Right Semi-Tensor Product for Matrices Over a Commutative Semiring

Pattrawut Chansangiam

Abstract


This paper generalizes the right semi-tensor product for real matrices to that for matrices over an arbitrary commutative semiring, and investigates its properties. This product is defined for any pair of matrices satisfying the matching-dimension condition. In particular, the usual matrix product and the scalar multiplication are its special cases. The right semi-tensor product turns out to be an associative bilinear map that is compatible with the transposition and the inversion. The product also satisfies certain identity-like properties and preserves some structural properties of matrices. We can convert between the right semi-tensor product of two matrices and the left semi-tensor product using commutation matrices. Moreover, certain vectorizations of the usual product of matrices can be written in terms of the right semi-tensor product.


Keywords


Right semi-tensor product; Kronecker product; Commutative semiring; Vector operator; Commutation matrix

Full Text:

PDF

References


P. Birkhovic, Max-algebra: the linear algebra of combinatorics, Linear Algebra and its Applications 367 (2003), 313 – 335, DOI: 10.1016/S0024-3795(02)00655-9.

C. C. Chang, Algebraic analysis of many valued logic, Transactions of the American Mathematical Society 88 (1958), 467 – 490, DOI: 10.1090/S0002-9947-1958-0094302-9.

P. Chansangiam, Fundamental properties of the box product for matrices over a commutative semiring and Johnson-Nylen transformation, KKU Science Journal 46(2) (2018), 372 – 382.

P. Chansangiam, Algebraic properties of the left semi-tensor product of matrices over a commutative semiring, Advances and Applications in Mathematical Sciences 17(6) (2018), 445 – 459.

P. Chansangiam and P. Gugaew, Algebraic properties of Hadamard product, Hadamard sum, and block Hadamard product for matrices over a commutative semiring, Journal of Science and Technology, Ubon Ratchathani University 21(1) (2019), 192 – 198.

D. Cheng, Semi-tensor product of matrices and its application to Morgen’s problem, Science in China (Series F) 44(3) (2001), 195 – 212, DOI: 10.1007/BF02714570.

D. Cheng, H. Qi and A. Xue, A survey on semi-tensor product of matrices, Journal of Systems Science and Complexity 20 (2007), 304 – 322, DOI: 10.1007/s11424-007-9027-0.

D. Cheng and H. Qi, Semi-tensor Product of Matrices: Theory and Applications, Science Press, Beijing (2007), DOI: 10.1016/B978-0-12-817801-0.00007-7.

D. Cheng, Semi-tensor product of matrices and its applications to dynamic systems, in New Directions and Applications in Control Theory, Lecture Notes in Control and Information Sciences, Springer, Netherlands (2005), 61 – 79, DOI: 10.1007/10984413_5.

D. Cheng, Semi-tensor product of matrices and its application: a survey, Higher Edu. Press, Int. Press (2007), 641 – 668.

D. Cheng, Input-state approach to Boolean networks, IEEE Transactions on Neural Network 20 (2009), 512 – 521, DOI: 10.1109/TNN.2008.2011359.

D. Cheng and H. Qi, Matrix expression of logic and fuzzy control, in Proc. 44th IEEE Conference on Decision and Control, Seville (2005), 3273 – 3278, DOI: 10.1109/CDC.2005.1582666.

R. L. O. Cignoli, I. M. L. D’ottaviano and D. Mundici, Algebraic Foundation of Many-Valued Reasoning, Kluwer Academic Publishers, Dordrecht (2000), DOI: 10.1007/978-94-015-9480-6.

R.A. Cuninghame-Green, Minimax algebra, Lecture Notes in Economics and Mathematical Systems, Springer, Berlin (1979), DOI: 10.1007/978-3-642-48708-8.

R. A. Cuninghame-Green and P. Birkhovic, Bases in max-algebra, Linear Algebra and its Applications 389 (2004), 107 – 120, DOI: 10.1016/j.laa.2004.03.022.

S. Ghosh, Matrices over semirings, Information Sciences 90 (1996), 221 – 230, DOI: 10.1016/0020-0255(95)00283-9.

J. S. Golan, Semirings and Their Applications, Kluwer Academic Publishers, Dordrecht (1999), DOI: 10.1007/978-94-015-9333-5.

M. Gondran and M. Minoux, Graphs Dioids and Semirings: New Models and Algorithms, Springer, New York (2008), DOI: 10.1007/978-0-387-75450-5.

K. H. Kim and F. W. Roush, Generalized fuzzy matrices, Fuzzy Sets and Systems 4 (1980), 293 – 315, DOI: 10.1016/0165-0114(80)90016-0.

S. Mei, F. Liu and A. Xue, Semi-tensor Product Approach to Transient Analysis of Power system, Tsinghua Uni. Press, Beijing (2010).

P. L. Poplin and R.E. Hartwig, Determinantal identities over commutative semirings, Linear Algebra and its Applications 387 (2004), 99 – 132, DOI: 10.1016/j.laa.2004.02.019.

C. Reutenauer and H. Straubing, Inversion of matrices over commutative semiring, Journal of Algebra 88 (1984), 350 – 360, DOI: 10.1016/0021-8693(84)90070-X.

R. Stangam and P. Chansangiam, Kronecker product of matrices over a commutative semiring, Thai Journal of Mathematics 14 (2016), Special Issue on ICMSA 2015, 21 – 38.

J. G. Zaborszky, On the phase portraits of a class of large nonlinear dynamic systems such as the power systems, IEEE Transactions on Automatic Control 33(1) (1998), 4 – 15, DOI: 10.1109/9.356.

S. Zhao and X. P. Wang, Invertible matrices and semilinear spaces over commutative semirings, Information Sciences 180 (2010), 5115 – 5124, DOI: 10.1016/j.ins.2010.08.033.

U. Zimmermann, Linear and combinatorial optimization in ordered algebraic structures, Annals of Discrete Mathematics 10 (1981), 30 – 40.




DOI: http://dx.doi.org/10.26713%2Fjims.v12i1.1217

eISSN 0975-5748; pISSN 0974-875X