Advertisement

Conservation of the number of the eigenvalues of two-parameter matrix problems in bounded domains under perturbations

  • Michael Gil’Email author
Article
  • 16 Downloads

Abstract

The paper deals with the two-parameter matrix eigenvalue problems \(T_jv_j - \lambda _1 A_{j1}v_j - \lambda _2 A_{j2}v_j=0\) and \(\tilde{T}_j\tilde{v}_j - \lambda _1 \tilde{A}_{j1}\tilde{v}_j - \lambda _2 \tilde{A}_{j2}\tilde{v}_j=0\), where \(\lambda _j, \tilde{\lambda }_j\in \mathbb {C}\); \(T_j, A_{jk},\tilde{T}_j, \tilde{A}_{jk}\;\;(j, k=1,2)\) are matrices. We derive the conditions, under which the problems have the same number of the eigenvalues in a given bounded domain. A two-parameter matrix eigenvalue problem is called a Schur–Cohn stable if its spectrum is inside the unit circle. It is said to be asymptotically stable if its spectrum is in the open left half-plane. As a consequence of the main result of the paper we obtain the tests for the Schur–Cohn and asymptotic stabilities. In addition, we suggest new localization results for the eigenvalues of the considered problems, which are sharp if the matrix coefficients are ”close” to triangular ones. Our main tool is a new perturbation result for the determinants of linear matrix pencils.

Keywords

Two parameter matrix eigenvalue problem Perturbations Localization of spectra Schur–Cohn stability Asymptotic stability Linear matrix pencils 

Mathematics Subject Classification

15A69 

Notes

References

  1. 1.
    Atkinson, F.V.: Multiparameter Eigenvalue Problems. Academic Press, New York (1972)zbMATHGoogle Scholar
  2. 2.
    Baruah, A.K., Changmai, J.: Approximations in a multiparameter eigenvalue problem under finite element procedures. Math. Forum 21, 100–122 (2010)MathSciNetGoogle Scholar
  3. 3.
    Binding, P., Browne, P.J.: A variational approach to multiparameter eigenvalue problems for matrices. SIAM J. Math. Anal. 8, 763–777 (1977)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Binding, P., Browne, P.J.: A variational approach to multiparameter eigenvalue problems in Hilbert space. SIAM J. Math. Anal. 9, 1054–1067 (1978)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cottin, N.: Dynamic model updating a multiparameter eigenvalue problem. Mech. Syst. Signal Process. 15, 649–665 (2001)CrossRefGoogle Scholar
  6. 6.
    Dookhitram, K., Tangman, Y.D., Bhuruth, M.: Convergence of Arnoldi’s method for generalized eigenvalue problems. Afr. Mat. 26, 485–501 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Gao, J., Li, C.: A new localization set for generalized eigenvalues. J. Inequal. Appl. 2017, 113 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gil’, M.I.: On spectral variation of two-parameter matrix eigenvalue problem. Publ. Math. Debr. 87(3–4), 269–278 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gil’, M.I.: Bounds for the spectrum of a two parameter matrix eigenvalue problem. Linear Algebra Appl. 498, 201–218 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gil’, M.I.: Bounds for Determinants of Linear Operators and their Applications. CRC Press, Boca Raton (2017)CrossRefGoogle Scholar
  11. 11.
    Gil’, M.I.: On the spectrum of a nonlinear two parameter matrix eigenvalue problem. In: Rassias, T.M. (ed.) Applications of Nonlinear Analysis, vol. 134. Springer, New York (2018)CrossRefGoogle Scholar
  12. 12.
    Hochstenbach, M.E., Muhic̆, A., Plestenjak, B.: On linearizations of the quadratic two-parameter eigenvalue problem. Linear Algebra Appl. 436(8), 2725–2743 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Hong, Y., Lim, D., Qi, F.: Some inequalities for generalized eigenvalues of perturbation problems on Hermitian matrices. J. Inequal. Appl. 2018, 6 (2018)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar
  15. 15.
    Isaev, G.A.: Lectures on Multiparameter Spectral Theory. Department of Mathematics and Statistics. The University of Calgary., Calgary (1985)Google Scholar
  16. 16.
    Jarlebring, E., Hochstenbach, M.E.: Polynomial two-parameter eigenvalue problems and matrix pencil methods for stability of delay-differential equations. Linear Algebra Appl. 431, 369–380 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Khazanov, V.B.: To solving spectral problems for multiparameter polynomial matrices. J. Math. Sci. 141, 1690–1700 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Kos̆ir, T.: Finite dimensional multiparameter spectral theory: the nonderogatory case. Linear Algebra Appl. 212(213), 45–70 (1994)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kostic, V., Cvetkovic̆, L.J., Varga, R.S.: Gershgorin-type localizations of generalized eigenvalues. Numer. Linear Algebra Appl. 16, 883–898 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Muhic, A., Plestenjak, B.: On the singular two-parameter eigenvalue problem. Electron. J. Linear Algebra 18, Article 34, 420–437 (2009)Google Scholar
  21. 21.
    Nakatsukasa, Y.: Gerschgorin’s theorem for generalized eigenvalue problem in the Euclidean metric. Math. Comput. 80(276), 2127–2142 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pons, A., Gutschmidt, S.: Nonlinear multiparameter eigenvalue problems in aeroelasticity. Int. J. Struct. Stab. Dyn. 19(05), 1941008 (2019)MathSciNetCrossRefGoogle Scholar

Copyright information

© Università degli Studi di Ferrara 2019

Authors and Affiliations

  1. 1.Department of MathematicsBen Gurion University of the NegevBeer-ShevaIsrael

Personalised recommendations