Condition numbers of gaussian random matrices

Let G(m x) (n) be an m x n real random matrix whose elements are independent and identically distributed standard normal random variables, and let kappa(2)(G(m x n)) be the 2-norm condition number of Gm x n. We prove that, for any m >= 2, n >= 2, and x >= vertical bar n - m vertical bar + 1, kappa(2)(G(m x n)) satisfes 1/root 2 pi (c/x)(|n-m|+1) < P(kappa(2)(G(m x n))/n/(vertical bar n - m vertical bar + 1) > x) < 1/root 2 pi (C/x)(vertical bar n - m vertical bar + 1), where 0.245 <= c <= 2.000 and 5.013 <= C <= 6.414 are universal positive constants independent of m, n, and x. Moreover, for any m >= 2 and n >= 2, E( log kappa(2)(G(m x n))) < log n/|n- m| + 1 + 2.258. A similar pair of results for complex Gaussian random matrices is also established.