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Monotone interval eigenproblem in max-min algebra. (English) Zbl 1202.15013
A triple $$({\mathcal B},\oplus,\otimes)$$ is called a max-min algebra if $${\mathcal B}$$ is a linearly ordered set, and $$\oplus=\max$$, $$\otimes=\min$$ are binary operations on $${\mathcal B}$$. These operations are extended to matrices and vectors in a formal way. The eigenproblem of a given $$n\times n$$ matrix $$A$$ over $${\mathcal B}$$ consists in finding a value $$\lambda\in{\mathcal B}$$ (= eigenvalue) and a vector $$x$$ (= eigenvector) with $$n$$ components from $${\mathcal B}$$ such that $$A\otimes x= \lambda\otimes x$$ holds. This problem can be reduced to $$A\otimes x= x$$ which is considered in the paper.
If $${\mathbf A}$$ is an $$n\times n$$ interval matrix and $${\mathbf X}$$ is an interval vector with $$n$$ components the interval eigenproblem for $${\mathbf A}$$ and $${\mathbf X}$$ consists in recognizing whether $$A\otimes x= x$$ holds true for $$A\in{\mathbf A}$$, $$x\in{\mathbf X}$$. In dependence on the applied qualifiers a classification containing six different types of interval eigenvectors $${\mathbf X}$$ is introduced, and a detailed characterization of all these types is given for so-called monotone eigenvectors. Moreover, relations between various types of such eigenvectors are presented, illustrated by a Hasse diagram. Simple examples show that certain other specific relations do not hold.

##### MSC:
 15A18 Eigenvalues, singular values, and eigenvectors 65G30 Interval and finite arithmetic 15A80 Max-plus and related algebras
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##### References:
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