On Cartesian product of Euclidean distance matrices

If A is an element of R-mxn and B is an element of R-nxn, we define the product A circle divide B as A circle divide B = A circle times J(n) + J(m) circle times B, where circle times denotes the Kronecker product and J(n) is the n x n matrix of all ones. We refer to this product as the Cartesian product of A and B since if D-1 and D-2 are the distance matrices of graphs G(1) and G(2) respectively, then D-1 circle divide D-2 is the distance matrix of the Cartesian product G(1)square G(2). We study Cartesian products of Euclidean distance matrices (EDMs). We prove that if A and B are EDMs, then so is the product A circle divide B. We show that if A is an EDM and U is symmetric, then A circle times U is an EDM if and only if U = cJ(n) for some c. It is shown that for EDMs A and B, A circle divide B is spherical if and only if both A and B are spherical. If A and B are EDMs, then we derive expressions for the rank and the Moore Penrose inverse of A circle divide B. In the final section we consider the product A circle divide B for arbitrary matrices. For A is an element of R-mxm, we show that all nonzero minors of A circle divide B of order m + n - 1 are equal. An explicit formula for a nonzero minor of order m + n - 1 is proved. The result is shown to generalize the familiar fact that the determinant of the distance matrix of a tree on n vertices does not depend on the tree and is a function only of n. (C) 2018 Elsevier Inc. All rights reserved.