Essentially Hermitian matrices revisited

Abstract

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,...,a_n and
c₁,...,c_n, respectively. Let κ be a non-real unimodular complex number, B = κC, b_j = κcj for
j = 1,...,n. Then

                                n                                                                                                                                    det(A − B) ∈ co｛ ∏ (a_j − b_σ(j)); σ ∈ S_n  ｝,                                                                                                                                 j=1

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