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Interval Eigenproblem in Max-Min Algebra

Chapter
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Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 677)

Abstract

The eigenvectors of square matrices in max-min algebra correspond to steady states in discrete events system in various application areas, such as design of switching circuits, medical diagnosis, models of organizations and information systems. Imprecise input data lead to considering interval version of the eigenproblem, in which interval eigenvectors of interval matrices in max-min algebra are investigated. Six possible types of an interval eigenvector of an interval matrix are introduced, using various combination of quantifiers in the definition. The previously known characterizations of the interval eigenvectors were restricted to the increasing eigenvectors, see [11]. In this chapter, the results are extended to the non-decreasing eigenvectors, and further to all possible interval eigenvectors of a given max-min matrix. Classification types of general interval eigenvectors are studied and characterization of all possible six types is presented.

Keywords

Interval Matrix Interval Version Interval Partition Eigenspace Structure Strongest Eigenvector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Cechlárová, K: Eigenvectors in bottleneck algebra. Lin. Algebra Appl. 175, 63–73 (1992)CrossRefGoogle Scholar
  2. 2.
    Cechlárová, K: Efficient computation of the greatest eigenvector in fuzzy algebra. Tatra Mt. Math. Publ. 12, 73–79 (1997)Google Scholar
  3. 3.
    Cechlárová, K: Solutions of interval linear systems in (max, +)-algebra. In: Proceedings of the 6th International Symposium on Operational Research Preddvor, pp. 321–326, Slovenia (2001)Google Scholar
  4. 4.
    Cechlárová, K., Cuninghame-Green, R.A.: Interval systems of max-separable linear equations. Lin. Algebra Appl. 340, 215–224 (2002)CrossRefGoogle Scholar
  5. 5.
    Cohen, G., Dubois, D., Quadrat, J.P., Viot, M.: A linear-system-theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Trans. Automat. Contr. AC-30, 210–220 (1985)CrossRefGoogle Scholar
  6. 6.
    Cuninghame-Green, R.A.: Describing industrial processes with interference and approximating their steady-state behavior. Oper. Res. Quart. 13, 95–100 (1962)CrossRefGoogle Scholar
  7. 7.
    Cuninghame-Green, R.A.: Minimax Algebra, Lecture Notes in Economics and Mathematical Systems, vol. 166. Springer, Berlin (1979)Google Scholar
  8. 8.
    Cuninghame-Green, R.A.: Minimax algebra and application. In: Hawkes, P.W. (ed.), Advances in Imaging and Electron Physics, vol. 90. Academic Press, New York (1995)Google Scholar
  9. 9.
    Fiedler, M., Nedoma, J., Ramík, J., Rohn, J., Zimmermann, K.: Linear Optimization Problems with Inexact Data. Springer, Berlin (2006)Google Scholar
  10. 10.
    Gavalec, M.: Monotone eigenspace structure in max–min algebra. Lin. Algebra Appl. 345, 149–167 (2002)CrossRefGoogle Scholar
  11. 11.
    Gavalec, M., Plavka, J.: Monotone interval eigenproblem in max-min algebra. Kybernetika 46, 387–396 (2010)Google Scholar
  12. 12.
    Gavalec, M., Zimmermann, K.: Classification of solutions to systems of two-sided equations with interval coefficients. Int. J. Pure Appl. Math. 45, 533–542 (2008)Google Scholar
  13. 13.
    Gondran, M.: Valeurs propres et vecteurs propres en classification hiérarchique. R. A. I. R. O. Informatique Théorique 10, 39–46 (1976)Google Scholar
  14. 14.
    Gondran, M., Minoux, M.: Eigenvalues and eigenvectors in semimodules and their interpretation in graph theory. In: Proc. 9th Prog. Symp., pp. 133–148 (1976)Google Scholar
  15. 15.
    Gondran, M., Minoux, M.: Valeurs propres et vecteurs propres en théorie des graphes. Colloques Internationaux, pp. 181–183. CNRS, Paris (1978)Google Scholar
  16. 16.
    Gondran, M., Minoux, M.: Dioïds and semirings: Links to fuzzy sets and other applications. Fuzzy Sets Syst. 158, 1273–1294 (2007)CrossRefGoogle Scholar
  17. 17.
    Olsder, G.: Eigenvalues of dynamic max–min systems. Discrete Events Dynamic Systems, vol. 1, pp. 177–201. Kluwer, Dordrecht (1991)Google Scholar
  18. 18.
    Rashid, I., Gavalec, M., Sergeev, S.: Eigenspace of a three-dimensional max-Łukasiewicz fuzzy matrix. Kybernetika 48, 309–328 (2012)Google Scholar
  19. 19.
    Rohn, J.: Systems of linear interval equations. Lin. Algebra Appl. 126, 39–78 (1989)CrossRefGoogle Scholar
  20. 20.
    Sanchez, E.: Resolution of eigen fuzzy sets equations. Fuzzy Sets Syst. 1, 69–74 (1978)CrossRefGoogle Scholar
  21. 21.
    Tan, Y.-J.: Eigenvalues and eigenvectors for matrices over distributive lattices. Lin. Algebra Appl. 283, 257–272 (1998)CrossRefGoogle Scholar
  22. 22.
    Tan, Y.-J.: On the powers of matrices over a distributive lattice. Lin. Algebra Appl. 336, 1–14 (2001)CrossRefGoogle Scholar
  23. 23.
    Zimmermann, U.: Linear and Combinatorial Optimization in Ordered Algebraic Structure. In: Ann. Discrete Math., vol. 10. North Holland, Amsterdam (1981)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.University of Hradec KraloveHradec KraloveCzech Republic
  2. 2.Silesian University in OpavaKarvinaCzech Republic
  3. 3.Charles University in PraguePragueCzech Republic

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