Several spectrally arbitrary ray patterns

Abstract

A ray pattern A of order n is said to be spectrally arbitrary if given any monic nth degree polynomial f(x) with coefficients from ℂ, there exists a matrix realization of A such that its characteristic polynomial is f(x). An n × n ray pattern A is said to be minimally spectrally arbitrary if replacing any nonzero entry of A by zero destroys this property. In this article, several families of ray patterns are presented and proved to be minimally spectrally arbitrary. We also show that for n ≥ 5, when A _n is spectrally arbitrary, then it is minimally spectrally arbitrary.

Keywords:

• spectrally arbitrary
• nilpotent matrix
• ray pattern

AMS Subject Classifications::

• 05C20
• 05C50

Acknowledgements

This research is supported by NSFC (61170311), Chinese Universities Specialized Research Fund for the Doctoral Program (20110185110020), Sichuan Province Sci. & Tech. Research Project (12ZC1802).