Abstract
The min-plus algebra is a commutative semiring with two operations: addition \(\varvec{a} \oplus \varvec{b := \min (a,b)}\) and multiplication \(\varvec{a} \otimes \varvec{b := a + b}\). In this paper, we discuss a min-plus algebraic counterpart of matrix diagonalization in conventional linear algebra. Due to the absence of subtraction in the min-plus algebra, few matrices admit such a canonical form. Instead, we consider triangulation of min-plus matrices in terms of algebraic eigenvectors, which is an extended concept of usual eigenvectors. We deal with two types of min-plus matrices: strongly diagonally dominant (SDD) and nearly diagonally dominant (NDD) matrices. For an SDD matrix, the roots of the characteristic polynomial coincide with its diagonal entries. On the other hand, for an NDD matrix, the roots except for the maximum one appear in diagonal entries. We show that SDD matrices admit upper triangulation whose diagonal entries are algebraic eigenvalues, while NDD matrices admit block upper triangulation. We exhibit applications of triangulation of min-plus matrices to traffic flow models.
Similar content being viewed by others
References
Akian M, Bapat R, Gaubert S (2006) Max-plus algebra. In: Hogben L (ed) Handbook of linear algebra. Chapman & Hall/CRC, Boca Raton, chap 25
Baccelli F, Cohen G, Olsder GJ et al (1992) Synchronization and linearity. Wiley, Chichester
Butkovič P (2003) Max-algebra: the linear algebra of combinatorics? Linear Algebra Appl 367:313–335. https://doi.org/10.1016/S0024-3795(02)00655-9
Butkovič P (2010) Max-linear systems: theory and algorithms. Springer-Verlag, London
Butkovič P, Schneider G, Sergeev S (2007) Generators, extremals and bases of max cones. Linear Algebra Appl 421:394–406. https://doi.org/10.1016/j.laa.2006.10.004
Butkovič P, Cuninghame-Green RA, Gaubert S (2009) Reducible spectral theory with applications to the robustness of matrices in max-algebra. SIAM J Matrix Anal Appl 31(3):1412–1431. https://doi.org/10.1137/080731232
Cohen G, Dubois D, Quadrat JP et al (1985) A linear-system-theoretic view of discrete-event processes and its use for performance evaluation in manufacturing. IEEE Trans Automat Control 30(3):210–220. https://doi.org/10.1109/TAC.1985.1103925
Cohen G, Gaubert S, Quadrat JP (1999) Max-plus algebra and system theory: where we are and where to go now. Annu Rev Control 23:207–219. https://doi.org/10.1016/S1367-5788(99)90091-3
Cuninghame-Green RA (1960) Process synchronization in a steelworks–a problem of feasibility. In: Proceedings of the 2nd international conference on operational research, Aix-en-Provence, pp 323–328
Cuninghame-Green RA (1962) Describing industrial processes with interface and approximating their steady-state behavior. Oper Res Quarterly 13:95–100. https://doi.org/10.1057/jors.1962.10
Cuninghame-Green RA (1979) Minimax algebra. Springer-Verlag, Berlin Hidelberg
Cuninghame-Green RA (1983) The characteristic maxpolynomial of a matrix. J Math Anal Appl 95:110–116. https://doi.org/10.1016/0022-247X(83)90139-7
Cuninghame-Green RA, Meijer PFJ (1980) An algebra for piecewise-linear minimax problems. Discrete Appl Math 2:267–294. https://doi.org/10.1016/0166-218X(80)90025-6
De Schutter B (2000) On the ultimate behavior of the sequence of consecutive powers of a matrix in the max-plus algebra. Linear Algebra Appl 307:103–117. https://doi.org/10.1016/S0024-3795(00)00013-6
De Schutter B, van den Boom T, Xu J et al (2020) Analysis and control of max-plus linear discrete-event systems: an introduction. Discrete Event Dyn Syst 30:25–54. https://doi.org/10.1007/s10626-019-00294-w
Di Loreto M, Gaubert S, Katz RD et al (2010) Duality between invariant spaces for max-plus linear discrete event systems. SIAM J Control Optim 48(8):5606–5628. https://doi.org/10.1137/090747191
