## 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:

 [1] Cechlárová, K.: Eigenvectors in bottleneck algebra. Lin. Algebra Appl. 175 (1992), 63-73. · Zbl 0756.15014 · doi:10.1016/0024-3795(92)90302-Q [2] Cechlárová, K.: Solutions of interval linear systems in $$(\operatorname{max},+)$$-algebra. Proc. 6th Internat. Symposium on Operational Research, Preddvor, Slovenia 2001, pp. 321-326. · Zbl 1016.93049 [3] Cechlárová, K., Cuninghame-Green, R. A.: Interval systems of max-separable linear equations. Lin. Algebra Appl. 340 (2002), 215-224. · Zbl 1004.15009 · doi:10.1016/S0024-3795(01)00405-0 [4] Cuninghame-Green, R. A.: Minimax Algebra. (Lecture Notes in Economics and Mathematical Systems 166.) Springer-Verlag, Berlin 1979. · Zbl 0739.90073 · doi:10.1016/0165-0114(91)90130-I [5] Fiedler, M., Nedoma, J., Ramík, J., Rohn, J., Zimmermann, K.: Linear Optimization Problems with Inexact Data. Springer-Verlag, Berlin 2006. · Zbl 1106.90051 · doi:10.1007/0-387-32698-7 [6] Gavalec, M.: Monotone eigenspace structure in max-min algebra. Lin. Algebra Appl. 345 (2002), 149-167. · Zbl 0994.15010 · doi:10.1016/S0024-3795(01)00488-8 [7] Gavalec, M., Zimmermann, K.: Classification of solutions to systems of two-sided equations with interval coefficients. Internat. J. Pure Applied Math. 45 (2008), 533-542. · Zbl 1154.65036 [8] Rohn, J.: Systems of linear interval equations. Lin. Algebra Appl. 126 (1989), 39-78. · Zbl 1061.15003 · doi:10.1137/S0895479801398955
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