On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application

Yimeng Xi; Zhihong Liu; Ying Li; Ruyu Tao; Tao Wang; Yimeng Xi; Zhihong Liu; Ying Li; Ruyu Tao; Tao Wang

doi:10.3934/math.20231427

AIMS Mathematics

2023, Volume 8, Issue 11: 27901-27923. doi: 10.3934/math.20231427

Previous Article Next Article

Research article

On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application

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

Received: 15 July 2023 Revised: 11 September 2023 Accepted: 18 September 2023 Published: 10 October 2023
MSC : 15A06

In this paper, we investigate the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints. With the help of $ \mathcal{L_C} $-representation and the properties of vector operator based on semi-tensor product of reduced biquaternion matrices, the reduced biquaternion matrix equation (1.1) can be transformed into linear equations. A systematic method, $ \mathcal{GH} $-representation, is proposed to decrease the number of variables of a special unknown reduced biquaternion matrix and applied to solve the least squares problem of linear equations. Meanwhile, we give the necessary and sufficient conditions for the compatibility of reduced biquaternion matrix equation (1.1) under sub-matrix constraints. Numerical examples are given to demonstrate the results. The method proposed in this paper is applied to color image restoration.
- reduced biquaternion,
- sub-matrix constraints,
- $ \mathcal{GH} $-representation,
- $ \mathcal{L_C} $-representation
Citation: Yimeng Xi, Zhihong Liu, Ying Li, Ruyu Tao, Tao Wang. On the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints and its application[J]. AIMS Mathematics, 2023, 8(11): 27901-27923. doi: 10.3934/math.20231427

Related Papers:

Abstract

In this paper, we investigate the mixed solution of reduced biquaternion matrix equation $ \sum\limits_{i = 1}^nA_iX_iB_i = E $ with sub-matrix constraints. With the help of $ \mathcal{L_C} $-representation and the properties of vector operator based on semi-tensor product of reduced biquaternion matrices, the reduced biquaternion matrix equation (1.1) can be transformed into linear equations. A systematic method, $ \mathcal{GH} $-representation, is proposed to decrease the number of variables of a special unknown reduced biquaternion matrix and applied to solve the least squares problem of linear equations. Meanwhile, we give the necessary and sufficient conditions for the compatibility of reduced biquaternion matrix equation (1.1) under sub-matrix constraints. Numerical examples are given to demonstrate the results. The method proposed in this paper is applied to color image restoration.

