Abstract

An ‘n’ x ‘n’ companion matrix is defined using ‘n’ complex numbers representing the trailing coefficients of its characteristic polynomial. In this paper, the set of unitarily irreducible complex 3 x 3 companion matrices whose numerical range has a flat portion on the boundary is explicitly described in terms of these three complex numbers. This result, combined with that of a 2008 paper by W. Calbeck, completes the set of conditions defining the shape of the numerical range of a 3 x 3 companion matrix.