Eigenspace of a circulant max-min matrix. (English) Zbl 1206.15008

The min-max algebra considered in the paper is a linearly ordered set \(\mathcal B\) equipped with binary operations \(a\oplus b=\max\{a,b\}\) and \(a\otimes b=\min\{a,b\}\). The authors deal with circulant matrices \(A\in\mathcal B^{n\times n}\) and its eigenvectors \(x\in\mathcal B^{n\times1}\) (i.e. \(Ax=x\)). Some sufficient and/or necessary conditions for \(x\) to be an eigenvector of \(A\) are given. The results are illustrated on some examples for \(n=12\), and a set \(\mathcal B\) of positive integers with the natural order.


15A18 Eigenvalues, singular values, and eigenvectors
15A80 Max-plus and related algebras
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[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] 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
[3] Cuninghame-Green, R. A.: Minimax Algebra and Application. Advances in Imaging and Electron Physics 90 (P. W. Hawkes, Academic Press, New York 1995. · Zbl 0739.90073
[4] 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
[5] Gavalec, M., Plavka, J.: Eigenproblem in extremal algebras. Proc. 9th Internat. Symposium Operations Research ’07, Nova Gorica, Slovenia 2007. · Zbl 1135.15004
[6] Gray, R. M.: Toeplitz and Circulant Matrices. Now Publishers, Delft 2006. · Zbl 1115.15021
[7] Plavka, J.: Eigenproblem for circulant matrices in max-algebra. Optimization 50 (2001), 477-483. · Zbl 1005.90054 · doi:10.1080/02331930108844576
[8] Plavka, J.: l-parametric eigenproblem in max algebra. Discrete Applied Mathematics 150 (2005), 16-28. · Zbl 1115.90048 · doi:10.1016/j.dam.2005.02.017
[9] Zimmermann, K.: Extremal Algebra (in Czech). Ekon. ústav ČSAV, Praha 1976.
[10] Zimmermann, U.: Linear and Combinatorial Optimization in Ordered Algebraic Structure. (Ann. Discrete Math. 10.) North Holland, Amsterdam 1981. · Zbl 0466.90045
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