Lucas' theorem and numerical range

Abstract

The Lucas' theorem is generalized to the numerical ranges of some 3 × 3 companion matrices. We determine monic polynomials of degree 3 which assert the generalization. Examples are provided to show the generalization is restricted which gives a negative answer to the question raised by Zemanek [J. Zemanek, The derivative and linear algebra, Kolo Mlodych Matematykow, Ogolnopolskie Warsztaty, dla Mlodych Matematykow, Krakow, 2003, pp. 207–212.].

Keywords:

• Lucas' theorem
• numerical range

AMS Subject Classification::

• 15A60

Acknowledgements

The research was supported in part by the Taiwan National Science Council. The author is also grateful to an anonymous referee for bringing 2 to his attention.