Control Theory and Technology

, Volume 17, Issue 1, pp 4–12 | Cite as

A new semi-tensor product of matrices

  • Daizhan ChengEmail author
  • Zequn Liu


A new matrix product, called the second semi-tensor product (STP-II) of matrices is proposed. It is similar to the classical semi-tensor product (STP-I). First, its fundamental properties are presented. Then, the equivalence relation caused by STP-II is obtained. Using this equivalence, a quotient space is also obtained. Finally, the vector space structure, the metric and the metric topology, the projection and subspaces, etc. of the quotient space are investigated in detail.


Second sime-tensor product (STP-II) equivalence class quotient space topology metric 


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  1. [1]
    D. Cheng. Semi-tensor product of matrices and its application to Morgan’s problem. Science in China–Series F: Information Sciences, 2001, 44(3): 195–212.zbMATHGoogle Scholar
  2. [2]
    S. Mei, F. Liu, A. Xue. Semi-tensor Product Method in Analysis of Transient Process of Power Systems. Beijing: Tsinghua University Press, 2010.Google Scholar
  3. [3]
    D. Cheng, H. Qi, Z. Li. Analysis and Control of Boolean Networks–A Semi-tensor Product Approach. London: Springer, 2011.CrossRefzbMATHGoogle Scholar
  4. [4]
    P. Guo, Y. Wang, H. Li. Algebraic formulation and strategy optimization for a class of evolutionary networked games via semi-tensor product method. Automatica, 2013, 49(11): 3384–3389.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    D. Cheng, F. He, H. Qi, et al. Modeling, analisis and control of networked evolutionary games. IEEE Transactions on Automatic Control, 2015, 60(9): 2402–2451.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    D. Cheng, Z. Xu, T. Shen. Equivalence-based model of dimensionvarying linear systems. arXiv, 2018: arXiv:1810.03520.Google Scholar
  7. [7]
    J. M. Howie. Fundamentals of Semigroup Theory. Oxford: The Clarendon Press, 1995.zbMATHGoogle Scholar
  8. [8]
    J. L. Kelley. General Topology. New York: Springer, 1975.zbMATHGoogle Scholar
  9. [9]
    S. Burris, H. Sankappanavar. A Course in Universal Algebra, New York: Springer, 1981.CrossRefzbMATHGoogle Scholar
  10. [10]
    D. Cheng. On equivalence of matrices. arXiv, 2016: arXiv:1605.09523.Google Scholar
  11. [11]
    L. Rade, B. Westergren. Mathematics Handbook for Science and Engineering. 4th ed. Berlin: Springer, 1998.Google Scholar
  12. [12]
    A. E. Taylar, D. C. Lay. Introduction to Functional Analysis. 2nd ed. New York: John Wiley & Sons, 1980.Google Scholar
  13. [13]
    J. Dieudonne. Foundation of Modern Analysis. New York: Academic Press, 1969.zbMATHGoogle Scholar

Copyright information

© Editorial Board of Control Theory & Applications, South China University of Technology and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.The Key Laboratory of Systems and Control, Academy of Mathematics and Systems SciencesChinese Academy of SciencesBeijingChina
  2. 2.University of Chinese Academy of SciencesBeijingChina

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