Completely positive factorizations associated with Euclidean distance matrices corresponding to an arithmetic progression

TL;DR

This paper explores the spectral properties of Euclidean distance matrices from arithmetic progressions to develop completely positive factorizations, including minimal translations for positive semidefiniteness and integer factorizations.

Contribution

It introduces methods for completely positive factorizations of Euclidean distance matrices from arithmetic progressions, highlighting minimal translations for positive semidefiniteness and integer factorizations.

Findings

Minimal translation yields a positive semidefinite matrix

Complete positive factorizations over integers are discussed

Methods can be applied to similar matrices with structural properties

Abstract

Euclidean distance matrices corresponding to an arithmetic progression have rich spectral and structural properties. We exploit those properties to develop completely positive factorizations of translations of those matrices. We show that the minimal translation that makes such a matrix positive semidefinite results in a completely positive matrix. We also discuss completely positive factorizations of such matrices over the integers. Methods developed in the paper can be used to find completely positive factorizations of other matrices with similar properties.