The least singular value of a random symmetric matrix

Marcelo Campos; Matthew Jenssen; Marcus Michelen; Julian Sahasrabudhe

doi:10.1017/fmp.2023.29

The least singular value of a random symmetric matrix

Marcelo Campos
, Matthew Jenssen
, Marcus Michelen
, Julian Sahasrabudhe^*

^*Corresponding author for this work

Department of Pure Mathematics and Mathematical Statistics
University of Illinois at Chicago

Research output: Contribution to journal › Article › peer-review

5 Citations (Scopus)

68 Downloads (Pure)

Abstract

Let A be an symmetric matrix with independent and identically distributed according to a subgaussian distribution. We show that where denotes the least singular value of A and the constants 0 $ ]]> depend only on the distribution of the entries of A. This result confirms the folklore conjecture on the lower tail of the least singular value of such matrices and is best possible up to the dependence of the constants on the distribution of. Along the way, we prove that the probability that A has a repeated eigenvalue is, thus confirming a conjecture of Nguyen, Tao and Vu [Probab. Theory Relat. Fields 167 (2017), 777-816].

Original language	English
Article number	e3
Number of pages	69
Journal	Forum of Mathematics, Pi
Volume	12
DOIs	https://doi.org/10.1017/fmp.2023.29
Publication status	Published - 23 Jan 2024

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10.1017/fmp.2023.29Licence: CC BY

The least singular value of a random symmetric matrix_Version of RecordFinal published version, 672 KBLicence: CC BY

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abstract = "Let A be an symmetric matrix with independent and identically distributed according to a subgaussian distribution. We show that where denotes the least singular value of A and the constants 0 \$ ]]> depend only on the distribution of the entries of A. This result confirms the folklore conjecture on the lower tail of the least singular value of such matrices and is best possible up to the dependence of the constants on the distribution of. Along the way, we prove that the probability that A has a repeated eigenvalue is, thus confirming a conjecture of Nguyen, Tao and Vu [Probab. Theory Relat. Fields 167 (2017), 777-816].",

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T1 - The least singular value of a random symmetric matrix

AU - Campos, Marcelo

AU - Jenssen, Matthew

AU - Michelen, Marcus

AU - Sahasrabudhe, Julian

PY - 2024/1/23

Y1 - 2024/1/23

N2 - Let A be an symmetric matrix with independent and identically distributed according to a subgaussian distribution. We show that where denotes the least singular value of A and the constants 0 $ ]]> depend only on the distribution of the entries of A. This result confirms the folklore conjecture on the lower tail of the least singular value of such matrices and is best possible up to the dependence of the constants on the distribution of. Along the way, we prove that the probability that A has a repeated eigenvalue is, thus confirming a conjecture of Nguyen, Tao and Vu [Probab. Theory Relat. Fields 167 (2017), 777-816].

AB - Let A be an symmetric matrix with independent and identically distributed according to a subgaussian distribution. We show that where denotes the least singular value of A and the constants 0 $ ]]> depend only on the distribution of the entries of A. This result confirms the folklore conjecture on the lower tail of the least singular value of such matrices and is best possible up to the dependence of the constants on the distribution of. Along the way, we prove that the probability that A has a repeated eigenvalue is, thus confirming a conjecture of Nguyen, Tao and Vu [Probab. Theory Relat. Fields 167 (2017), 777-816].

UR - https://www.scopus.com/pages/publications/85184176010

U2 - 10.1017/fmp.2023.29

DO - 10.1017/fmp.2023.29

M3 - Article

AN - SCOPUS:85184176010

SN - 2050-5086

VL - 12

JO - Forum of Mathematics, Pi

JF - Forum of Mathematics, Pi

M1 - e3

ER -

The least singular value of a random symmetric matrix

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