Geodesic
On the weak robustness of fuzzy matrices
Kybernetika, Tome 49 (2013) no. 1, pp. 128-140 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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A matrix $A$ in $(\max,\min)$-algebra (fuzzy matrix) is called weakly robust if $A^k\otimes x $ is an eigenvector of $A$ only if $x$ is an eigenvector of $A$. The weak robustness of fuzzy matrices are studied and its properties are proved. A characterization of the weak robustness of fuzzy matrices is presented and an $O(n^2)$ algorithm for checking the weak robustness is described.
A matrix $A$ in $(\max,\min)$-algebra (fuzzy matrix) is called weakly robust if $A^k\otimes x $ is an eigenvector of $A$ only if $x$ is an eigenvector of $A$. The weak robustness of fuzzy matrices are studied and its properties are proved. A characterization of the weak robustness of fuzzy matrices is presented and an $O(n^2)$ algorithm for checking the weak robustness is described.
Classification : 08A72, 90B35, 90C47
Keywords: weak robustness; fuzzy matrices 
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Plavka, Ján. On the weak robustness of fuzzy matrices. Kybernetika, Tome 49 (2013) no. 1, pp. 128-140. http://geodesic.mathdoc.fr/item/KYB_2013_49_1_a8/

[1] Cechlárová, K.: Eigenvectors in bottleneck algebra. Linear Algebra Appl. 174 (1992), 63-73. | MR | Zbl

[2] Cechlárová, K.: Unique solvability of $\max-\min$ fuzzy equations and strong regularity of matrices over fuzzy algebra. Fuzzy Sets and Systems 75 (1995), 165-177. | MR | Zbl

[3] Nola, A. Di, Sessa, S., Pedrycz, W., Sanchez, E.: Fuzzy Relation Equations and Their Application to Knowledge Engineering. Kluwer, Dordrecht 1989. | MR

[4] Gavalec, M.: Computing matrix period in max-min algebra. Discrete Appl. Math. 75 (1997), 63-70. | DOI | MR | Zbl

[5] Gavalec, M.: Solvability and unique solvability of $\max$-$\min$ fuzzy equations. Fuzzy Sets and Systems 124 (2001), 385-393. | DOI | MR | Zbl

[6] Gondran, M., Minoux, M.: Valeurs propres et vecteurs propres en théorie des graphes. Colloques Internationaux, C.N.R.S., Paris 1978, pp. 181-183.

[7] Gondran, M., Minoux, M.: Graphs, Dioids and Semirings: New Models and Algorithms. Springer 2008. | MR | Zbl

[8] Horvath, T., Vojtáš, P.: Induction of fuzzy and annotated logic programs. Inductive Logic Programming 4455 (2007), 260-274. | Zbl

[9] Molnárová, M., Myšková, H., Plavka, J.: The robustness of interval fuzzy matrices. Linear Algebra and Its Applications

[10] Myšková, H.: Interval systems of max-separable linear equations. Linear Algebra Appl. 403 (2005), 263-272. | MR | Zbl

[11] Myšková, H.: Control solvability of interval systems of max-separable linear equations. Linear Algebra Appl. 416 (2006), 215-223. | MR | Zbl

[12] Myšková, M.: Max-min interval systems of linear equations with bounded solution. Kybernetika 48 (2012), 2, 299-308. | MR

[13] Plavka, J., Szabó, P.: On the $\lambda$-robustness of matrices over fuzzy algebra. Discrete Appl. Math. 159(2011), 5, 381-388. | DOI | MR | Zbl

[14] Plavka, J.: On the $O(n^3)$ algorithm for checking the strong robustness of interval fuzzy matrices. Discrete Appl. Math. 160 (2012), 640-647. | DOI | MR

[15] Plavka, J., Vojtáš, P.: On the computing the maximal multiple users preferences using strong robustness of interval fuzzy matrices. Submitted.

[16] Sanchez, E.: Resolution of eigen fuzzy sets equations. Fuzzy Sets and Systems 1 (1978), 69-74. | DOI | MR | Zbl

[17] Sanchez, E.: Medical diagnosis and composite relations. In: Advances in Fuzzy Set Theory and Applications (M. M. Gupta, R. K. Ragade, R. R. Yager, eds.), North-Holland, Amsterdam - New York 1979, pp. 437-444. | MR

[18] Tan, Yi-Jia: Eigenvalues and eigenvectors for matrices over distributive lattices. Lin. Algebra Appl. 283 (1998), 257-272. | DOI | MR | Zbl

[19] Tan, Yi-Jia: On the eigenproblem of matrices over distributive lattices. Lin. Algebra Appl. 374 (2003), 96-106. | MR

[20] Terano, T., Tsukamoto, Y.: Failure diagnosis by using fuzzy logic. In: Proc. IEEE Conference on Decision Control, New Orleans 1977, pp. 1390-1395.

[21] Zimmernann, K.: Extremální algebra (in Czech). Ekonom. ústav ČSAV, Praha 1976.

[22] Zimmermann, U.: Linear and Combinatorial Optimization in Ordered Algebraic Structure. North Holland, Amsterdam 1981. | MR