### Electronic Research Archive

2021, Issue 3: 2249-2267. doi: 10.3934/era.2020114
Special Issues

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

• Received: 01 May 2020 Revised: 01 August 2020 Published: 19 October 2020
• Primary: 15A06; Secondary: 15A03, 15A04

• 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[J]. Electronic Research Archive, 2021, 29(3): 2249-2267. doi: 10.3934/era.2020114

### Related Papers:

• 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.

 [1] Solving periodic Lyapunov matrix equations via finite steps iteration. IET Control Theory Appl. (2012) 6: 2111-2119. [2] (2002) Matrix and Polynomial Approach to Dynamics Control Systems.Science Press. [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 [4] Evolutionarily stable strategy of networked evloutionary games. IEEE Transactions on Neural Networks and Learning (2014) 25: 1335-1345. [5] Semi-tensor product of matrices: A new convenient tool. Chinese Science Bulletin (2011) 56: 2664-2674. [6] General decomposition of fuzzy relations: Semi-tensor product approach. Fuzzy Sets and Systems (2020) 384: 75-90. [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 [8] (2014) Study on Several Kinds of Cryptographic Algorithm Based on the Semi-Tensor Product.Beijing Jiaotong University Press. [9] (1991) Topics in Matrix Analysis.Cambridge University Press. [10] (2004) The Linear Algebra System and Control Science.Science Press. [11] An overview on the solutions of the algebraic matrix Riccati equation and related problems. Large Scale Systems (1980) 1: 167-192. [12] G. G. Jesus, Block Toeplitz Matrices: Asymptotic Results and Applications, Now Publishers, Hanover, 2012. [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. [14] Electromagnetic modeling of composite metallic and dielectric structures. Microwave Theory Tech (1999) 47: 1021-1032. [15] On solution of the linear matrix equations. Journal of Automation and Information Sciences (2015) 47: 1-9. [16] (2010) A Tensor Product in Power System Transient Analysis Method.Tsinghua University Press. [17] Two iterative methods of decomposition of a fuzzy relation for image compression/decompres-sion processing. Soft Comput (2004) 8: 698-704. [18] Fast solving method of fuzzy relational equation and its application to lossy image compression/reconstruction. IEEE Trans. Fuzzy Syst. (2000) 18: 325-334. [19] G. W. Stagg and A. H. El-Abiad, Computer Methods in Power System Analysis, McGraw-Hill, New York, 1968. [20] Robust graph coloring based on the matrix semi-tensor product with application to examination timetabling. Control Theory Technol. (2014) 12: 187-197. [21] On solutions of the matrix equation $AX=B$ with respect to semi-tensor product. J. Franklin Inst. (2016) 353: 1109-1131. [22] Block decoupling of Boolean control networks. IEEE Trans. Automat. Control (2019) 64: 3129-3140. [23] Solving the mixed Sylvester matrix equations by matrix decompositions. C. R. Math. Acad. Sci. Paris (2015) 353: 1053-1059.
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142