Invertibility via distance for noncentered random matrices with continuous distributions

Abstract

Let A be an n×n random matrix with independent rows R1(A),…,Rn(A), and assume that for any i ≤ n and any three‐dimensional linear subspace the orthogonal projection of Ri(A) onto F has distribution density satisfying (x∈F) for some constant C1>0. We show that for any fixed n×n real matrix M we have urn:x-wiley:rsa:media:rsa20920:rsa20920-math-0004 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.