On the eigenvalues of quaternion matrices

Abstract

This article is a continuation of the article [F. Zhang, Geršgorin type theorems for quaternionic matrices, Linear Algebra Appl. 424 (2007), pp. 139–153] on the study of the eigenvalues of quaternion matrices. Profound differences in the eigenvalue problems for complex and quaternion matrices are discussed. We show that Brauer's theorem for the inclusion of the eigenvalues of complex matrices cannot be extended to the right eigenvalues of quaternion matrices. We also provide necessary and sufficient conditions for a complex square matrix to have infinitely many left eigenvalues, and analyse the roots of the characteristic polynomials for 2 × 2 matrices. We establish a characterisation for the set of left eigenvalues to intersect or be part of the boundary of the quaternion balls of Geršgorin.

Keywords:

• Brauer's theorem
• Geršgorin theorem
• left eigenvalue
• quaternion
• quaternion matrix
• right eigenvalue

AMS Subject Classifications::

• 15A18
• 15A33

Acknowledgements

We would like to thank the referee for carefully reading the manuscript and for the suggestions that improved the exposition of this article. F.O. Farid would like to thank the library of UBC Okanagan for the research facilities they provided. The work of Q.-W. Wang was supported by grants from the Ph.D. Programs Foundation of Ministry of Education of China (20093108110001) and Key Disciplines of Shanghai Municipality (S30104). The work of F. Zhang was partially supported by an NSU 2010 President's Faculty Research and Development Grant.