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August  2021, 29(3): 2249-2267. doi: 10.3934/era.2020114

Solvability of the matrix equation $ AX^{2} = B $ with semi-tensor product

1. 

School of Mathematics, Shandong University, Jinan 250100, China

2. 

School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, China

* Corresponding author: Jin Wang

Received  May 2020 Revised  August 2020 Published  August 2021 Early access  October 2020

Fund Project: The first author is supported by ZR2019MF002, NSFC 11971181, NNSF 61773371 and 61877036

We investigate the solvability of the matrix equation $ AX^{2} = B $ in which the multiplication is the semi-tensor product. Then compatible conditions on the matrices $ A $ and $ B $ are established in each case and necessary and sufficient condition for the solvability is discussed. In addition, concrete methods of solving the equation are provided.

Citation: Jin Wang, Jun-E Feng, Hua-Lin Huang. Solvability of the matrix equation $ AX^{2} = B $ with semi-tensor product. Electronic Research Archive, 2021, 29 (3) : 2249-2267. doi: 10.3934/era.2020114
References:
[1]

G.-B. Cai and C.-H. Hu, Solving periodic Lyapunov matrix equations via finite steps iteration, IET Control Theory Appl., 6 (2012), 2111-2119.  doi: 10.1049/iet-cta.2011.0560.  Google Scholar

[2] D. Z. Cheng, Matrix and Polynomial Approach to Dynamics Control Systems, Science Press, Beijing, 2002.   Google Scholar
[3]

D. Cheng, H. Qi and Y. Zhao, An Introduction to Semi-tensor Product of Matrices and its Application, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/8323.  Google Scholar

[4]

D. Z. ChengT. Xu and H. S. Qi, Evolutionarily stable strategy of networked evloutionary games, IEEE Transactions on Neural Networks and Learning, 25 (2014), 1335-1345.   Google Scholar

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D. Cheng and Y. Zhao, Semi-tensor product of matrices: A new convenient tool, Chinese Science Bulletin, 56 (2011), 2664-2674.   Google Scholar

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H. FanJ.-E. FengM. Meng and B. Wang, General decomposition of fuzzy relations: Semi-tensor product approach, Fuzzy Sets and Systems, 384 (2020), 75-90.  doi: 10.1016/j.fss.2018.12.012.  Google Scholar

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J.-E. Feng, J. Yao and P. Cui, Singular Boolean network: Semi-tensor product approach, Sci. China Inf. Sci., 56 (2013), 112203, 14 pp. doi: 10.1007/s11432-012-4666-8.  Google Scholar

[8] B. Gao, Study on Several Kinds of Cryptographic Algorithm Based on the Semi-Tensor Product, Beijing Jiaotong University Press, 2014.   Google Scholar
[9] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511840371.  Google Scholar
[10] L. Huang, The Linear Algebra System and Control Science, Science Press, Beijing, 2004.   Google Scholar
[11]

M. Jamshidi, An overview on the solutions of the algebraic matrix Riccati equation and related problems, Large Scale Systems, 1 (1980), 167-192.   Google Scholar

[12]

G. G. Jesus, Block Toeplitz Matrices: Asymptotic Results and Applications, Now Publishers, Hanover, 2012. Google Scholar

[13]

P. Jiang, Y. Z. Wang and R. M. Xu, Mobile Robot Odor Source Localization Via Semi-Tensor Product, The Thirty-Fourth China Conference on Control, Hangzhou, 2015. Google Scholar

[14]

B. M. Kolundzija, Electromagnetic modeling of composite metallic and dielectric structures, Microwave Theory Tech, 47 (1999), 1021-1032.   Google Scholar

[15]

V. B. Larin, On solution of the linear matrix equations, Journal of Automation and Information Sciences, 47 (2015), 1-9.  doi: 10.1615/JAutomatInfScien.v47.i9.10.  Google Scholar

[16] S. W. MeiF. Liu and A. C. Xue, A Tensor Product in Power System Transient Analysis Method, Tsinghua University Press, Beijing, 2010.   Google Scholar
[17]

H. NobuharaK. HirotaW. Pedrycz and S. Sessa, Two iterative methods of decomposition of a fuzzy relation for image compression/decompres-sion processing, Soft Comput, 8 (2004), 698-704.   Google Scholar

[18]

H. Nobuhara and W. Pedrycz, Fast solving method of fuzzy relational equation and its application to lossy image compression/reconstruction, IEEE Trans. Fuzzy Syst., 18 (2000), 325-334.   Google Scholar

[19]

G. W. Stagg and A. H. El-Abiad, Computer Methods in Power System Analysis, McGraw-Hill, New York, 1968. Google Scholar

[20]

M. XuY. Wang and A. Wei, Robust graph coloring based on the matrix semi-tensor product with application to examination timetabling, Control Theory Technol., 12 (2014), 187-197.  doi: 10.1007/s11768-014-0153-7.  Google Scholar

[21]

J. YaoJ.-E. Feng and M. Meng, On solutions of the matrix equation $AX=B$ with respect to semi-tensor product, J. Franklin Inst., 353 (2016), 1109-1131.  doi: 10.1016/j.jfranklin.2015.04.004.  Google Scholar

[22]

