ELA, Volume 15, pp. 285-296, November 2006, abstract.
Essentially Hermitian Matrices Revisited
Stephen W. Drury
The following case of the Determinantal Conjecture
of Marcus and de Oliveira is established. Let A and
C be hermitian nxn matrices with prescribed eigenvalues
a_1,...,a_n and c_1,...,c_n, respectively. Let k be
a non-real unimodular complex number, B=kC,
b_j = k c_j for j=1,...,n. Then det(A-B) belongs to
\co{\prod_{j=1}^n (a_j-b_{\sigma(j)});\sigma \in S_n},
where S_n denotes the group of all permutations of
{1,...,n} and co the convex hull taken in the complex
plane.