Mathematics Faculty Articles
On the Eigenvalues of Quaternion Matrices
Document Type
Article
Publication Date
2011
Publication Title
Linear and Multilinear Algebra
Keywords
Brauer's theorem, Gersgorin theorem, Left eigenvalue quaternion, Quaternion matrix, Right eigenvalue, 15A18, 15A33
ISSN
0308-1087
Volume
59
Issue/No.
4
First Page
451
Last Page
473
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.
NSUWorks Citation
Farid, F. O.; Wang, Qing-Wen; and Zhang, Fuzhen, "On the Eigenvalues of Quaternion Matrices" (2011). Mathematics Faculty Articles. 87.
https://nsuworks.nova.edu/math_facarticles/87
DOI
10.1080/03081081003739204
Comments
AMS Subject Classifications:: 15A18, 15A33