Introducing Discrepancy Values of Matrices with Application to Bounding Norms of Commutators

TL;DR

This paper introduces discrepancy values for matrices, inspired by spectral spread, and demonstrates their utility in deriving tight bounds on commutator norms, offering new tools for matrix analysis.

Contribution

It defines discrepancy values, explores their properties, and applies them to establish bounds on commutator norms, advancing matrix spectral analysis.

Findings

Discrepancy values share properties with eigenvalues and singular values.

Representation theorems and majorization inequalities for discrepancy values are established.

Tight bounds on commutator norms are derived using discrepancy values.

Abstract

We introduce discrepancy values, quantities inspired by the notion of the spectral spread of Hermitian matrices. We define them as the discrepancy between two consecutive Ky-Fan-like seminorms. As a result, discrepancy values share many properties with singular values and eigenvalues, yet are substantially different to merit their own study. We describe key properties of discrepancy values, and establish several tools such as representation theorems, majorization inequalities, convex formulations, etc., for working with them. As an important application, we illustrate the role of discrepancy values in deriving tight bounds on the norms of commutators.