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

For $n$ points $A_i$, $i=1,2,\dots,n$, in Euclidean space $\mathbb R^m$, the distance matrix is defined as a matrix of the form $D=(D_{i,j})_{\substack{i=1,n\\j=1,n}}$, where the $D_{i,j}$ are the distances between the points $A_i$ and $A_j$ . Two configurations of points $A_i$, $i=1,2,\dots,n$, are considered. These are the configurations of points all lying on a circle or on a line and of points at the vertices of an $m$-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.