Eigenvalues of symmetric matrices over integral domains

TL;DR

This paper characterizes which elements over an integral domain can be eigenvalues of symmetric matrices, providing a criterion that is both sufficient and necessary in certain cases, and explores the matrix size growth related to algebraic number fields.

Contribution

It introduces a criterion for eigenvalues of symmetric matrices over integral domains, especially for algebraic number fields, and addresses a question about matrix size growth.

Findings

The criterion is necessary and sufficient over rings of integers in algebraic number fields.

Matrix size grows linearly with the degree of the eigenvalue.

The work resolves a question posed by Bass, Estes, and Guralnick.

Abstract

Given an integral domain A we consider the set of all integral elements over A that can occur as an eigenvalue of a symmetric matrix over A. We give a sufficient criterion for being such an element. In the case where A is the ring of integers of an algebraic number field this sufficient criterion is also necessary and we show that the size of matrices grows linearly in the degree of the element. The latter result settles a questions raised by Bass, Estes and Guralnick.