Distance matrices for points on a line, on a circle, and at the vertices of ann-dimensional cube

Abstract

Forn pointsA _i ,i=1, 2, ...,n, in Euclidean space ℝ^m, the distance matrix is defined as a matrix of the form D=(D _i ,j) _i ,j=1,...,n, where theD _i ,j are the distances between the pointsA _i andA _j . Two configurations of pointsA _i ,i=1, 2,...,n, are considered. These are the configurations of points all lying on a circle or on a line and of points at the vertices of anm-dimensional cube. In the first case, the inverse matrix is obtained in explicit form. In the second case, it is shown that the complete set of eigenvectors is composed of the columns of the Hadamard matrix of appropriate order. Using the fact that distance matrices in Euclidean space are nondegenerate, several inequalities are derived for solving the system of linear equations whose matrix is a given distance matrix.