Eigenvalues and condition numbers of random matrices
SIAM Journal on Matrix Analysis and Applications
Matrix computations (3rd ed.)
The Expected Norm of Random Matrices
Combinatorics, Probability and Computing
Smoothed analysis of algorithms: Why the simplex algorithm usually takes polynomial time
Journal of the ACM (JACM)
Smoothed Analysis of the Condition Numbers and Growth Factors of Matrices
SIAM Journal on Matrix Analysis and Applications
The condition number of a randomly perturbed matrix
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Spectral norm of random matrices
Combinatorica
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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 investigation reveals a new and interesting fact that, unlike the gaussian case, in the general case the "core" matrix M does play a role in the tail bounds for the least singular value of M + N n . Consequently, our estimate involves the norm ||M || and it is a challenging question to determine the right magnitude of this involvement. When ||M || is relatively small, our estimate is nearly optimal and extends or refines several existing result.