Abstract
A detailed account of the max-algebraic eigenvalue-eigenvector theory for square matrices over reals extended by minus infinity is provided. This theory is similar (but not identical) with the Perron-Frobenius theory for non-negative matrices in linear algebra. The algorithms presented and proved here enable us to find all eigenvalues and bases of all eigenspaces in almost linear time. These results are of fundamental importance for solving the reachability problems in Chap. 8 and elsewhere.
The chapter starts with definitions and basic properties of the eigenproblem, then continues by proving one of the most important results in max-algebra, namely that for every matrix the maximum cycle mean is the greatest (principal) eigenvalue. It is shown how the corresponding (principal) eigenspace is described. Next a spectral theorem is presented, that fully describes all eigenvalues of a matrix and enables us to find them. It also makes it possible to characterize matrices with finite eigenvectors. Finally, it is explained how to efficiently describe all eigenvectors of a matrix.
A number of properties, such as the existence of a common eigenvector of commuting matrices, is proved using the presented eigenvalue-eigenvector theory.
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References
Akian, M., Gaubert, S., & Walsh, C. (2005). Discrete max-plus spectral theory. ESI Preprint 1485. In G. L. Litvinov & V. P. Maslov (Eds.), Contemporary mathematics series : Vol. 377. Idempotent mathematics and mathematical physics (pp. 53–77). Providence: AMS.
Bapat, R. B. (1995). Pattern properties and spectral inequalities in max algebra. SIAM Journal on Matrix Analysis and Applications, 16, 964–976.
Bapat, R. B., & Raghavan, T. E. S. (1997). Nonnegative matrices and applications. Cambridge: Cambridge University Press.
Bapat, R. B., Stanford, D., & van den Driessche, P. (1993). The eigenproblem in max algebra (DMS-631-IR). University of Victoria, British Columbia.
Berman, A., & Plemmons, R. J. (1979). Nonnegative matrices in the mathematical sciences. New York: Academic Press.
Brualdi, R. A., & Ryser, H. (1991). Combinatorial matrix theory. Cambridge: Cambridge University Press.
Butkovič, P. (2008). Permuted max-algebraic (tropical) eigenvector problem is NP-complete. Linear Algebra and Its Applications, 428, 1874–1882.
Butkovič, P., & Cuninghame-Green, R. A. (1992). An algorithm for the maximum cycle mean of an n×n bivalent matrix. Discrete Applied Mathematics, 35, 157–162.
Butkovič, P., & Plávka, J. (1989). On the dependence of the maximum cycle mean of a matrix on permutations of the rows and columns. Discrete Applied Mathematics, 23, 45–53.
Butkovič, P., Gaubert, S., & Cuninghame-Green, R. A. (2009). Reducible spectral theory with applications to the robustness of matrices in max-algebra. SIAM Journal on Matrix Analysis and Applications, 31(3), 1412–1431.
Chen, W., Qi, X., & Deng, S. (1990). The eigenproblem and period analysis of the discrete event systems. Systems Science and Mathematical Sciences, 3(3), 243–260.
Cuninghame-Green, R. A. (1962). Describing industrial processes with interference and approximating their steady-state behaviour. Operations Research Quarterly, 13, 95–100.
Cuninghame-Green, R. A. (1979). Lecture notes in economics and math systems : Vol. 166. Minimax algebra. Berlin: Springer. (Downloadable from http://web.mat.bham.ac.uk/P.Butkovic/).
Cuninghame-Green, R. A. (1981). Minimax algebra. Bulletin—Institute of Mathematics and Its Applications, 17(4), 66–69.
Cuninghame-Green, R. A. (1995). Minimax algebra and applications. In Advances in imaging and electron physics (Vol. 90, pp. 1–121). New York: Academic Press.
Cuninghame-Green, R. A., & Butkovič, P. (1995). Extremal eigenproblem for bivalent matrices. Linear Algebra and Its Applications, 222, 77–89.
Cuninghame-Green, R. A., & Butkovič, P. (2008). Generalised eigenproblem in max algebra. IEEE Xplore. Discrete Event Systems (pp. 236–241).
Elsner, L., & van den Driessche, P. (2008). Bounds for the Perron root using max eigenvalues. Linear Algebra and Its Applications, 428, 2000–2005.
Gaubert, S. (1992). Théorie des systèmes linéaires dans les dioïdes. Thèse, Ecole des Mines de Paris.
Gondran, M., & Minoux, M. (1977). Valeurs propres et vecteur propres dans les dioïdes et leur interprétation en théorie des graphes. Bulletin de la Direction des Etudes et Recherches Serie C Mathematiques et Informatiques (2), 25–41.
Heidergott, B., Olsder, G. J., & van der Woude, J. (2005). Max plus at work: modeling and analysis of synchronized systems. A course on max-plus algebra. Princeton: Princeton University Press.
Katz, R., Schneider, H., & Sergeev, S. (2010). Commuting matrices in max-algebra (Preprint 2010/03). University of Birmingham, School of Mathematics.
Rosen, K. H. et al. (2000). Handbook of discrete and combinatorial mathematics. New York: CRC Press.
Schneider, H. (1988). The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: a survey. Linear Algebra and Its Applications, 84, 161–189.
Tarjan, R. E. (1972). Depth-first search and linear graph algorithms. SIAM Journal on Computing, 1(2), 146–160.
Vorobyov, N. N. (1967). Extremal algebra of positive matrices. Elektronische Informationsverarbeitung und Kybernetik, 3, 39–71 (in Russian).
Vorobyov, N. N. (1970). Extremal algebra of nonnegative matrices. Elektronische Informationsverarbeitung und Kybernetik, 6, 303–311 (in Russian).
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Butkovič, P. (2010). Eigenvalues and Eigenvectors. In: Max-linear Systems: Theory and Algorithms. Springer Monographs in Mathematics. Springer, London. https://doi.org/10.1007/978-1-84996-299-5_4
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DOI: https://doi.org/10.1007/978-1-84996-299-5_4
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