Advances in Applied Clifford Algebras

, Volume 26, Issue 4, pp 1095–1125

Bounds for Eigenvalues of Matrix Polynomials Over Quaternion Division Algebra

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Abstract

Localization theorems are discussed for the left and right eigenvalues of block quaternionic matrices. Basic definitions of the left and right eigenvalues of quaternionic matrices are extended to quaternionic matrix polynomials. Furthermore, bounds on the absolute values of the left and right eigenvalues of quaternionic matrix polynomials are devised and illustrated for the matrix p norm, where \({p = 1, 2, \infty, F}\). The above generalizes the bounds on the absolute values of the eigenvalues of complex matrix polynomials, which give sharper bounds to the bounds developed in [LAA, 358, pp. 5–22 2003] for the case of 1, 2, and \({\infty}\) matrix norms.

Keywords

Skew field Quaternionic matrix Left and right eigenvalues Quaternionic matrix polynomials Quaternionic block companion matrix 

Mathematics Subject Classification

12E15 34L15 15A18 15A66 

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References

  1. 1.
    Adler S.L.: Quaternionic quantum mechanics and quantum fields. Oxford University Press, New York (1995)MATHGoogle Scholar
  2. 2.
    Ahmad, S.S., Ali, I.: Localization theorems of matrices and bounds for the zeros of polynomials over quaternion division algebra (2016) (preprint)Google Scholar
  3. 3.
    Baker A.: Right eigenvalues for quaternionic matrices: a topological approach. Linear Algebra Appl. 286, 303–309 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bini D.A., Noferini V., Sharify M.: Locating the eigenvalues of matrix polynomials. SIAM J. Matrix Anal. Appl. No. 4(34), 1708–1727 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Brenner J.L.: Matrices of quaternions. Pacific J. Math. 1, 329–335 (1951)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Conway, J.H., Smith, D.A.: On quaternions and octonions: their geometry, arithmetic, and symmetry. A K Peters Natick (2002)Google Scholar
  7. 7.
    Feingold D.G., Varga R.S.: Block diagonally dominant matrices and generalizations of the Gerschgorin circle theorem. Pacific J. Math. 12, 1241–1250 (1962)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gerstner A.B., Byers R., Mehrmann V.: A quaternion QR algorithm. Numerih. Mathek. 55, 83–95 (1989)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gohberg I., Lancaster P., XRodman P.: Matrix polynomials. Academic Press, New York (1982)Google Scholar
  10. 10.
    Hankins T.L.: Sir William Rowan Hamilton. The Johns Hopkins University Press, Baltimore (1980)MATHGoogle Scholar
  11. 11.
    Higham N.J., Tisseur F.: Bounds for eigenvalues of matrix polynomials. Linear Algebra Appl. 358, 5–22 (2003)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Huang L., So W.: On left eigenvalues of a quaternionic matrix. Linear Algebra Appl. 323, 105–116 (2001)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Kalantari B.: Algorithms for quaternion polynomial root-finding. J. complex. 29, 302–322 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Kamberov, G., Norman, P., Pedit, F., Pinkall, U.: Quaternions, spinors, and surfaces, contemporary mathematics, vol. 299. Amer. Math. Soc., Province (2002)Google Scholar
  15. 15.
    Kierzkowski J., Smoktunowicz A.: Block normal matrices and Gershgorin-type discs. Electr. J. Linear Algebra 22, 1059–1069 (2011)MathSciNetMATHGoogle Scholar
  16. 16.
    Lee, H.C.: Eigenvalues of cannonical forms of matrices with quaternion coefficients. Proc. R. I. A. 52(sec A), 253–260 (1949)Google Scholar
  17. 17.
    Liping H.: The matrix equation AXB-GXD = E over the quaternion field. Linear Algebra Appl. 234, 197–208(1996)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Melman A.: Generalization and variations of Pellet’s theorem for matrix polynomials. Linear Algebra Appl. 439, 1550–1567 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Opfer G.: Polynomials and Vandermonde matrices over the field of quaternions. Electr. Trans. Num. Anal. 36, 9–16 (2009)MathSciNetMATHGoogle Scholar
  20. 20.
    Pereira, R.: Quaternionic polynomials and behavioral systems. Ph.D. thesis, University of Aveiro (2006)Google Scholar
  21. 21.
    Pereira R., Rocha P.: On the determinant of quaternionic polynomial matrices and its application to system stability. Math Methods Appl. Sci. 31, 99–122 (2008)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Pereira R., Rocha P., Vettori P.: Algebraic tools for the study of quaternionic behavioral systems. Linear Algebra Appl. 400, 121–140 (2005)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Rodman L.: Canonical forms for mixed symmetric-skewsymmetric quaternion matrix pencils. Linear Algebra Appl. 424, 184–221 (2007)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rodman L.: Pairs of Hermitian and skew-Hermitian quaternionic matrices: canonical forms and their applications. Linear Algebra Appl. 429, 981–1019 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Rodman L.: Stability of invariant subspaces of quaternion matrices. Complex Anal. Oper. Theory 6, 1069–1119 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Tisseur F., Higham N.J.: Structured pseudospectra for polynomial eigenvalue problems with applications. SIAM J. Matrix Anal. Appl. 23(1), 187–208 (2001)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Wood R.M.W.: Quaternionic eigenvalues. Bull. Lond. Math. Soc. 17, 137–138 (1985)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Wu J.: Distribution and estimation for eigenvalues of real quaternion matrices. Comp. Math. Appl. 55, 1998–2004 (2008)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zhang F.: Quaternions and matrices of quaternions. Linear Algebra Appl. 251, 21–57 (1997)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zhang F.: Geršgorin type theorems for quaternionic matrices. Linear Algebra Appl. 424, 139–155 (2007)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Zou L., Jiang Y., Wu J.: Location for the right eigenvalues of quaternion matrices. J. Appl. Math. Comput. 38, 71–83 (2012)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.School of Basic Sciences, Discipline of MathematicsIndian Institute of Technology IndoreIndoreIndia
  2. 2.School of Basic Sciences, Discipline of MathematicsIndian Institute of Technology IndoreIndoreIndia

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