# Small Ball Probability for the Condition Number of Random Matrices

@article{Litvak2019SmallBP, title={Small Ball Probability for the Condition Number of Random Matrices}, author={Alexander E. Litvak and Konstantin E. Tikhomirov and Nicole Tomczak-Jaegermann}, journal={Lecture Notes in Mathematics}, year={2019} }

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…

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