Premium
Invertibility via distance for noncentered random matrices with continuous distributions
Author(s) -
Tikhomirov Konstantin
Publication year - 2020
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20920
Subject(s) - mathematics , combinatorics , random matrix , constant (computer programming) , matrix (chemical analysis) , distribution (mathematics) , row , subspace topology , projection (relational algebra) , gaussian , identity matrix , cover (algebra) , discrete mathematics , eigenvalues and eigenvectors , mathematical analysis , algorithm , physics , mechanical engineering , materials science , quantum mechanics , database , computer science , engineering , composite material , programming language
Let A be an n × n random matrix with independent rows R 1 ( A ),…, R n ( A ), and assume that for any i ≤ n and any three‐dimensional linear subspace F ⊂ R nthe orthogonal projection of R i ( A ) onto F has distribution density ρ ( x ) : F → R +satisfying ρ ( x ) ≤ C 1 / max ( 1 , ‖ x ‖ 2 2000 ) ( x ∈ F ) for some constant C 1 >0. We show that for any fixed n × n real matrix M we haveP { s min ( A + M ) ≤ t n − 1 / 2 } ≤ C ′t ,t > 0 , where C ′ >0 is a universal constant. In particular, the above result holds if the rows of A are independent centered log‐concave random vectors with identity covariance matrices. Our method is free from any use of covering arguments, and is principally different from a standard approach involving a decomposition of the unit sphere and coverings, as well as an approach of Sankar‐Spielman‐Teng for noncentered Gaussian matrices.