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.

LaTeX, 9 pages