On Cartesian product of Euclidean distance matrices

Abstract If A ∈ R m × m and B ∈ R n × n , we define the product A ⊘ B as A ⊘ B = A ⊗ J n + J m ⊗ B , where ⊗ denotes the Kronecker product and J n is the n × 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 ⊘ D 2 is the distance matrix of the Cartesian product G 1 □ 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 ⊘ B . We show that if A is an EDM and U is symmetric, then A ⊗ U is an EDM if and only if U = c J n 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 ∈ R m × m , B ∈ R n × 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.