Small Ball Probability for the Condition Number of Random Matrices

  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},
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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