Properties of Euclidean and Non-Euclidean Distance Matrices

A distance matrix D of order n is symmetric with elements idfj, where d,, = 0. D is Euclidean when the in(n 1) quantities dij can be generated as the distances between a set of n points, X (n X p), in a Euclidean space of dimension p. The dimensionality of D is defined as the least value of p = rank(X) of any generating X; in general p + 1 and p +2 are also acceptable but may include imaginary coordinates, even when D is Euclidean. Basic properties of Euclidean distance matrices are established; in particular, when p = rank(D) it is shown that, depending on whether erD-e is not or is zero, the generating points lie in either p = p 1 dimensions, in which case they lie on a hypersphere, or in p = p 2 dimensions, in which case they do not. (The notation e is used for a vector all of whose values are one.) When D is non-Euclidean its dimensionality p = r + s will comprise r real and s imaginary columns of X, and (T, s) are invariant for all generating X of minimal rank. Higher-ranking representations can arise only from p + 1= (r + 1) + s or p + 1 = r + (s + 1) or p + 2 = (r + 1) + (s + l), so that not only are r, s invariant, but they are both minimal for all admissible representations X.