A Permanental Inequality—The Case of Equality

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Abstract
In (3) we proved that if A is a complex n-square normal matrix with characteristic roots α1 … , αn,then 1 If A is positive semi-definite hermitian, the inequality (1) becomes 2 This inequality partially answers the problem of determining the maximum permanent of a positive semi-definite hermitian matrix with prescribed characteristic roots (6). In (1), Brualdi and Newman proved that (2) also holds when A is an n-square circulant with non-negative entries. In a recent conversation Dr. Newman raised the question of determining the cases of equality in (1). In the present note we answer this question.

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