# Article

 Title: On the vectors associated with the roots of max-plus characteristic polynomials (English) Author: Nishida, Yuki Author: Watanabe, Sennosuke Author: Watanabe, Yoshihide Language: English Journal: Applications of Mathematics ISSN: 0862-7940 (print) ISSN: 1572-9109 (online) Volume: 65 Issue: 6 Year: 2020 Pages: 785-805 Summary lang: English . Category: math . Summary: We discuss the eigenvalue problem in the max-plus algebra. For a max-plus square matrix, the roots of its characteristic polynomial are not its eigenvalues. In this paper, we give the notion of algebraic eigenvectors associated with the roots of characteristic polynomials. Algebraic eigenvectors are the analogues of the usual eigenvectors in the following three senses: (1) An algebraic eigenvector satisfies an equation similar to the equation $A\otimes \boldsymbol {x} = \lambda \otimes \boldsymbol {x}$ for usual eigenvectors. Under a suitable assumption, the equation has a nontrivial solution if and only if $\lambda$ is a root of the characteristic polynomial. (2) The set of algebraic eigenvectors forms a max-plus subspace called algebraic eigenspace. (3) The dimension of each algebraic eigenspace is at most the multiplicity of the corresponding root of the characteristic polynomial. (English) Keyword: max-plus algebra Keyword: eigenvalue Keyword: eigenvector Keyword: characteristic polynomial MSC: 15A18 MSC: 15A80 idZBL: Zbl 07285957 idMR: MR4191369 DOI: 10.21136/AM.2020.0374-19 . Date available: 2020-11-18T09:38:57Z Last updated: 2021-04-08 Stable URL: http://hdl.handle.net/10338.dmlcz/148397 . Reference: [1] Akian, M., Bapat, R., Gaubert, S.: Max-plus algebra.Handbook of Linear Algebra L. Hogben et al. Discrete Mathematics and Its Applications 39. Chapman & Hall/CRC, Boca Raton (2007), 35-1, 18 pages. Zbl 1122.15001, MR 2279160 Reference: [2] Akian, M., Gaubert, S., Guterman, A.: Tropical polyhedra are equivalent to mean payoff games.Int. J. Algebra Comput. 22 (2012), Article ID 1250001, 43 pages. Zbl 1239.14054, MR 2900854, 10.1142/S0218196711006674 Reference: [3] Baccelli, F., Cohen, G., Olsder, G. J., Quadrat, J. P.: Synchronization and Linearity: An Algebra for Discrete Event Systems.Wiley Series on Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, Chichester (1992). Zbl 0824.93003, MR 1204266 Reference: [4] Butkovič, P.: Max-Linear Systems: Theory and Algorithms.Springer Monographs in Mathematics. Springer, London (2010). Zbl 1202.15032, MR 2681232, 10.1007/978-1-84996-299-5 Reference: [5] Butkovič, P., Schneider, H., Sergeev, S.: Generators, extremals and bases of max cones.Linear Algebra Appl. 421 (2007), 394-406. Zbl 1119.15018, MR 2294351, 10.1016/j.laa.2006.10.004 Reference: [6] Cuninghame-Green, R. A.: The characteristic maxpolynomial of a matrix.J. Math. Anal. Appl. 95 (1983), 110-116. Zbl 0526.90098, MR 0710423, 10.1016/0022-247X(83)90139-7 Reference: [7] Cuninghame-Green, R. A.: Minimax algebra and applications.Adv. Imaging Electron Phys. 90 (1994), 1-121. MR 0618736, 10.1016/S1076-5670(08)70083-1 Reference: [8] Heidergott, B., Olsder, G. J., Woude, J. Van der: Max Plus at Work. Modeling and Analysis of Synchronized Systems: A Course on Max-Plus Algebra and Its Applications.Princeton Series in Applied Mathematics. Princeton University Press, Princeton (2006). Zbl 1130.93003, MR 2188299 Reference: [9] Izhakian, Z., Rowen, L.: Supertropical matrix algebra.Isr. J. Math. 182 (2011), 383-424. Zbl 1215.15018, MR 2783978, 10.1007/s11856-011-0036-2 Reference: [10] Izhakian, Z., Rowen, L.: Supertropical matrix algebra II: Solving tropical equations.Isr. J. Math. 186 (2011), 69-96. Zbl 1277.15013, MR 2852317, 10.1007/s11856-011-0133-2 Reference: [11] Izhakian, Z., Rowen, L.: Supertropical matrix algebra III: Powers of matrices and their supertropical eigenvalues.J. Algebra 341 (2011), 125-149. Zbl 1283.15055, MR 2824513, 10.1016/j.jalgebra.2011.06.002 Reference: [12] Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry.Graduate Studies in Mathematics 161. American Mathematical Society, Providence (2015). Zbl 1321.14048, MR 3287221, 10.1090/gsm/161 Reference: [13] Nishida, Y., Sato, K., Watanabe, S.: A min-plus analogue of the Jordan canonical form associated with the basis of the generalized eigenspace.(to appear) in Linear Multilinear Algebra. 10.1080/03081087.2019.1700892 Reference: [14] Niv, A., Rowen, L.: Dependence of supertropical eigenspaces.Commun. Algebra 45 (2017), 924-942. Zbl 1378.15006, MR 3573348, 10.1080/00927872.2016.1172603 .

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