Numerical ranges of companion matrices: flat portions on the boundary

Abstract

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.

Keywords:

• numerical range
• companion matrix

AMS Subject Classification:

• 15A60

Acknowledgements

A large part of JE's research was done while an undergraduate at the College of William and Mary. His research was supported by The James Monroe Scholars Program.

Notes

1. All numerical ranges are plotted using the program by C. Cowen and E. Harel, available at http://www.math.iupui.edu/~ccowen/Downloads/33NumRange.html