Determinants of Sums of Normal Matrices

Abstract

Recent efforts in matrix theory have been concerned with describing invariants of matrices with "nice" properties. In this dissertation, we address a conjecture on the determinant of the sum of a pair of normal matrices. Reducing this conjecture to the problem of providing non-negative solutions for a system of linear equations without full rank, we use tools from representation theory and combinatorics to describe the modifications required to provide such solutions, and we suggest a statistical approach to the problem more broadly.