris 346 (2008), no. 15-16, 853--856. MR2441920
  • Adamczak, Radosław; Guédon, Olivier; Litvak, Alexander E.; Pajor, Alain; Tomczak-Jaegermann, Nicole. Condition number of a square matrix with i.i.d. columns drawn from a convex body. Proc. Amer. Math. Soc. 140 (2012), no. 3, 987--998. MR2869083
  • Bourgain, Jean; Vu, Van H.; Wood, Philip Matchett. On the singularity probability of discrete random matrices. J. Funct. Anal. 258 (2010), no. 2, 559--603. MR2557947
  • Costello, Kevin P.; Vu, Van. Concentration of random determinants and permanent estimators. SIAM J. Discrete Math. 23 (2009), no. 3, 1356--1371. MR2556534
  • Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer. Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. IMRN 2010, no. 3, 436--479. MR2587574
  • Farrell Brendan; Vershynin, Roman. Smoothed analysis of symmetric random matrices with continuous distributions. Preprint available at http://arxiv.org/abs/1212.3531
  • Maltsev, Anna; Schlein, Benjamin. A Wegner estimate for Wigner matrices. Entropy and the quantum II, 145--160, Contemp. Math., 552, Amer. Math. Soc., Providence, RI, 2011. MR2868046
  • Nguyen, Hoi H. Inverse Littlewood-Offord problems and the singularity of random symmetric matrices. Duke Math. J. 161 (2012), no. 4, 545--586. MR2891529
  • Nguyen, Hoi H. On the least singular value of random symmetric matrices. Electron. J. Probab. 17 (2012), no. 53, 19 pp. MR2955045
  • Nguyen, H.; Vu, V. Small ball probability, inverse theorems, and applications. Preprint available at http://arxiv.org/abs/1301.0019
  • Rudelson, Mark. Invertibility of random matrices: norm of the inverse. Ann. of Math. (2) 168 (2008), no. 2, 575--600. MR2434885
  • Rudelson, Mark; Vershynin, Roman. The Littlewood-Offord problem and invertibility of random matrices. Adv. Math. 218 (2008), no. 2, 600--633. MR2407948
  • Sankar, Arvind; Spielman, Daniel A.; Teng, Shang-Hua. Smoothed analysis of the condition numbers and growth factors of matrices. SIAM J. Matrix Anal. Appl. 28 (2006), no. 2, 446--476 (electronic). MR2255338
  • Tao, Terence; Vu, Van. On random $\pm1$ matrices: singularity and determinant. Random Structures Algorithms 28 (2006), no. 1, 1--23. MR2187480
  • Tao, Terence; Vu, Van H. Inverse Littlewood-Offord theorems and the condition number of random discrete matrices. Ann. of Math. (2) 169 (2009), no. 2, 595--632. MR2480613
  • Tao, Terence; Vu, Van. On the permanent of random Bernoulli matrices. Adv. Math. 220 (2009), no. 3, 657--669. MR2483225
  • Tao, Terence; Vu, Van. Random matrices: the distribution of the smallest singular values. Geom. Funct. Anal. 20 (2010), no. 1, 260--297. MR2647142
  • Tao, Terence; Vu, Van. Smooth analysis of the condition number and the least singular value. Math. Comp. 79 (2010), no. 272, 2333--2352. MR2684367
  • Vershynin, Roman. Invertibility of symmetric random matrices. To appear in Random Structures and Algorithms. Preprint available at http://arxiv.org/abs/1102.0300, 2012.

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.