THE EXACT DISTRIBUTION OF THE CONDITION NUMBER OF A GAUSSIAN MATRIX

Suppose G(pxn) is a real random matrix whose elements are independent and identically distributed standard normal random variables. Let W-pxp = G(pxn)G(nxp)(T), which is the usual Wishart matrix. In addition, let lambda(1) > lambda(2) > ... > lambda(p) > 0 and sigma(1) > sigma(2) > ... > sigma(p) > 0 denote the distinct eigenvalues of the matrix Wpxp and singular values of G(pxn), respectively. The 2-norm condition number of G(pxn) is kappa(2)(G(pxn)) = root 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(pxn)).