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 $m\ge 2$, $n\ge 2$ and $x\ge \mathrm{\mid }n-m\mathrm{\mid }+1$, ${\kappa }_2(G_{m\times n})$ satisfies $ \frac{1}{\sqrt{2\pi}} ({c}/{x})^{|n-m|+1} < P(\frac{\kappa_2(G_{m \times n})} {{n}/{(|n-m|+1)}}> x) < \frac{1}{\sqrt{2\pi}} ({C}/{x})^{|n-m|+1}, $where$0.245 \leq c \leq 2.000$and$ 5.013 \leq C \leq 6.414$areuniversalpositiveconstantsindependentof$m$,$n$and$x$\mathrm{.}Moreover,forany$m \geq 2$and$n \geq 2$,$ E(\log\kappa_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.