Mathematics Faculty Articles

Title

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.

Comments

AMS Subject Classifications:: 15A18, 15A33

DOI

10.1080/03081081003739204

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