The distance matrix eigensystem of an equally spaced row of points Original Research Article

The distance matrix of an equally spaced row of points can be taken to be the matrix with ijth entry i−j. Denote the characteristic polynomial of this matrix by ch(n,x). Because this matrix is symmetric and centrosymmetric, it is similar to a 2 by 2 block diagonal matrix. This corresponds to a factorization of ch(n,x) into two characteristic polynomials, sym(n,x) and ant(n,x). The eigenvalues from sym have symmetric eigenvectors and those from ant have antisymmetric eigenvectors. Expansion by minors gives recursions for sym and ant. A rich system of relations between sym(n,x), ant(n,x) and a third set din(n,x) is derived. These relations allow the simple recursive calculation of the polynomials. They are used to show that the eigenvalues are simple and to determine when eigenvalues for different n can be the same. Eigenvectors built by repeating smaller eigenvectors cause eigenvalues for one n to be repeated for multiples of n. An unexpected result following from this is that the ant and din polynomials have factorizations that are parallel to the factorization of xn−1 into cyclotomic polynomials.