Invertibility via distance for noncentered random matrices with continuous distributions

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 the orthogonal projection of R i ( A ) onto F has distribution density satisfying ( x ∈ F ) for some constant C 1 >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.