Condition numbers of Gaussian random matrices

Let G _m×n be an m _m×n real random matrix whose elements are Independent and identically distributed standard normal random variables, and let Ka(G _m×n) be the 2-norm condition number of G _m×n. We prove that, for any m ≥ 2, n ≥ 2, and x ≥|n - m| + 1, k ₂(G _m×n) satisfies 1/√(c/x) ^|n-m|+1 < P(k ₂(G _m×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(logk ₂(G _m×n)) < log n/|n-m|+1 + 2.268. A similar pair of results for complex Gaussian random matrices is also established.