Condition Numbers of Gaussian Random Matrices

Abstract. Let Gm×n be an m × n real random matrix whose elements are independent and identically distributed standard normal random variables, and let κ2(Gm×n) be the 2-norm condition number of Gm×n. We prove that, for any m ≥ 2, n ≥ 2, and x ≥ |n − m| + 1, κ2(Gm×n) satisfies 1 √ 2π (c/x)|n−m|+1 < P ( κ2(Gm×n) n/(|n−m|+1) > x) < 1 √ 2π (C/x)|n−m|+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 κ2(Gm×n)) < log n |n−m|+1 + 2.258. A similar pair of results for complex Gaussian random matrices is also established.