Essentially Hermitian matrices revisited

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Stephen W. Drury


The following case of the Determinantal Conjecture of Marcus and de Oliveira is
established. Let A and C be hermitian n × n matrices with prescribed eigenvalues a1,...,an and
c1,...,cn, respectively. Let κ be a non-real unimodular complex number, B = κC, bj = κcj for
j = 1,...,n. Then

                                n                                                                                                                                    det(A − B) ∈ co{ ∏ (aj − bσ(j)); σ ∈ Sn  },                                                                                                                                 j=1

where Sn denotes the group of all permutations of {1,...,n} and co the convex hull taken in the
complex plane.

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