Characteristics of Eigenvalues Realized by Path-Connected Sets of Matrices

TL;DR

This paper explores how path-connected sets of matrices influence eigenvalue multiplicities and introduces new characterizations and computational methods, especially focusing on convex hulls of matrices.

Contribution

It provides a novel analysis of eigenvalue behavior in path-connected matrix sets and offers new characterizations and computational tools for studying these phenomena.

Findings

Path-connected sets induce specific eigenvalue paths.

Equivalence relations relate to eigenvalue multiplicities.

Convex hulls exhibit unique eigenvalue properties.

Abstract

We consider path-connected sets of matrices and the induced paths between eigenvalues. We discuss the equivalence relation generated by these paths, and how it relates to the presence of higher multiplicity eigenvalues realized by the set. Particular interest is applied to the convex hull of matrices, where additional characterizations are provided of this phenomena, and computational methods are given for further study.