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Let M be an arbitrary n by n matrix. We study the conditionnumber a random perturbation M+Nn of M, where Nn is arandom matrix. It is shown that, under very general conditions on M and Mn, the condition number of M+Nn is polynomial in nwith very high probability. The main novelty here is that we allow Nn to have discrete distribution.
Vu et al. (Mon,) studied this question.