Let F be an N x 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)| <= \prod_{ j=1}^N|f_j|. Here we prove a sharp upper bound on the permanent of F, which is |perm(F)| <= N!N^{ -N/2} \prod_{ j=1}^N|f_j|, and we determine all of the cases of equality.
Categories: math.CA math.FA math.OA