OMC for scalar multiples of unitaries

Abstract

We establish the following case of the Determinantal Conjecture of Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973) 1137-1149] and de Oliveira [G.N. de Oliveira, Research problem: Normal matrices, Linear and Multilinear Algebra 12 (1982) 153-154]. Let A and B be unitary n × n matrices with prescribed eigenvalues a1, ..., an and b1, ..., bn, respectively. Then for any scalars t and sdet (tA - sB) ∈ co fenced(underover(∏, j = 1, n) (taj - sbσ (j)) ; σ ∈ Sn),where Sn denotes the group of all permutations of {1, ..., n} and co the convex hull taken in the complex plane. © 2006 Elsevier Inc. All rights reserved.