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.