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An Inequality of Hadamard Type for Permanents

Eric Carlen·Elliott H. Lieb·Michael Loss
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Abstract

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.

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Eric Carlen, Elliott H. Lieb, Michael Loss. 2005-08-04. An Inequality of Hadamard Type for Permanents. https://arxiv.org/abs/math/0508096

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