Applied Modeling Techniques and Data Analysis 2

14Properties of the Extreme Points of the Joint Eigenvalue Probability Density Function of the Wishart Matrix

We will examine some properties of the extreme points of the probability density distribution of the Wishart matrix, using properties of the Vandermonde determinant and showing examples of the applications of these properties.

14.1. Introduction

The Gaussian ensembles of random matrices have been extensively investigated and date back to the works on the statistical distribution of widths and spacings of nuclear resonance levels (Wigner 1951) and the statistical theory of the energy levels of complex systems (Dyson 1962). Thus, random matrix theory (as thoroughly discussed in Mehta (1967), König (2005) and Forester (2010)) has proved to be pivotal in high dimensional and/or multivariate statistical analysis, plus many other applications based on the Wishart matrix (Anderson 2003), as well as orthogonal polynomials (Szego 1939). Therefore, we attempt to investigate the properties of the extreme points of the joint eigenvalue probability density function of the random Wishart matrix optimized over the unit p-sphere (Muhumuza et al. 2018b). We also apply the techniques of Vandermonde polynomial optimizations (Lundengård and Silvestrov 2013), matrix factorization (Oruç and Phillips 2000) and eigenvalue optimization (Golub and Van Loan 1996), matrix norms (Demmel 1997) and condition number (Edelman 1988).

Investigating the characteristic properties of the extreme or optimal ...