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- W4288623674 abstract "Let $A$ be an $ntimes n$ 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 $${mathbb P}big{s_{max}(A)/s_{min}(A)leq n/tbig}leq 2exp(-c t^2),quad tgeq 1,$$ 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 $A$, ${mathbb P}big{s_{n-k+1}(A)leq ck/sqrt{n}big}leq 2 exp(-c k^2), quad 1leq kleq n,$ obtained (under some additional assumptions) by Nguyen." @default.
- W4288623674 created "2022-07-30" @default.
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- W4288623674 date "2019-01-24" @default.
- W4288623674 modified "2023-09-26" @default.
- W4288623674 title "Small ball probability for the condition number of random matrices" @default.
- W4288623674 doi "https://doi.org/10.48550/arxiv.1901.08655" @default.
- W4288623674 hasPublicationYear "2019" @default.
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