A connection between positive semidefinite and Euclidean distance matrix completion problems

The positive semidefinite and Euclidean distance matrix completion problems have received a lot of attention in the literature. Results have been obtained for these two problems that are very similar in their formulations. Although there is a strong relationship between positive semidefinite matrices and Euclidean distance matrices, it was not clear (as noted by Johnson et al.) how to link the two completion problems. The purpose of this note is twofold. First, we show how the results for the Euclidean distance matrix completion problem can be derived from the corresponding results for the positive semidefinite completion problem, using a functional transform introduced by Schoenberg. Second, we introduce a new set of necessary conditions that are stronger than some previously known ones and we identify the graphs for which these conditions suffice for ensuring completability. (C) 1998 Elsevier Science Inc.