The Exact Distribution of the Condition Number of a Gaussian Matrix

• Authors: William Anderson;Martin T. Wells
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 2009

Suppose $G_{p\times n}$ is a real random matrix whose elements are independent and identically distributed standard normal random variables. Let $W_{p\times p}=G_{p\times n}^{}G_{n\times p}^{{\scriptscriptstyle\mathsf{T}}}$, which is the usual Wishart matrix. In addition, let $\lambda_{1}\lambda_{2}\cdots\lambda_{p}0$ and $\sigma_{1}\sigma_{2}\cdots\sigma_{p}0$ denote the distinct eigenvalues of the matrix $W_{p\times p}$ and singular values of $G_{p\times n}$, respectively. The 2-norm condition number of $G_{p\times n}$ is $\kappa_{2}(G_{p\times n})=\sqrt{\lambda_{1}/\lambda_{p}}=\sigma_{1}/\sigma_{p}$, the square root of the ratio of largest to smallest eigenvalues of the Wishart matrix. In this article we derive an exact expression, albeit somewhat complex, for the probability density function of $\kappa_{2}(G_{p\times n})$.