Numerical ranges of companion matrices: flat portions on the boundary
Jeffrey Eldred, Leiba Rodman, Ilya M. Spitkovsky · Linear and Multilinear Algebra · 2011
Criterion for a companion matrix to have a certain number of flat portions on the boundary of its numerical range is given. The criterion is specialized to the cases of 3 × 3 and 4 × 4 matrices. In the latter case, it is proved that a 4 × 4 unitarily irreducible companion matrix cannot have three flat portions on the boundary of its numerical range. Numerical examples are given to illustrate the main results.