Condition Numbers of Gaussian Random Matrices

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(logκ₂(Gm × n)) < log n/|n-m|+1 + 2.258. A similar pair of results for complex Gaussian random matrices is also established.