Optimization of the Wishart Joint Eigenvalue Probability Density Distribution Based on the Vandermonde Determinant

A number of models from mathematics, physics, probability theory and statistics can be described in terms of Wishart matrices and their eigenvalues. The most prominent example being the Laguerre ensembles of the spectrum of Wishart matrix. We aim to express extreme points of the joint eigenvalue probability density distribution of a Wishart matrix using optimisation techniques for the Vandermonde determinant over certain surfaces implicitly defined by univariate polynomials.