On Cartesian product of Euclidean distance matrices

Abstract

If A∈Rm×m and B∈Rn×n, we define the product A⊘B as A⊘B=A⊗Jn+Jm⊗B, where ⊗ denotes the Kronecker product and Jn is the n×n matrix of all ones. We refer to this product as the Cartesian product of A and B since if D1 and D2 are the distance matrices of graphs G1 and G2 respectively, then D1⊘D2 is the distance matrix of the Cartesian product G1□G2. We study Cartesian products of Euclidean distance matrices (EDMs). We prove that if A and B are EDMs, then so is the product A⊘B. We show that if A is an EDM and U is symmetric, then A⊗U is an EDM if and only if U=cJn for some c. It is shown that for EDMs A and B, A⊘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⊘B. In the final section we consider the product A⊘B for arbitrary matrices. For A∈Rm×m,B∈Rn×n, we show that all nonzero minors of A⊘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.