Small ball probability for the condition number of random matrices

Abstract

Let $A$ be an $n\times n$ random matrix with i.i.d. entries of zero mean, unit variance and a bounded subgaussian moment. We show that the condition number $s_{\max }(A)/s_{\min }(A)$ satisfies the small ball probability estimate $$\mathbb{P}\{s_{\max }(A)/s_{\min }(A)\le n/t\}\le 2\exp (-c{t}^2),t\ge 1,$$ where $c>0$ may only depend on the subgaussian moment. Although the estimate can be obtained as a combination of known results and techniques, it was not noticed in the literature before. As a key step of the proof, we apply estimates for the singular values of $A$, $\mathbb{P}\{s_{n-k+1}(A)\le ck/\sqrt{n}\}\le 2\exp (-c{k}^2),1\le k\le n,$ obtained (under some additional assumptions) by Nguyen.

Some changes according to the Referee's comments