The Binet–Cauchy theorem for the hyperdeterminant of boundary format multi-dimensional matrices

Abstract Let A , B be multi-dimensional matrices of boundary format ∏ i =0 p ( k i +1), ∏ j =0 q ( l j +1), respectively. Assume that k p = l 0 so that the convolution A∗B is defined. We prove that Det (A∗B)= Det (A) α · Det (B) β where α = l 0 !/( l 1 !… l q !), β =( k 0 +1)!/( k 1 !… k p −1 !( k p +1)!), and Det is the hyperdeterminant. When A , B are square matrices, this formula is the usual Binet–Cauchy Theorem computing the determinant of the product A · B . It follows that A∗B is nondegenerate if and only if A and  B are both nondegenerate. We show by a counterexample that the assumption of boundary format cannot be dropped.