The distribution and moments of the smallest eigenvalue of a random matrix of wishart type

Abstract

Given a random rectangular m × n matrix with elements from a normal distribution, what is the distribution of the smallest singular value? To pose an equivalent question in the language of multivariate statistics, what is the distribution of the smallest eigenvalue of a matrix from the central Wishart distribution in the null case? We give new results giving the distribution as a simple recursion. This includes the more difficult case when n – m is an even integer, without resorting to zonal polynomials and hypergeometric functions of matrix arguments. With the recursion, one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials, exponentials, and at worst ordinary hypergeometric functions. We further elaborate on the special cases when n – m = 0, 1, 2, and 3 and give a numerical table of the expected values for 2 ⩽ m ⩽ 25 and 0 ⩽ n – m ⩽ 25.