On the diagonal scaling of Euclidean distance matrices to doubly stochastic matrices

Abstract

We consider the problem of scaling a nondegenerate predistance matrix A to a doubly stochastic matrix B. If A is nondegenerate, then there exists a unique positive diagonal matrix C such that B = CAC. We further demonstrate that, if A is a Euclidean distance matrix, then B is a spherical Euclidean distance matrix. Finally, we investigate how scaling a nondegenerate Euclidean distance matrix A to a doubly stochastic matrix transforms the points that generate A. We find that this transformation is equivalent to an inverse stereographic projection.