Vol. 16, No. 3, 1966

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Permanent of the direct product of matrices

Richard Anthony Brualdi

Vol. 16 (1966), No. 3, 471–482

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 × B of A with B. Specifically we prove that

per(A ×B ) ≧ (per(A ))n(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 A × 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 Km,n exists such that

per(A × B ) ≦ Km,n(per(A))n(per(B))m

for all nonnegative matrices A and B of orders m and n respectively. A conjecture is given for the value of Km,n.

Mathematical Subject Classification
Primary: 15.20
Received: 23 October 1964
Published: 1 March 1966
Richard Anthony Brualdi