2011/07/15 by Eldred, Jeffrey, Rodman, Leiba, Spitkovsky, Ilya M.
#15 #Algebraic Geometry (math.AG) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.1107.3070
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-by-3 and 4-by-4 matrices. In the latter case, it is proved that a 4-by-4 unitarily irreducible companion matrix cannot have 3 flat portions on the boundary of its numerical range. Numerical examples are given to illustrate the main results.