## 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.

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