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.
The chapter was written by “Martin Gavalec”
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Cechlárová, K: Eigenvectors in bottleneck algebra. Lin. Algebra Appl. 175, 63–73 (1992)
Cechlárová, K: Efficient computation of the greatest eigenvector in fuzzy algebra. Tatra Mt. Math. Publ. 12, 73–79 (1997)
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)
Cechlárová, K., Cuninghame-Green, R.A.: Interval systems of max-separable linear equations. Lin. Algebra Appl. 340, 215–224 (2002)
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)
Cuninghame-Green, R.A.: Describing industrial processes with interference and approximating their steady-state behavior. Oper. Res. Quart. 13, 95–100 (1962)
Cuninghame-Green, R.A.: Minimax Algebra, Lecture Notes in Economics and Mathematical Systems, vol. 166. Springer, Berlin (1979)
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)
Fiedler, M., Nedoma, J., Ramík, J., Rohn, J., Zimmermann, K.: Linear Optimization Problems with Inexact Data. Springer, Berlin (2006)
Gavalec, M.: Monotone eigenspace structure in max–min algebra. Lin. Algebra Appl. 345, 149–167 (2002)
Gavalec, M., Plavka, J.: Monotone interval eigenproblem in max-min algebra. Kybernetika 46, 387–396 (2010)
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)
Gondran, M.: Valeurs propres et vecteurs propres en classification hiérarchique. R. A. I. R. O. Informatique Théorique 10, 39–46 (1976)
Gondran, M., Minoux, M.: Eigenvalues and eigenvectors in semimodules and their interpretation in graph theory. In: Proc. 9th Prog. Symp., pp. 133–148 (1976)
Gondran, M., Minoux, M.: Valeurs propres et vecteurs propres en théorie des graphes. Colloques Internationaux, pp. 181–183. CNRS, Paris (1978)
Gondran, M., Minoux, M.: Dioïds and semirings: Links to fuzzy sets and other applications. Fuzzy Sets Syst. 158, 1273–1294 (2007)
Olsder, G.: Eigenvalues of dynamic max–min systems. Discrete Events Dynamic Systems, vol. 1, pp. 177–201. Kluwer, Dordrecht (1991)
Rashid, I., Gavalec, M., Sergeev, S.: Eigenspace of a three-dimensional max-Łukasiewicz fuzzy matrix. Kybernetika 48, 309–328 (2012)
Rohn, J.: Systems of linear interval equations. Lin. Algebra Appl. 126, 39–78 (1989)
Sanchez, E.: Resolution of eigen fuzzy sets equations. Fuzzy Sets Syst. 1, 69–74 (1978)
Tan, Y.-J.: Eigenvalues and eigenvectors for matrices over distributive lattices. Lin. Algebra Appl. 283, 257–272 (1998)
Tan, Y.-J.: On the powers of matrices over a distributive lattice. Lin. Algebra Appl. 336, 1–14 (2001)
Zimmermann, U.: Linear and Combinatorial Optimization in Ordered Algebraic Structure. In: Ann. Discrete Math., vol. 10. North Holland, Amsterdam (1981)
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Gavalec, M., Ramík, J., Zimmermann, K. (2015). Interval Eigenproblem in Max-Min Algebra. In: Decision Making and Optimization. Lecture Notes in Economics and Mathematical Systems, vol 677. Springer, Cham. https://doi.org/10.1007/978-3-319-08323-0_5
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