# On Unit Spherical Euclidean Distance Matrices Which Differ in One Entry

## TL;DR

This paper characterizes the set of all unit spherical Euclidean distance matrices that differ in only one entry, revealing conditions under which this set is discrete or continuous, using two different mathematical approaches.

## Contribution

It provides a novel characterization of how a single entry change affects unit spherical EDMs, including conditions for discreteness or continuity of the set.

## Key findings

- The set can be either discrete with one or two elements or continuous.
- Characterization is achieved through two different approaches, including Cayley-Menger matrices.
- Results deepen understanding of the structure of spherical EDMs with a single entry variation.

## Abstract

A unit spherical Euclidean distance matrix (EDM) D is a matrix whose entries can be realized as the interpoint (squared) Euclidean distances of n points on a unit sphere. In this paper, given such a D and 1 \leq k < l \leq n, we present a characterization of the set of all unit spherical EDMs whose entries agree with those of D except possibly with the entry in the klth and lkth positions. As a result, we show that this set can be discrete, consisting of one or two elements, or it can be continuous. The results are derived using two alternative approaches, the second of which is based on Cayley-Menger matrices.