Y. YuJ.-E. FengJ. Pan and D. Cheng, Block decoupling of Boolean control networks, IEEE Trans. Automat. Control, 64 (2019), 3129-3140.  doi: 10.1109/TAC.2018.2880411.  Google Scholar

[23]

Y. Yuan, Solving the mixed Sylvester matrix equations by matrix decompositions, C. R. Math. Acad. Sci. Paris, 353 (2015), 1053-1059.  doi: 10.1016/j.crma.2015.08.010.  Google Scholar

show all references

References:
[1]

G.-B. Cai and C.-H. Hu, Solving periodic Lyapunov matrix equations via finite steps iteration, IET Control Theory Appl., 6 (2012), 2111-2119.  doi: 10.1049/iet-cta.2011.0560.  Google Scholar

[2] D. Z. Cheng, Matrix and Polynomial Approach to Dynamics Control Systems, Science Press, Beijing, 2002.   Google Scholar
[3]

D. Cheng, H. Qi and Y. Zhao, An Introduction to Semi-tensor Product of Matrices and its Application, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. doi: 10.1142/8323.  Google Scholar

[4]

D. Z. ChengT. Xu and H. S. Qi, Evolutionarily stable strategy of networked evloutionary games, IEEE Transactions on Neural Networks and Learning, 25 (2014), 1335-1345.   Google Scholar

[5]

D. Cheng and Y. Zhao, Semi-tensor product of matrices: A new convenient tool, Chinese Science Bulletin, 56 (2011), 2664-2674.   Google Scholar

[6]

H. FanJ.-E. FengM. Meng and B. Wang, General decomposition of fuzzy relations: Semi-tensor product approach, Fuzzy Sets and Systems, 384 (2020), 75-90.  doi: 10.1016/j.fss.2018.12.012.  Google Scholar

[7]

J.-E. Feng, J. Yao and P. Cui, Singular Boolean network: Semi-tensor product approach, Sci. China Inf. Sci., 56 (2013), 112203, 14 pp. doi: 10.1007/s11432-012-4666-8.  Google Scholar

[8] B. Gao, Study on Several Kinds of Cryptographic Algorithm Based on the Semi-Tensor Product, Beijing Jiaotong University Press, 2014.   Google Scholar
[9] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511840371.  Google Scholar
[10] L. Huang, The Linear Algebra System and Control Science, Science Press, Beijing, 2004.   Google Scholar
[11]

M. Jamshidi, An overview on the solutions of the algebraic matrix Riccati equation and related problems, Large Scale Systems, 1 (1980), 167-192.   Google Scholar

[12]

G. G. Jesus, Block Toeplitz Matrices: Asymptotic Results and Applications, Now Publishers, Hanover, 2012. Google Scholar

[13]

P. Jiang, Y. Z. Wang and R. M. Xu, Mobile Robot Odor Source Localization Via Semi-Tensor Product, The Thirty-Fourth China Conference on Control, Hangzhou, 2015. Google Scholar

[14]

B. M. Kolundzija, Electromagnetic modeling of composite metallic and dielectric structures, Microwave Theory Tech, 47 (1999), 1021-1032.   Google Scholar

[15]

V. B. Larin, On solution of the linear matrix equations, Journal of Automation and Information Sciences, 47 (2015), 1-9.  doi: 10.1615/JAutomatInfScien.v47.i9.10.  Google Scholar

[16] S. W. MeiF. Liu and A. C. Xue, A Tensor Product in Power System Transient Analysis Method, Tsinghua University Press, Beijing, 2010.   Google Scholar
[17]

H. NobuharaK. HirotaW. Pedrycz and S. Sessa, Two iterative methods of decomposition of a fuzzy relation for image compression/decompres-sion processing, Soft Comput, 8 (2004), 698-704.   Google Scholar

[18]

H. Nobuhara and W. Pedrycz, Fast solving method of fuzzy relational equation and its application to lossy image compression/reconstruction, IEEE Trans. Fuzzy Syst., 18 (2000), 325-334.   Google Scholar

[19]

G. W. Stagg and A. H. El-Abiad, Computer Methods in Power System Analysis, McGraw-Hill, New York, 1968. Google Scholar

[20]

M. XuY. Wang and A. Wei, Robust graph coloring based on the matrix semi-tensor product with application to examination timetabling, Control Theory Technol., 12 (2014), 187-197.  doi: 10.1007/s11768-014-0153-7.  Google Scholar

[21]

J. YaoJ.-E. Feng and M. Meng, On solutions of the matrix equation $AX=B$ with respect to semi-tensor product, J. Franklin Inst., 353 (2016), 1109-1131.  doi: 10.1016/j.jfranklin.2015.04.004.  Google Scholar

[22]

Y. YuJ.-E. FengJ. Pan and D. Cheng, Block decoupling of Boolean control networks, IEEE Trans. Automat. Control, 64 (2019), 3129-3140.  doi: 10.1109/TAC.2018.2880411.  Google Scholar

[23]

Y. Yuan, Solving the mixed Sylvester matrix equations by matrix decompositions, C. R. Math. Acad. Sci. Paris, 353 (2015), 1053-1059.  doi: 10.1016/j.crma.2015.08.010.  Google Scholar

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