On the singular values of random matrices

We present an approach that allows one to bound the largest and smallest singular values of an N × n random matrix with iid rows, distributed according to a measure on Rn that is supported in a relatively small ball and linear functionals are uniformly bounded in Lp for some p > 8, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1 ± c √ n/N not only in the case of the above mentioned measure, but also when the measure is log-concave or when it a product measure of iid random variables with “heavy tails”.

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