On the nullspace, the rangespace and the characteristic polynomial of Euclidean distance matrices

We present a characterization of the nullspace and the rangespace of a Euclidean distance matrix (EDM) D in terms of the vector of all ones, and in terms of the Gale subspace G(D) and the realization matrix P corresponding to D. This characterization is then used to compute the characteristic polynomial of D. We also present some results concerning EDMs generated by regular figures and EDMs generated by centrally symmetric points.