Rank deficiency of random matrices

Vishesh Jain; Ashwin Sah; Mehtaab Sawhney

doi:10.1214/22-ECP455

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Home > Journals > Electron. Commun. Probab. > Volume 27 > Article

2022 Rank deficiency of random matrices

Vishesh Jain, Ashwin Sah, Mehtaab Sawhney

Author Affiliations +

Vishesh Jain,¹ Ashwin Sah,² Mehtaab Sawhney²
¹Department of Statistics, Stanford University, Stanford CA 94305, USA
²Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Electron. Commun. Probab. 27: 1-9 (2022). DOI: 10.1214/22-ECP455

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Abstract

Let ${M_{n}}$ be a random $n\times n$ matrix with i.i.d. $\operatorname{Bernoulli}(1/ 2)$ entries. We show that for fixed $k\ge 1$ ,

$\underset{n\to \mathrm{\infty }}{\lim }\frac{1}{n}{\log _{2}}\mathbb{P}[\operatorname{corank}{M_{n}}\ge k]=-k.$

Acknowledgments

We thank Jordan Ellenberg for a Twitter question which motivated the present work. We also thank Han Huang for sharing a preliminary version of [6] with us. Finally we are grateful to the anonymous referees for their careful reading of our paper and for numerous comments which have improved the presentation.

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Vishesh Jain. Ashwin Sah. Mehtaab Sawhney. "Rank deficiency of random matrices." Electron. Commun. Probab. 27 1 - 9, 2022. https://doi.org/10.1214/22-ECP455

Information

Received: 12 March 2021; Accepted: 12 February 2022; Published: 2022

First available in Project Euclid: 23 February 2022

arXiv: 2103.02467

MathSciNet: MR4368695

zbMATH: 1484.15040

Digital Object Identifier: 10.1214/22-ECP455

Subjects:

Primary: 60B20

Keywords: corank , invertibility , random matrices

Rights: Creative Commons Attribution 4.0 International License.

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Electron. Commun. Probab.

Vol.27 • 2022

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Vishesh Jain, Ashwin Sah, Mehtaab Sawhney "Rank deficiency of random matrices," Electronic Communications in Probability, Electron. Commun. Probab. 27(none), 1-9, (2022)

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