Gassner E, Klinz B (2010) A fast parametric assignment algorithm with applications in max-algebra. Networks 55(2):61–77. https://doi.org/10.1002/net.20288
Goverde RMP (1998) The max-plus algebra approach to railway timetable design. WIT Trans Built Environ 37:339–350. https://doi.org/10.2495/CR980331
Heidergott B, Olsder GJ, van der Woude J (2005) Max plus at work: modeling and analysis of synchronized systems: a course on max-plus algebra and its applications. Princeton University Press, Princeton
Izhakian Z, Rowen L (2011a) Supertropical matrix algebra II: solving tropical equations. Israel J Math 186:60–96. https://doi.org/10.1007/s11856-011-0133-2
Izhakian Z, Rowen L (2011b) Supertropical matrix algebra III: powers of matrices and their supertropical eigenvalues. J Algebra 341:125–149. https://doi.org/10.1016/j.jalgebra.2011.06.002
Johnson DB (1975) Finding all the elementary circuits of a directed graph. SIAM J Comput 4(1):77–84. https://doi.org/10.1137/0204007
Katz RD (2007) Max-plus \((a, b)\)-invariant spaces and control of timed discrete-event systems. IEEE Trans Automat Control 52:229–241. https://doi.org/10.1109/TAC.2006.890478
Lotito PA, Mancinelli EM, Quadrat JP (2005) A min-plus derivation of the fundamental car-traffic law. IEEE Trans Automat Control 50:699–705. https://doi.org/10.1109/TAC.2005.848336
Mandel A, Simon I (1977) On finite semigroups of matrices. Theor Comput Sci 5(2):101–111. https://doi.org/10.1016/0304-3975(77)90001-9
Nagel K, Schreckenberg M (1992) A cellular automaton model for freeway traffic. J Physique I France 2:2221–2229. https://doi.org/10.1051/jp1:1992277
Nishida Y, Watanabe S, Watanabe Y (2020) On the vectors associated with the roots of max-plus characteristic polynomial. Appl math 65:785–805. https://doi.org/10.21136/AM.2020.0374-19
Nishida Y, Sato K, Watanabe S (2021) A min-plus analogue of the jordan canonical form associated with the basis of the generalized eigenspace. Linear Multilinear Algebra 69:2933–2943. https://doi.org/10.1080/03081087.2019.1700892
Nishida Y, Watanabe S, Watanabe Y (2024) Independence and orthogonality of algebraic eigenvectors. https://arxiv.org/abs/2110.00285
Nishinari K, Takahashi D (1999) A new deterministic ca model for traffic flow with multiple states. J Phys A 32:93–104. https://doi.org/10.1088/0305-4470/32/1/010
Pin JE (2019) The influence of Imre Simon’s work in the theory of automata, languages and semigroups. Semigroup Forum 98:1–8. https://doi.org/10.1007/s00233-019-09999-8
Sergeev S (2009) Max algebraic powers of irreducible matrices in the periodic regime: an application of cyclic classes. Linear Algebra Appl 431:1325–1339. https://doi.org/10.1016/j.laa.2009.04.027
Simon I (1988) Recognizable sets with multiplicities in the tropical semiring. In: Proceedings of 13th symposium, mathematical foundations of computer science. Carlsbad, Czechoslovakia, pp 107–120
Takayasu M, Takayasu H (1993) \(1/f\) noise in a traffic model. Fractals 1:860–866. https://doi.org/10.1142/S0218348X93000885
Wolfram S (1994) Cellular automata and complexity: collected papers. Westview Press, Boulder
Acknowledgements
The authors thank anonymous referees for their helpful comments. This work is supported by JSPS KAKENHI Grant No.22K13964.
Funding
This work is supported by JSPS KAKENHI Grant No.22K13964.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no competing interests to declare that are relevant to the content of this article.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Nishida, Y., Watanabe, S. & Watanabe, Y. Triangulation of diagonally dominant min-plus matrices. Discrete Event Dyn Syst (2024). https://doi.org/10.1007/s10626-024-00397-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10626-024-00397-z