Max-Plus algebra on tensors and its properties

Document Type: Research Paper


1 Department of Mathematics, Vali-e-Asr University, Rafsanjan, Islamic Republic of Iran

2 Department of Mathematics, Faculty of Sciences, Imam Hossein Comprehensive University, Tehran, Islamic Republic of Iran


In this paper we generalize the max plus algebra system of real matrices to the class of real tensors and derive its fundamental properties. Also we give some basic properties for the left (right) inverse, under the new system. The existence of order 2 left (right) inverses of tensors is characterized.


[1] H.R. Afshin, A.R. Shojaeifard, A max version of Perron Frobenuos theorem for nonnegative tensor, Ann. Funct. Anal., 6 (2015).
[2] M. Akian, R. Bapat, and S. Gaubert, Max-plus algebras, in Handbook of Linear Algebra, Discrete Mathematics and Its Applications 39, L. Hogben, ed., Chapman and Hall/CRC, Boca Raton, FL, 2006.
[3] F. Baccelli, G. Cohen. G. Olsder. J. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event
Systems, Wiley, Chichester. 1992.
[4] P. Butkovic, Max-linear Systems: Theory and Algorithms, Springer Monogr. Math., SpringerVerlag, London,
[5] C. Bu, X. Zhang, J. Zhou, W. Wang, Y. Wei, The inverse, rank and product of tensors, Linear Algebra Appl.,
446 (2014) 269-280.
[6] R.A. Cuninghame-Green, Minimax Algbera, Lecture notes in Economics and Mathematical Systems, 166
Springer, 1979.
[7] B.D. Shutter, On the ultimate behavior of the sequence of consecutive powers of a matrix in the max-plus
algebra, Linear Algebra Appl., 30 (2000), 103-117.
[8] S. Gaubert, Methods and applications of (max,+) linear algebra, Lecure Notes in Computer Science 500,
Springer Verlag, Berlin, 1997, 261-282.
[9] N. Ghasemizadeh and Gh. Aghamollaei, Some results on matrix polynomials in the max algebra, Banach J. Math. Anal., 40 (2015), 17-26.
[10] R.G. Halburd, N.J. Southall, Tropical nevanlinna theory and ultradisctete equations, Loughborough University, 2007.
[11] B. Heidergott, G. Olsder and J. Van Der Woude, Max Plus at Work: Modeling and Analysis of Synchronized Systems, Princeton University Press, 2005.
[12] L.H. Lim, Singular values and eigenvalues of tensors: a variational approach, Proceedings 1st IEEE International Workshop on Computational Advances of Multitensor Adaptive Processing, (2005), 129-132.
[13] L. Qi, Eigenvalues of a real supersymmetric tensor. J. Symbolic Comput., 40 (2005), 1302-1324.
[14] J.Y. Shao, A general product of tensors with applications, Linear Algebra Appl., 439 (2013), 2350-2366.