Tails of condition number distributions

Alan Edelman and Brian D. Sutton

SIAM J. Matrix Anal. Appl. 27 (2005), no. 2, 547–560.

Let kappa be the condition number of an m-by-n matrix with independent standard Gaussian entries, either real (beta = 1) or complex (beta = 2). The major result is the existence of a constant C (depending on m, n, beta) such that P [kappa > x] < C x^(-beta) for all x. As x -> infinity, the bound is asymptotically tight. An analytic expression is given for the constant C, and simple estimates are given, one involving a Tracy-Widom largest eigenvalue distribution. All of the results extend beyond real and complex entries to general beta.