Bohemian Matrix Geometry

TL;DR

This paper explores the properties of Bohemian matrices, focusing on their eigenvalue distributions, through computational methods to answer open questions about their spectral characteristics.

Contribution

It introduces computational analysis of specific Bohemian matrix families, addressing open questions about eigenvalue density distributions.

Findings

Eigenvalue distributions of certain Bohemian matrices are characterized.

Computational methods effectively answer open questions about spectral properties.

Symmetric, Hessenberg, and Toeplitz Bohemian matrices exhibit distinct eigenvalue behaviors.

Abstract

A Bohemian matrix family is a set of matrices all of whose entries are drawn from a fixed, usually discrete and hence bounded, subset of a field of characteristic zero. Originally these were integers -- hence the name, from the acronym BOunded HEight Matrix of Integers (BOHEMI) -- but other kinds of entries are also interesting. Some kinds of questions about Bohemian matrices can be answered by numerical computation, but sometimes exact computation is better. In this paper we explore some Bohemian families (symmetric, upper Hessenberg, or Toeplitz) computationally, and answer some open questions posed about the distributions of eigenvalue densities.