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

Let A, B be multi-dimensional matrices of boundary format Pi(i=0)(p)(k(i) + 1), Pi(j=0)(q()l(j) + 1), respectively. Assume that k(p) = I-0 so that the convolution A * B is defined. We prove that Det(A * B) =Det(A)(alpha) (.) Det(B)(beta) where alpha = l(0)!/(l(1)!...l(q)!), beta = (k(0) + l)!/(k(l)!...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. (C) 2002 Elsevier Science (USA). All rights reserved.