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

✍ Scribed by Carla Dionisi; Giorgio Ottaviani

• Book ID: 104140450
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 95 KB
• Volume: 259
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(02)00537-9

✦ Synopsis

Let A, B be multi-dimensional matrices of boundary format p i=0 (k i + 1), q j =0 (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.