Permanents of direct products
Marvin D. Marcus · Proceedings of the American Mathematical Society · 1966
1. Results. It is well known [2] that if A and B are n and msquare matrices respectively then (1) det(A 0 B) = (det(A))m(det(B))n where A 0B is the tensor or direct product of A and B. By taking absolute values on both sides of (1) we can rewrite the equality as (2) | det(A 0 B) 12 = (det(AA*))m(det(B*B))n, where A* is the conjugate transpose of A. The main result is a direct extension of (2) to permanents. In general, equality will not be maintained, and the cases of equality will require a somewhat delicate analysis. THEOREM 1. If A and B are n-square and m-square complex matrices respectively then (3) | per(A 0 B) 12 (+) (i) (per(A))m(per(B))n.