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The condition number of a randomly perturbed matrix

Terence Tao·Van Vu
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Abstract

Let $M$ be an arbitrary $n$ by $n$ matrix. We study the condition number a random perturbation $M+N_n$ of $M$, where $N_n$ is a random matrix. It is shown that, under very general conditions on $M$ and $M_n$, the condition number of $M+N_n$ is polynomial in $n$ with very high probability. The main novelty here is that we allow $N_n$ to have discrete distribution.

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Terence Tao, Van Vu. 2007-03-11. The condition number of a randomly perturbed matrix. https://arxiv.org/abs/math/0703307

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