An Inequality of Hadamard Type for Permanents
Eric Carlen, Elliott H. Lieb, Michael Loss
Abstract
Let F be an N x N complex matrix whose jth column is the vector fj in CN. Let |fj|2 denote the sum of the absolute squares of the entries of fj. Hadamard's inequality for determinants states that |(F)| <= Πj=1N|fj|. Here we prove a sharp upper bound on the permanent of F, which is |perm(F)| <= N!N-N/2 Πj=1N|fj|, and we determine all of the cases of equality.
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