Linear Algebra, Random Matrices and Lie Theory

Abstract

This thesis bridges the gap between pure and applied mathematics. The first part of this thesis focuses on the theory and computation of various Lie groups. The classical Lie groups as well as the automorphism groups of the bilinear and sesquilinear forms are discussed with numerical examples. In particular, we present a general approach for computing a basis of the tangent space of the automorphism group. In the second part, we derive a series of matrix factorizations from the generalized Cartan decomposition introduced by Flensted-Jensen and Hoogenboom. The generalized Cartan decomposition applied to structured matrices proves the existence of several known matrix factorizations at once and at the same time reveals a number of new matrix factorizations. Finally in the last part we derive the joint eigenvalue-like densities of the classical random matrices associated with the matrix factorizations. The Jacobian of the generalized Cartan decomposition computes the classical joint densities with various parameters using root systems. We complete the link between classical random matrices and symmetric spaces by introducing this generalized approach. Furthermore, two new families of the Jacobi ensemble parameters are obtained as a result.