Tails of Condition Number Distributions

Let κ be the condition number of an m-by-n matrix with independent standard Gaussian entries, either real (β = 1) or complex (β = 2). The major result is the existence of a constant C (depending on m, n, and β) such that P[κ > x] < C \, x-β for all x. As x → ∞, 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 β.