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 $${\mathbb P}\big\{s_{\max}(A)/s_{\min}(A)\leq n/t\big\}\leq 2\exp(-c t^2),\quad t\geq 1,$$ 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 $A$, ${\mathbb P}\big\{s_{n-k+1}(A)\leq ck/\sqrt{n}\big\}\leq 2 \exp(-c k^2), \quad 1\leq k\leq n,$ obtained (under some additional assumptions) by Nguyen.