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On the Eigenvalues of Quaternion Matrices
Linear and Multilinear Algebra
  • F. O. Farid
  • Qing-Wen Wang, Shanghai University
  • Fuzhen Zhang, Nova Southeastern University; Shenyang Normal University
Document Type
Publication Date
  • Brauer's theorem,
  • Gersgorin theorem,
  • Left eigenvalue quaternion,
  • Quaternion matrix,
  • Right eigenvalue,
  • 15A18,
  • 15A33
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.

AMS Subject Classifications:: 15A18, 15A33

Citation Information
F. O. Farid, Qing-Wen Wang and Fuzhen Zhang. "On the Eigenvalues of Quaternion Matrices" Linear and Multilinear Algebra Vol. 59 Iss. 4 (2011) p. 451 - 473 ISSN: 0308-1087
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