Simple and exact extreme eigenvalue distributions of finite Wishart matrices

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.