Numerical range of a nilpotent matrix and related central force

Abstract

In this paper, we introduce a new class of upper triangular nilpotent matrices, etermine the eigenvalues of the Hermitian part of rotating the type of nilpotent matrices and relate the numerical range of that typical type of matrices to the orbit of a point mass under a central force.

Keywords:

• numerical range
• nilpotent matrix
• central force

AMS Subject Classification:

• 15A60
• 70K40
• 14Q05

Acknowledgments

The authors are grateful to an anonymous referee for valuable comments and, in particular, for providing an alternative proof of Theorem 2.1 which is given in the Appendix. The work of the first author was partially supported by Taiwan National Science Council under NSC101-2115-M-031-002, the second author was supported in part by Japan Society for Promotion of Science, Project Number 23540180.