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Keywords:
max-min algebra; interval; eigenspace; simple image set
max-min algebra; interval; eigenspace; simple image set
Summary:
A matrix $A$ is said to have \mbox{\boldmath$X$}-simple image eigenspace if any eigenvector $x$ belonging to the interval $\mbox{\boldmath$X$}=\{x\colon \underline x\leq x\leq\overline x\}$ containing a constant vector is the unique solution of the system $A\otimes y=x$ in \mbox{\boldmath$X$}. The main result of this paper is an extension of \mbox{\boldmath$X$}-simplicity to interval max-min matrix $\mbox{\boldmath$A$}=\{A\colon \underline A\leq A\leq\overline A\}$ distinguishing two possibilities, that at least one matrix or all matrices from a given interval have \mbox{\boldmath$X$}-simple image eigenspace. \mbox{\boldmath$X$}-simplicity of interval matrices in max-min algebra are studied and equivalent conditions for interval matrices which have \mbox{\boldmath$X$}-simple image eigenspace are presented. 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.
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