Least singular value and condition number of a square random matrix with i.i.d. rows

We consider a square random matrix made by i.i.d. rows with any distribution and prove that, for any given dimension, the probability for the least singular value to be in [0; $ε$) is at least of order $ε$. This allows us to generalize a result about the expectation of the condition number that was proved in the case of centered gaussian i.i.d. entries: such an expectation is always infinite. Moreover, we get some additional results for some well-known random matrix ensembles, in particular for the isotropic log-concave case, which is proved to have the best behaving in terms of the well conditioning.

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