Spectra of Convex Hulls of Matrix Groups

TL;DR

This paper investigates the eigenvalues achievable by convex combinations of matrices within various groups, extending classical results and providing bounds and exact characterizations for specific classes of matrix groups.

Contribution

It introduces the concept of hull spectra for matrix groups, establishes their properties, and determines the spectra for key classes of groups and their representations.

Findings

Hull spectra share many algebraic properties.

Bounds on hull spectra are established.

Exact spectra are determined for important matrix groups.

Abstract

The still-unsolved problem of determining the set of eigenvalues realized by $n$-by-$n$ doubly stochastic matrices, those matrices with row sums and column sums equal to $1$, has attracted much attention in the last century. This problem is somewhat algebraic in nature, due to a result of Birkhoff demonstrating that the set of doubly stochastic matrices is the convex hull of the permutation matrices. Here we are interested in a general matrix group $G \subseteq GL_n(\mathbb{C})$ and the hull spectrum $\text{HS}(G)$ of eigenvalues realized by convex combinations of elements of $G$. We show that hull spectra of matrix groups share many nice properties. Moreover, we give bounds on the hull spectra of matrix groups, determine $\text{HS}(G)$ exactly for important classes of matrix groups, and study the hull spectra of representations of abstract groups.