The Binet-Cauchy Theorem for the Hyperdeterminant of boundary format multidimensional Matrices

Abstract

Let $A$, $B$ be multidimensional matrices of boundary format respectively $\prod_{i=0}^p(k_i+1)$, $\prod_{j=0}^q(l_j+1)$. Assume that $k_p=l_0$ so that the convolution $A\ast B$ is defined. We prove that $Det (A\ast B)=Det(A)^{\alpha}\cdot Det(B)^{\beta}$ where $\alpha= \frac {l_0!}{l_1!... l_q!}$, $\beta=\frac {(k_0+1)!}{k_1! ... k_{p-1}!(k_p+1)!}$ 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 $A\cdot B$. It follows that $A \ast B$ 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.