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A new semi-tensor product of matrices

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Abstract

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.

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Author information

Authors and Affiliations

  1. The Key Laboratory of Systems and Control, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing, 100190, China

    Daizhan Cheng & Zequn Liu

  2. University of Chinese Academy of Sciences, Beijing, 100049, China

    Zequn Liu

Authors
  1. Daizhan Cheng
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  2. Zequn Liu
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Corresponding author

Correspondence to Daizhan Cheng.

Additional information

This work was supported in part by the National Natural Science Foundation of China (Nos. 61733018, 61333001, 61773371).

Daizhan CHENG (SM’01-F’06) received the B.Sc. degree from Department of Mechanics, Tsinghua University, in 1970, received the M.Sc. degree from Graduate School of Chinese Academy of Sciences in 1981, the Ph.D. degree from Washington University, St. Louis, in 1985. Since 1990, he is a Professor with Institute of Systems Science, Academy of Mathematics and Systems Science, Chinese Academy of Sciences. He is the author/coauthor of over 200 journal papers, 9 books and 100 conference papers. He was Associate Editor of the International Journal of Mathematical Systems, Estimation and Control (1990–1993); Automatica (1999–2002); the Asian Journal of Control (2001–2004); Subject Editor of the International Journal of Robust and Nonlinear Control (2005–2008). He is currently Editor-in-Chief of the J. Control Theory and Applications and Deputy Editor-in-Chief of Control and Decision. He was the Chairman of IEEE CSS Beijing Chapter (2006–2008), Chairman of Technical Committee on Control Theory, Chinese Association of Automation, Program Committee Chair of annual Chinese Control Conference (2003–2010), IEEE Fellow (2005–) and IFAC Fellow (2008–). Prof. Cheng’s research interests include nonlinear system control, hamiltonian system, numerical method in system analysis and control, complex systems.

Zequn LIU received the B.Sc. degree in Mathematics and Applied Mathematics from Shandong University, Jinan, China, in 2011. He is currently a Ph.D. candidate in Academy of Mathematics and Systems Science, Chinese Academy of Sciences. His research interests include game theory and Boolean control networks.

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Cheng, D., Liu, Z. A new semi-tensor product of matrices. Control Theory Technol. 17, 4–12 (2019). https://doi.org/10.1007/s11768-019-8161-2

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  • DOI: https://doi.org/10.1007/s11768-019-8161-2

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