Cyclotomic matrices over real quadratic integer rings

TL;DR

This paper classifies cyclotomic matrices over real quadratic integer rings and related matrices with eigenvalues in [-2,2], providing a comprehensive understanding of their structure and properties.

Contribution

It provides a complete classification of cyclotomic matrices over real quadratic integer rings and enumerates related matrices with specific eigenvalue properties.

Findings

Classification of all cyclotomic matrices over real quadratic integer rings

Equivalence of classification over individual rings and their compositum

Enumeration of symmetric matrices with eigenvalues in [-2,2] but non-integer characteristic polynomials

Abstract

We classify all cyclotomic matrices over real quadratic integer rings and we show that this classification is the same as classifying cyclotomic matrices over the compositum all real quadratic integer rings. Moreover, we enumerate a related class of symmetric matrices; those matrices whose eigenvalues are contained inside the interval [-2,2] but whose characteristic polynomials are not in Z[x].