Characteristic polynomials of distance matrices of one dimensional sets

Properties of the eigenvalues of the distance matrix of a one dimensional point set are derived from identities involving the characteristic polynomial and some related polynomials called the Ant, Sym, Din and Sof polynomials of the point set. Let A⊕B denote the concatenation of the lists A and B and MS[i,j]=M[m+1-i,n+1-j] the spin of the m by n matrix M. The Ant and Sym polynomials come from a factorization of the characteristic polynomial of the distance matrix of the set -AS⊕A obtained by reflecting A about the origin. The roots of Ant are the eigenvalues with antisymmetric eigenvectors and the roots of Sym are the eigenvalues with symmetric eigenvectors. Given a square matrix M and a vector A, we say that v≠0 is an eigenvector of M relative to A iff Mv=λv+kA and A·v=0. Some basic properties of relative eigenvectors are developed. The roots of Din are the eigenvalues of the distance matrix of A relative to the vector of 1’s and the roots of Sof are the eigenvalues relative to A itself. Some simple recursions for these polynomials obtained using expansion by minors are elaborated into an extensive series of identities relating polynomials of lists to polynomials of concatenations of the lists. These identities are then used to derive a linear time algorithm for computing the polynomials and proving some results about location, distinctness and interlacing of eigenvalues.