Characterizing matrices with XX-simple image eigenspace in max-min semiring

A matrix $A$ is said to have $X$-simple image eigenspace if any eigenvector $x$ belonging to the interval $X=\{x\colon \underline{x}\leq x\leq\overline{x}\}$ is the unique solution of the system $A\otimes y=x$ in $X$. 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.Comment: 23 page

X-simple image eigencones of tropical matrices

We investigate max-algebraic (tropical) one-sided systems $A\otimes x=b$ where $b$ is an eigenvector and $x$ lies in an interval $X$. A matrix $A$ is said to have $X$-simple image eigencone associated with an eigenvalue $\lambda$, if any eigenvector $x$ associated with $\lambda$ and belonging to the interval $X$ is the unique solution of the system $A\otimes y=\lambda x$ in $X$. We characterize matrices with $X$-simple image eigencone geometrically and combinatorially, and for some special cases, derive criteria that can be efficiently checked in practice.Comment: 25 page