Elliptic numerical ranges of 3x3 companion matrices

A companion matrix is determined by the zeros of its characteristic polynomial. We determine the location of the zeros which yields an elliptic numerical range. In particular we show that given any two complex numbers z(1) and z(2) there exists a third complex number z(3) such that the companion matrix of the polynomial (z-z(1)) (z-z(2)) (z-z(3)) will have elliptic numerical range with foci at z(1) and z(2). (C) 2008 Elsevier Inc. All rights reserved.