Smooth analysis of the condition number and the least singular value
  Let $\a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $\a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $\a$ is gaussian.   Our investigation reveals an interesting fact that the "core" matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n} $. This does not occur in Spielman-Teng studies when $\a$ is gaussian.   Consequently, our general estimate involves the norm $\|M\|$.   In the special case when $\|M\|$ is relatively small, this estimate is nearly optimal and extends or refines existing results. 
Terence Tao, Van Vu