Simple and exact extreme eigenvalue distributions of finite Wishart matrices

Abstract

The authors provide compact and exact expressions for the extreme eigenvalues of finite Wishart matrices with arbitrary dimensions. Using a combination of earlier results, which they refer to as the James–Edelman–Dighe framework, not only an original expression for the cumulative distribution function (CDF) of the ‘smallest’ eigenvalue is obtained, but also the CDF of the ‘largest’ eigenvalue and the probability density functions of both are expressed in a similar and convenient matrix form. These compact expressions involve only inner products of exponential vectors, vectors of monomials and certain coefficient matrices which therefore assume a key role of carrying all the required information to build the expressions. The computation of these all-important coefficient matrices involves the evaluation of a determinant of a Hankel matrix of incomplete gamma functions. They offer a theorem which proves that the latter matrix has ‘catalectic’ properties, such that the degree of its determinant is surprisingly small. The theorem also implies a closed-form and numerical procedure (no symbolic calculations required) to build the coefficient matrices.