Permanents of Direct Products1

1. Results. It is well known [2] that if A and B are n and msquare matrices respectively then (1) det(^ ® B) = (det(A))m(det(B))» where A®B 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) I det(4 B) |2 = (det(4^*))'»(det(73*P))«, 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 I. If A and B are n-square and m-square complex matrices respectively then (3) I per(4 ® B) |2 ^ (per(AA*))m(per(B*B))n. Equality holds in (3) if and only if either (a) A has a zero row or B has a zero column, or (b) A and B are both generalized permutation matrices, i.e., each of A and B is a product of a diagonal matrix and a permutation matrix. The inequality (1) should also be compared to a recent abstract [l] in which the following result is announced: (4) per(4 B) ^ (per(^))TM(per(7i))» where A and B are assumed to have non-negative entries. A lower bound of the type (4) is also available for positive semidefinite hermitian matrices. Theorem 2. If A and B are positive semi-definite hermitian n-square and m-square matrices respectively then