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 ?2(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, ?2(Gm × n) satisfies "Multiple line equation(s) cannot be represented in ASCII text", 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 ?2(Gm × n)) < log n/∣n-m∣+1 +2.258. A similar pair of results for complex Gaussian random matrices is also established. [ABSTRACT FROM AUTHOR]