Geodesic
Characterizing matrices with ${\bf {X}}$-simple image eigenspace in max-min semiring
Kybernetika, Tome 52 (2016) no. 4, pp. 497-513
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A matrix $A$ is said to have \mbox{\boldmath$X$}-simple image eigenspace if any eigenvector $x$ belonging to the interval $\mbox{\boldmath$X$}=\{x\colon \underline x\leq x\leq\overline x\}$ is the unique solution of the system $A\otimes y=x$ in $\mbox{\boldmath$X$}$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.
A matrix $A$ is said to have \mbox{\boldmath$X$}-simple image eigenspace if any eigenvector $x$ belonging to the interval $\mbox{\boldmath$X$}=\{x\colon \underline x\leq x\leq\overline x\}$ is the unique solution of the system $A\otimes y=x$ in $\mbox{\boldmath$X$}$. The main result of this paper is a combinatorial characterization of such matrices in the linear algebra over max-min (fuzzy) semiring. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.
DOI : 10.14736/kyb-2016-4-0497
Classification : 08A72, 15A18, 15A80
Keywords: max-min algebra; interval; weakly robust; weakly stable; eigenspace; simple image set 
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Plavka, Ján; Sergeev, Sergeĭ. Characterizing matrices with ${\bf {X}}$-simple image eigenspace in max-min semiring. Kybernetika, Tome 52 (2016) no. 4, pp. 497-513. doi: 10.14736/kyb-2016-4-0497

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