Two theorems on Euclidean distance matrices and Gale transform Original Research Article

We present a characterization of those Euclidean distance matrices (EDMs) D which can be expressed as D=λ(E−C) for some nonnegative scalar λ and some correlation matrix C, where E is the matrix of all ones. This shows that the cones image where image is the elliptope (set of correlation matrices) and image is the (closed convex) cone of EDMs. The characterization is given using the Gale transform of the points generating D. We also show that given points image , for any scalars λ1,λ2,…,λn such that image ∑j=1nλjpj=0, ∑j=1nλj=0,we have image ∑j=1nλjshort parallelpi−pjshort parallel2=α forall i=1,…,n,for some scalar α independent of i.