Generalized Euclidean distance matrices

TL;DR

This paper introduces generalized Euclidean distance matrices (GDMs), extending properties of EDMs to a broader class, and demonstrates their applications in matrix construction and analysis.

Contribution

It extends key properties of Euclidean distance matrices to GDMs, including eigenvalues, inverses, and inequalities, and presents new applications in matrix construction.

Findings

Many EDM properties extend to GDMs

GDMs include non-symmetric, nonnegative matrices

Application in constructing infinitely divisible matrices

Abstract

Euclidean distance matrices (EDM) are symmetric nonnegative matrices with several interesting properties. In this article, we introduce a wider class of matrices called generalized Euclidean distance matrices (GDMs) that include EDMs. Each GDM is an entry-wise nonnegative matrix. A GDM is not symmetric unless it is an EDM. By some new techniques, we show that many significant results on Euclidean distance matrices can be extended to generalized Euclidean distance matrices. These contain results about eigenvalues, inverse, determinant, spectral radius, Moore-Penrose inverse and some majorization inequalities. We finally give an application by constructing infinitely divisible matrices using generalized Euclidean distance matrices.