Let $ x $ 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 $ x $ and let $ 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 $ x $ is gaussian.

A few years ago, Spielman and Teng initiated the study of Smooth analysis of the condition
number and the least singular value of a matrix. Let x be a complex variable with mean zero
and bounded variance. Let N n be the random matrix of sie n whose entries are iid copies of
x and M a deterministic matrix of the same size. The goal of smooth analysis is to estimate
the condition number and the least singular value of M+ N n. Spielman and Teng considered
the case when x is gaussian. We are going to study the general case when x is arbitrary. Our …