Document Type : Research Paper
1 Department of Mathematics and Statistics, Faculty of Basic Sciences, Imam Hossein Comprehensive University, Tehran, Iran.
2 Department of Mathematics, Faculty of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Islamic Republic of Iran.
Recently we generalized the max algebra system to the class of nonnegative tensors. In this paper we give some basic properties for the left (right) inverse, under the new system. The existence of order 2 left (right) inverse of tensors is characterized. Also we generalize the direct product of matrices to the direct product of tensors (of the same order, but may be different dimensions) and investigate its properties relevant to the spectral theory.
 R. Bapat, A max version of the Perron Frobenius theorem, Linear Algebra Appl., (1998), pp. 3-18.
 F. Baccelli, G. Cohen, G. Olsder, and J. Quadrat, Synchronization and Linearity: An Algebra for Discrete Event Systems, Wiley, 1992.
 P. Butkovic and M. Fiedler, Tropical tensor product and beyond, School of Mathematics University of Birmingham, 2011.
 C. Bu, X. Zhang, J. Zhou, W. Wang, and Y. Wei, The inverse, rank and product of tensors, Linear Algebra Appl., 446 (2014), pp. 269-280.
 K.C. Chang, K. Pearson, and T. Zhang, Perron Frobenius theorem for nonnegative tensors, Commun. Math. Sci., 6 (2008), pp. 507-520.
 R.A. Cuninghame-Green, Minimax Algbera, Springer-Verlag, 1979.
 V. Loan, Future directions in tensor based computation and modeling, NSF Workshop Report in Arlington, Virginia, USA, 2009.
 K. Pearson, Essentially positive tensors, Int. J. Algebra., 4 (2010), pp. 421-427.
 L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), pp. 1302-1324.
 J.Y. Shao, A general product of tensors with applications, Linear Algebra Appl., 439 (2013), pp. 2350-2366.