Realizing Suleimanova Spectra via Permutative Matrices

TL;DR

This paper introduces a constructive approach using permutative matrices to realize Suleimanova spectra and demonstrates that spectra with up to four elements can be realized by such matrices or their direct sums.

Contribution

It provides a constructive method for Suleimanova spectra and extends realizability results to spectra with up to four elements using permutative matrices.

Findings

Constructive realization of Suleimanova spectra via permutative matrices.

All spectra with up to four elements can be realized by permutative matrices or their direct sums.

Strengthens existing spectral realization results.

Abstract

A permutative matrix is a square matrix such that every row is a permutation of the first row. A constructive version of a result attributed to Suleimanova is given via permutative matrices. In addition, we strengthen a well-known result by showing that all realizable spectra containing at most four elements can be realized by a permutative matrix or by a direct sum of permutative matrices. We conclude by posing a problem.