Discrepancy Values and their Applications

Abstract

In this thesis, we introduce the concept of discrepancy values for the square matrices. We prove several properties of such objects, together with their relationship with other operators of matrices, such as singular values. After creating a toolbox of results about the discrepancy values, we proceed by introducing some ways to compute the discrepancy values efficiently. We then employ these objects to achieve tight bounds on the norms of the commutator of two matrices and several other quantities. We finish the thesis by putting forward some conjectures regarding discrepancy values and discussing their implications.