Solving reduced biquaternion matrices equation $ \sum\limits_{i = 1}^{k}A_iXB_i = C $ with special structure based on semi-tensor product of matrices

Wenxv Ding; Ying Li; Anli Wei; Zhihong Liu; Wenxv Ding; Ying Li; Anli Wei; Zhihong Liu

doi:10.3934/math.2022181

AIMS Mathematics

2022, Volume 7, Issue 3: 3258-3276. doi: 10.3934/math.2022181

Previous Article Next Article

Research article Special Issues

Solving reduced biquaternion matrices equation $ \sum\limits_{i = 1}^{k}A_iXB_i = C $ with special structure based on semi-tensor product of matrices

1.
College of Mathematical Sciences, Liaocheng University, Liaocheng 252000, China
2.
Research Center of Semi-tensor Product of Matrices: Theory and Applications, Liaocheng 252000, China

Received: 18 September 2021 Accepted: 22 November 2021 Published: 29 November 2021
MSC : 15A06

In this paper, we propose a real vector representation of reduced quaternion matrix and study its properties. By using this real vector representation, Moore-Penrose inverse, and semi-tensor product of matrices, we study some kinds of solutions of reduced biquaternion matrix equation (1.1). Several numerical examples show that the proposed algorithm is feasible at last.
- semi-tensor product of matrices,
- reduced biquaternion,
- matrix equation,
- real vector representation
Citation: Wenxv Ding, Ying Li, Anli Wei, Zhihong Liu. Solving reduced biquaternion matrices equation $ \sum\limits_{i = 1}^{k}A_iXB_i = C $ with special structure based on semi-tensor product of matrices[J]. AIMS Mathematics, 2022, 7(3): 3258-3276. doi: 10.3934/math.2022181

Related Papers:

Abstract

In this paper, we propose a real vector representation of reduced quaternion matrix and study its properties. By using this real vector representation, Moore-Penrose inverse, and semi-tensor product of matrices, we study some kinds of solutions of reduced biquaternion matrix equation (1.1). Several numerical examples show that the proposed algorithm is feasible at last.

References

[1]	S. C. Pei, J. H. Chang, J. J. Ding, Commutative reduced biquaternions and their Fourier transform for signal and image processing applications, IEEE T. Signal. Proces., 52 (2004), 2012–2031. doi: 10.1109/TSP.2004.828901. doi: 10.1109/TSP.2004.828901
[2]	H. D. Sch$\ddot{u}$tte, J. Wenzel, Hypercomplex numbers in digital signal processing, IEEE Int. Symp. Circuits Syst., 2 (1990), 1557–1560. doi: 10.1109/ISCAS.1990.112431. doi: 10.1109/ISCAS.1990.112431
[3]	S. C. Pei, J. H. Chang, J. J. Ding, M. Y. Chen, Eigenvalues and singular value decompositions of reduced biquaternion matrices, IEEE T. Circuits Syst. I, 55 (2008), 2673–2685. doi: 10.1109/TCSI.2008.920068. doi: 10.1109/TCSI.2008.920068
[4]	T. Isokawa, H. Nishimura, N. Matsui, Commutative quaternion and multistate hopfield neural networks, In: The 2010 international joint conference on neural networks, 2010. doi: 10.1109/IJCNN.2010.5596736.
[5]	H. H. K$\ddot{o}$sal, Least-squares solutions of the reduced biquaternion matrix equation $AX = B$ and their applications in colour image restoration, J. Mod. Optic., 66 (2019), 1802–1810. doi: 10.1080/09500340.2019.1676474. doi: 10.1080/09500340.2019.1676474
[6]	S. F. Yuan, Y. Tian, M. Z. Li, On Hermitian solutions of the reduced biquaternion matrix equation $(AXB, CXD) = (E, G)$, Linear Multlinear Algebra, 68 (2020), 1355–1373. doi: 10.1080/03081087.2018.1543383. doi: 10.1080/03081087.2018.1543383
[7]	I. I. Kyrchei, Determinantal representation of general and (skew-) Hermitian solutions to the generalized sylvester-type quaternion matrix equation, Abst. Appl. Anal., 2019 (2019), 5926832. doi: 10.1155/2019/5926832. doi: 10.1155/2019/5926832
[8]	I. I. Kyrchei, Determinantal representations of solutions and Hermitian solutions to some system of two-sided quaternion matrix equations, J. Math., 2018 (2018), 6294672. doi: 10.1155/2018/6294672. doi: 10.1155/2018/6294672
[9]	J. S. Respondek, Recursive numerical recipes for the high efficient inversion of the confluent Vandermonde matrices, Appl. Math. Comput., 225 (2013), 718–730. doi: 10.1016/j.amc.2013.10.018. doi: 10.1016/j.amc.2013.10.018
[10]	F. X. Zhang, M. S. Wei, Y. Li, J. L. Zhao, An efficient method for least squares problem of the quaternion matrix equation $X-A\hat{X}B = C$, Linear Multlinear Algebra, 2020, 1–13. doi: 10.1080/03081087.2020.1806197.
[11]	D. Z. Cheng, Q. H. Qi, Z. Q. Liu, From STP to game-based control, Sci. China Inform. Sci., 61 (2018), 010201. doi: 10.1007/s11432-017-9265-2. doi: 10.1007/s11432-017-9265-2
[12]	Y. Z. Wang, C. H. Zhang, Z. B. Liu, A matrix approach to graph maximum stable set and coloring problems with application to multi-agent systems, Automatica, 48 (2012), 1227–1236. doi: 10.1016/j.automatica.2012.03.024. doi: 10.1016/j.automatica.2012.03.024
[13]	J. Q. Lu, H. T. Li, Y. Liu, F. F. Li, Survey on semi-tensor product method with its applications in logical networks and other finite-valued systems, IET Control Theory Appl., 11 (2017), 2040–2047. doi: 10.1049/iet-cta.2016.1659. doi: 10.1049/iet-cta.2016.1659
[14]	D. Z. Cheng, H. S. Qi, Z. Q. Li, Analysis and control of Boolean networks: A semi-tensor product approach, London: Springer, 2011. doi: 10.1007/978-0-85729-097-7.
[15]	D. Z. Cheng, Z. Q. Liu, Z. H. Xu, T. L. Shen, Generalised semi-tensor product of matrices, IET Control Theory Appl., 14 (2020), 85–95. doi: 10.1049/iet-cta.2019.0337. doi: 10.1049/iet-cta.2019.0337
[16]	G. H. Golub, C. F. VanLoan, Matrix computations, 4 Eds., Baltimore MD: The Johns Hopkins University Press, 2013.

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)