Condition Numbers of Gaussian Random Matrices

Abstract

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 $x\ge |n-m|+1$, ${\kappa }_2(G_{m\times n})$ 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 $0.245\le c\le 2.000$ and $5.013\le C\le 6.414$ are universal positive constants independent of $m$, $n$ and $x$. Moreover, for any $m\ge 2$ and $n\ge 2$, $E(\log {\kappa }_2(G_{m\times n}))<\log \frac{n}{|n-m|+1}+\mathrm{2.258.}$ A similar pair of results for complex Gaussian random matrices is also established.