Solution of the embedding problem and decomposition of symmetric matrices.

A solution of the problem of calculating cartesian coordinates from a matrix of interpoint distances (the embedding problem) is reported. An efficient and numerically stable algorithm for the transformation of distances to coordinates is then obtained. It is shown that the embedding problem is intimately related to the theory of symmetric matrices, since every symmetric matrix is related to a general distance matrix by a one-to-one transformation. Embedding of a distance matrix yields a decomposition of the associated symmetric matrix in the form of a sum over outer products of a linear independent system of coordinate vectors. It is shown that such a decomposition exists for every symmetric matrix and that it is numerically stable. From this decomposition, the rank and the numbers of positive, negative, and zero eigenvalues of the symmetric matrix are obtained directly.