A Note on an Inequality of M. Marcus and M. Newman

Abstract

with equality if and only if A = J., the matrix all of whose entries are 1/n. This result is a partial answer to a conjecture of van der Waerden [3] stating that if S is any doubly stochastic matrix then p(S) > p(Jn) = n!/n'1 with equality if and only if S= Jn. In the present note we extend the result of Marcus and Newman to all positive semi-definite hermitian matrices which have e = (1, 1, * *, 1) as a characteristic vector and prove it by a new method. We first establish a lemma which is a weakened version of Theorem 2 in [1] and use it in conjunction with the following inequality also due to Marcus and Newman (Theorem 5 in [2]). If A and B are complex n-square matrices then