Permanent of the direct product of matrices
Richard A. Brualdi · Pacific Journal of Mathematics · 1966
Let A and B be nonnegative matrices of orders m and n respectively.In this paper we derive some properties of the permanent of the direct product A x B of A with B. Specifically we prove that per (A x B) ^ (per (^)) π (per (B)) m with equality if and only if A or B has at most one nonzero term in its permanent expansion.We also show that every term in the permanent expansion of Ax B is expressible as the product of n terms in the permanent expansion of A and m terms in the permanent expansion of B, and conversely.This implies that a minimal positive number K m , n exists such that per (A x B) ^ JΓ«,»(per (A))*(per (£))• for all nonnegative matrices A and B of orders m and n respectively.A conjecture is given for the value of K m , n .Definitions* Let A = [a iS ] be a matrix of order m with entries from a field F. The permanent of A is defined by