On Unit Spherical Euclidean Distance Matrices Which Differ in One Entry

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.