An Inequality of Hadamard Type for Permanents

Abstract

Let F be an N ×N complex matrix whose jth column is the vector f j in C N. Let | f j | 2 denote the sum of the absolute squares of the entries of f j. Hadamard's inequality for determinants states that | det(F)| ≤ É N j=1 | f j |. Here we prove a sharp upper bound on the permanent of F , which is |perm(F)| ≤ N ! N N/2 N j=1 | f j |, and we determine all of the cases of equality. We also discuss the case in which | f j | is replaced by the ℓp norm of the vector f considered as a function on {1, 2,. .. , N }. We note a simple sharp inequality for p = 1, and obtain bounds for intermediate p by interpolation.