Distance matrices for points on a line, on a circle, and at the vertices of an \(n\)-dimensional cube (Q1916607)

For \(n\) points \(A_i\), \(i = 1, 2, \dots, n\), in the space \(\mathbb{R}^m\), the distance matrix \(D = (D_{ij})\), \(i,j = 1, \dots, n\), is defined, where the \(D_{ij}\) are distances between the points \(A_i\), \(A_j\). The author considers two 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 and 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. Finally, several inequalities are derived for solving the system of linear equations whose matrix is a given distance matrix.