Condition Numbers of Gaussian Random Matrices

Let $G_{m times n}$ be an $m times n$ real random matrix whose elements are independent and identically distributed standard normal random variables, and let $kappa_2(G_{m times n})$ be the 2-norm condition number of $G_{m times n}$. We prove that, for any $m geq 2$, $n geq 2$ and $x geq |n-m|+1$, $kappa_2(G_{m times n})$ satisfies $ frac{1}{sqrt{2pi}} ({c}/{x})^{|n-m|+1} < P(frac{kappa_2(G_{m times n})} {{n}/{(|n-m|+1)}}> x) < frac{1}{sqrt{2pi}} ({C}/{x})^{|n-m|+1}, $ where $0.245 leq c leq 2.000$ and $ 5.013 leq C leq 6.414$ are universal positive constants independent of $m$, $n$ and $x$. Moreover, for any $m geq 2$ and $n geq 2$, $ E(logkappa_2(G_{m times n})) < log frac{n}{|n-m|+1} + 2.258. $ A similar pair of results for complex Gaussian random matrices is also established.