Least squares solutions of matrix equation $ AXB = C $ under semi-tensor product

Jin Wang; Jin Wang

doi:10.3934/era.2024136

Electronic Research Archive

2024, Volume 32, Issue 5: 2976-2993. doi: 10.3934/era.2024136

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Research article

Least squares solutions of matrix equation $ AXB = C $ under semi-tensor product

Jin Wang ^,

Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China

Received: 24 January 2024 Revised: 22 March 2024 Accepted: 12 April 2024 Published: 19 April 2024

This paper mainly studies the least-squares solutions of matrix equation $ AXB = C $ under a semi-tensor product. According to the definition of the semi-tensor product, the equation is transformed into an ordinary matrix equation. Then, the least-squares solutions of matrix-vector and matrix equations respectively investigated by applying the derivation of matrix operations. Finally, the specific form of the least-squares solutions is given.
- matrix equations,
- semi-tensor product,
- least squares solution
Citation: Jin Wang. Least squares solutions of matrix equation $ AXB = C $ under semi-tensor product[J]. Electronic Research Archive, 2024, 32(5): 2976-2993. doi: 10.3934/era.2024136

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Abstract

This paper mainly studies the least-squares solutions of matrix equation $ AXB = C $ under a semi-tensor product. According to the definition of the semi-tensor product, the equation is transformed into an ordinary matrix equation. Then, the least-squares solutions of matrix-vector and matrix equations respectively investigated by applying the derivation of matrix operations. Finally, the specific form of the least-squares solutions is given.

