Small Ball Probability for the Condition Number of Random Matrices

Abstract

Let A be an n × n random matrix with i.i.d. entries of zero mean, unit variance and a bounded sub-Gaussian moment. We show that the condition number \(s_{\max }(A)/s_{\min }(A)\) satisfies the small ball probability estimate $$\displaystyle {\mathbb P}\big \{s_{\max }(A)/s_{\min }(A)\leq n/t\big \}\leq 2\exp (-c t^2),\quad t\geq 1, $$ where c > 0 may only depend on the sub-Gaussian 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.