References

[1]	H. Lin, T. Maekawa, C. Deng, Survey on geometric iterative methods and their applications, Comput. Aided Des., 95 (2018), 40–51. https://doi.org/10.1016/j.cad.2017.10.002 doi: 10.1016/j.cad.2017.10.002
[2]	M. Liu, B. Li, Q. Guo, C. Zhu, P. Hu, Y. Shao, Progressive iterative approximation for regularized least square bivariate B-spline surface fitting, J. Comput. Appl. Math., 327 (2018), 175–187. https://doi.org/10.1016/j.cam.2017.06.013 doi: 10.1016/j.cam.2017.06.013
[3]	Z. Tian, Y. Wang, N. C. Wu, Z. Liu, On the parameterized two-step iteration method for solving the matrix equation $AXB = C$, Appl. Math. Comput., 464 (2024), 128401. https://doi.org/10.1016/j.amc.2023.128401 doi: 10.1016/j.amc.2023.128401
[4]	N. C. Wu, C. Z. Liu, Q. Zuo, On the Kaczmarz methods based on relaxed greedy selection for solving matrix equation $AXB = C$, J. Comput. Appl. Math., 413 (2022), 114374. https://doi.org/10.1016/j.cam.2022.114374 doi: 10.1016/j.cam.2022.114374
[5]	Z. Tian, X. Li, Y. Dong, Z. Liu, Some relaxed iteration methods for solving matrix equation $AXB = C$, Appl. Math. Comput., 403 (2021), 126189. https://doi.org/10.1016/j.amc.2021.126189 doi: 10.1016/j.amc.2021.126189
[6]	F. Chen, T. Li, Two-step AOR iteration method for the linear matrix equation $AXB = C$, Comput. Appl. Math., 40 (2021), 89. https://doi.org/10.1007/s40314-021-01472-z doi: 10.1007/s40314-021-01472-z
[7]	Z. Liu, Z. Li, C. Ferreira, Y. Zhang, Stationary splitting iterative methods for the matrix equation $AXB = C$, Appl. Math. Comput., 378 (2020), 125195. https://doi.org/10.1016/j.amc.2020.125195 doi: 10.1016/j.amc.2020.125195
[8]	Y. Xu, Linear Algebra and Matrix Theory, Beijing, Higher Education Press, 1992.
[9]	Y. Ding, About matrix equations $AXB = C$, Math. Bull., 2 (1997), 43–45.
[10]	Q. Li, Numeric Analysis, Tsinghua University Press, Beijing, 2008.
[11]	Z. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation $AXB = C$, Appl. Math. Comput., 170 (2005), 711–723. https://doi.org/10.1016/j.amc.2004.12.032 doi: 10.1016/j.amc.2004.12.032
[12]	Y. X. Peng, X. Y. Hu, L. Zhang, An iteration mathod for the symmetric solutions and the optimal approximation solution of the matrix equation $AXB = C$, Appl. Math. Comput., 3 (2005), 763–777. https://doi.org/10.1016/j.amc.2003.11.030 doi: 10.1016/j.amc.2003.11.030
[13]	Y. Yuan, H. Dai, Generalized reflexive solutions of the matrix equation $AXB = C$ and an associated optimal approximation problem, Math. Appl., 6 (2008), 1643–1649. https://doi.org/10.1016/j.camwa.2008.03.015 doi: 10.1016/j.camwa.2008.03.015
[14]	G. X. Huang, F. Ying, K. Guo, An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation $AXB = C$, J. Comput. Appl. Math., 212 (2008), 231–244. https://doi.org/10.1016/j.cam.2006.12.005 doi: 10.1016/j.cam.2006.12.005
[15]	Y. Zhang, An iterative method for the bisymmetric least-squares solutions and the optimal approximation of the matrix equation $AXB = C$, Chin. J. Eng. Math., 4 (2009), 753–756. https://doi.org/10.3969/j.issn.1005-3085.2009.04.023 doi: 10.3969/j.issn.1005-3085.2009.04.023
[16]	Z. Peng, New matrix iterative methods for constraint solutions of the matrix equation $AXB = C$, J. Comput. Appl. Math., 3 (2010), 726–735. https://doi.org/10.1016/j.cam.2010.07.001 doi: 10.1016/j.cam.2010.07.001
[17]	X. Wang, Y. Li, L. Dai, On Hermitian and skew-Hermitian splitting iteration methods for the linear matrix $AXB = C$, Comput. Math. Appl. Int. J., 65 (2013), 657–664. https://doi.org/10.1016/j.camwa.2012.11.010 doi: 10.1016/j.camwa.2012.11.010
[18]	T. Xu, M. Tian, Z. Liu, T. Xu, The Jacobi and Gauss-Seidel-type iteration methods for the matrix equation $AXB = C$, Appl. Math. Comput., 292 (2017), 63–75. https://doi.org/10.1016/j.amc.2016.07.026 doi: 10.1016/j.amc.2016.07.026
[19]	D. Cheng, Semi-tensor product of matrices and its application to Morgen's problem, Sci. China Inf. Sci., 44 (2001), 195–212. https://doi.org/10.1007/BF02714570 doi: 10.1007/BF02714570
[20]	D. Cheng, Y. Zhao, Semi-tensor product of matrices–-A convenient new tool, Sci. China Inf. Sci., 56 (2011), 2664–2674. https://doi.org/10.1360/972011-1262 doi: 10.1360/972011-1262
[21]	M. Ramadan, A. Bayoumi, Explicit and iterative methods for solving the matrix equation $AV + BW = EVF + C$, Asian J. Control, 13 (2015), 1070–1080. https://doi.org/10.1002/asjc.965 doi: 10.1002/asjc.965
[22]	H. Li, G. Zhao, M. Meng, J. Feng, A survey on applications of semi-tensor product method in engineering, Sci. China Inf. Sci., 61 (2018), 28–44. https://doi.org/10.1007/s11432-017-9238-1 doi: 10.1007/s11432-017-9238-1
[23]	J. E. Feng, J. Yao, P. Cui, Singular Boolean network: Semi-tensor product approach, Sci. China Inf. Sci., 56 (2013), 1–14. https://doi.org/10.1007/s11432-012-4666-8 doi: 10.1007/s11432-012-4666-8
[24]	Y. Yu, J. Feng, J. Pan, Ordinal potential game and its application in agent wireless networks, Control Decis., 32 (2017), 393–402. https://doi.org/10.13195/j.kzyjc.2016.0183 doi: 10.13195/j.kzyjc.2016.0183
[25]	M. Xu, Y. Wang, A. Wei, Robust graph coloring based on the matrix semi-tensor product with application to examination time tabling, Control Theory Technol., 2 (2014), 187–197. https://doi.org/10.1007/s11768-014-0153-7 doi: 10.1007/s11768-014-0153-7
[26]	H. Fan, J. Feng, M, Meng, B. Wang, General decomposition of fuzzy relations: Semi-tensor product approach, Fuzzy Sets Syst., 384 (2020), 75–90. https://doi.org/10.1016/j.fss.2018.12.012 doi: 10.1016/j.fss.2018.12.012
[27]	Y. Yan, D. Cheng, J. E. Feng, H. Li, J. Yue, Survey onapplications of algebraic statespace theory of logicalsystems to finite statemachines, Sci. China Inf. Sci., 66 (2023), 111201. https://doi.org/10.1007/s11432-022-3538-4 doi: 10.1007/s11432-022-3538-4
[28]	J. Yao, J. Feng, M. Meng, On solutions of the matrix equation $AX = B$ with respect to semitensor product, J. Franklin Inst., 353 (2016), 1109–1131. https://doi.org/10.1016/j.jfranklin.2015.04.004 doi: 10.1016/j.jfranklin.2015.04.004
[29]	J. Wang, J. Feng, H. Huang, Solvability of the matrix equation $AX^2 = B$ with semi-tensor product, Electorn. Res. Arch., 29 (2020), 2249–2267. https://doi.org/10.3934/era.2020114 doi: 10.3934/era.2020114
[30]	J. Wang, On Solutions of the matrix equation $A\circ_{l}X = B$ with respect to MM-2 semi-tensor product, J. Math., 2021 (2021), 651434. https://doi.org/10.1155/2021/6651434 doi: 10.1155/2021/6651434
[31]	N. Wang, Solvability of the sylvester equation $AX-XB = C$ under left semi-tensor product, Math. Modell. Control, 2 (2022), 81–89. http://dx.doi.org/10.3934/mmc.2022010 doi: 10.3934/mmc.2022010
[32]	Y. Li, H. Li, X. Ding, G. Zhao, Leader-follower consensus of multiagent systems with time delays over finite fields, IEEE Trans. Cybern., 49 (2018), 3203–3208. https://doi.org/10.1109/TCYB.2018.2839892 doi: 10.1109/TCYB.2018.2839892
[33]	Z. Ji, J. Li, X. Zhou, F. Duan, T. Li, On solutions of matrix equation $AXB = C$ under semi-tensor product, Linear Multilinear Algebra, (2019), 1650881. https://doi.org/10.1080/03081087.2019.1650881
[34]	R. Horn, C. Johnson, Topicsin Matrix Analysis, Cambridge University Press, New York, 1991.

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