References

[1]	Z. Al-Zhour, Some new linear representations of matrix quaternions with some applicationss, J. King Saud Univ. Sci., 31 (2019), 42–47. https://doi.org/10.1016/j.jksus.2017.05.017 doi: 10.1016/j.jksus.2017.05.017
[2]	Z. Al-Zhour, The general solutions of singular and non-singular matrix fractional time-varying descriptor systems with constant coefficient matrices in Caputo sense, Alex. Eng. J., 55 (2016), 1675–1681. https://doi.org/10.1016/j.aej.2016.02.024 doi: 10.1016/j.aej.2016.02.024
[3]	A. El-Ajou, Z. Al-Zhour, A vector series solution for a class of hyperbolic system of Caputo time-fractional partial differential equations with variable coefficients, Front. Phys., 9 (2021), 5252501. https://doi.org/10.3389/fphy.2021.525250 doi: 10.3389/fphy.2021.525250
[4]	A. Sarhan, A. Burqan, R. Saadeh, Z. Al-Zhour, Analytical solutions of the nonlinear time-fractional coupled Boussinesq-burger equations using Laplace residual power series technique, Fractal Fract., 6 (2022), 631. https://doi.org/10.3390/fractalfract6110631 doi: 10.3390/fractalfract6110631
[5]	N. Le Bihan, J. Mars, Singular value decomposition of quaternion matrices: A new tool for vector-sensor signal processing, Signal Process., 84 (2004), 1177–1199. https://doi.org/10.1016/j.sigpro.2004.04.001 doi: 10.1016/j.sigpro.2004.04.001
[6]	S. De Leo, G. Scolarici, Right eigenvalue equation in quaternionic quantum mechanics, J. Phys. A: Math. Gen., 33 (2000), 2971–2995. https://doi.org/10.1088/0305-4470/33/15/306 doi: 10.1088/0305-4470/33/15/306
[7]	F. X. Zhang, M. S. Wei, Y. Li, J. L. Zhao, Special least squares solutions of the quaternion matrix equation $AX = B$ with applications, Appl. Math. Comput., 270 (2015), 425–433. https://doi.org/10.1016/j.amc.2015.08.046 doi: 10.1016/j.amc.2015.08.046
[8]	S. F. Yuan, Q. W. Wang, X. F. Duan, On solutions of the quaternion matrix equation $AX = B$ and their applications in color image restoration, Appl. Math. Comput., 221 (2013), 10–20. https://doi.org/10.1016/j.amc.2013.05.069 doi: 10.1016/j.amc.2013.05.069
[9]	L. Fortuna, G. Muscato, M. G. Xibilia, A comparison between HMLP and HRBF for attitude control, IEEE. T. Neural Networ., 12 (2001), 318–328. https://doi.org/10.1109/72.914526 doi: 10.1109/72.914526
[10]	T. Li, Q. W. Wang, Structure preserving quaternion full orthogonalization method with applications, Numer. Linear Algebr., 30 (2023), e2495. https://doi.org/10.1002/nla.2495 doi: 10.1002/nla.2495
[11]	A. Ben-Israel, T. N. E. Greville, Generalized inverses: Theory and applications, New York: Springer, 2003. https://doi.org/10.1007/b97366
[12]	Q. W. Wang, H. S. Zhang, S. W. Yu, On solutions to the quaternion matrix equation $AXB+CYD = E$, Electron. J. Linear Al., 17 (2008), 343–358. https://doi.org/10.13001/1081-3810.1268 doi: 10.13001/1081-3810.1268
[13]	X. L. Xu, Q. W. Wang, The consistency and the general common solution to some quaternion matrix equations, Ann. Funct. Anal., 14 (2023), 53. https://doi.org/10.1007/s43034-023-00276-y doi: 10.1007/s43034-023-00276-y
[14]	S. F. Yuan, Q. W. Wang, Y. B. Yu, Y. Tian, On Hermitian solutions of the split quaternion matrix equation $AXB+ CXD = E$, Adv. Appl. Clifford Algebras, 27 (2017), 3235–3258. https://doi.org/10.1007/s00006-017-0806-y doi: 10.1007/s00006-017-0806-y
[15]	C. Q. Song, G. L. Chen, On solutions of matrix equation $XF-AX = C$ and $XF-A\widetilde{X} = C$ over quaternion field, J. Appl. Math. Comput., 37 (2011), 57–68. https://doi.org/10.1007/s12190-010-0420-9 doi: 10.1007/s12190-010-0420-9
[16]	A. P. Liao, Z. Z. Bai, Least-squares solution of $AXB = D$ over symmetric positive semidefinite matrices X, J. Comput. Math., 21 (2003), 175–182.
[17]	A. P. Liao, Z. Z. Bai, Y. Lei, Best approximate solution of matrix equation $AXB+CYD = E$, SIAM J. Matrix Anal. Appl., 27 (2005), 675–688. https://doi.org/10.1137/040615791 doi: 10.1137/040615791
[18]	B. Y. Ren, Q. W. Wang, X. Y. Chen, The $\eta$-anti-Hermitian solution to a system of constrained matrix equations over the generalized segre quaternion algebra, Symmetry, 15 (2003), 592. https://doi.org/10.3390/sym15030592 doi: 10.3390/sym15030592
[19]	H. D. Schtte, J. Wenzel, Hypercomplex numbers in digital signal processing, 1990 IEEE International Symposium on Circuits and Systems (ISCAS), 2 (1990), 1557–1560. https://doi.org/10.1109/ISCAS.1990.112431 doi: 10.1109/ISCAS.1990.112431
[20]	K. Ueda, S. I. Takahashi, Digital filters with hypercomplex coefficients, Electronics and Communications in Japan (Part III: Fundamental Electronic Science), 76 (1993), 85–98. https://doi.org/10.1002/ecjc.4430760909 doi: 10.1002/ecjc.4430760909
[21]	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. https://doi.org/10.1109/TSP.2004.828901 doi: 10.1109/TSP.2004.828901
[22]	S. C. Pei, J. H. Chang, J. J. Ding, M. Y. Chen, Eigenvalues and singular value decompositions of reduced biquaternion matrices, IEEE T. Circ. Syst., 55 (2008), 2673–2685. https://doi.org/10.1109/TCSI.2008.920068 doi: 10.1109/TCSI.2008.920068
[23]	S. F. Yuan, Y. Tian, M. Z. Li, On Hermitian solutions of the reduced biquaternion matrix equation $(AXB, CXD) = (E, G)$, Linear Multilinear A., 68 (2020), 1355–1373. https://doi.org/10.1080/03081087.2018.1543383 doi: 10.1080/03081087.2018.1543383
[24]	H. H. Kösal, 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. https://doi.org/10.1080/09500340.2019.1676474 doi: 10.1080/09500340.2019.1676474
[25]	X. Y. Chen, Q. W. Wang, The $\eta$-(anti-)Hermitian solution to a constrained Sylvester-type generalized commutative quaternion matrix equation, Banach J. Math. Anal., 17 (2023), 40. https://doi.org/10.1007/s43037-023-00262-5 doi: 10.1007/s43037-023-00262-5
[26]	L. Gong, X. Hu, L. Zhang, The expansion problem of anti-symmetric matrix under a linear constraint and the optimal approximation J. Comput. Appl. Math., 197 (2006), 44–52. https://doi.org/10.1016/j.cam.2005.10.021 doi: 10.1016/j.cam.2005.10.021
[27]	L. Zhao, X. Hu, L. Zhang, Least squares solutions to $AX = B$ for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation, Linear Algebra Appl., 428 (2008), 871–880. https://doi.org/10.1016/j.laa.2007.08.019 doi: 10.1016/j.laa.2007.08.019
[28]	J. F. Li, X. Y. Hu, L. Zhang, The submatrix constraint problem of matrix equation AXB+CYD = E, Appl. Math. Comput., 215 (2009), 2578–2590. https://doi.org/10.1016/j.amc.2009.08.051 doi: 10.1016/j.amc.2009.08.051
[29]	D. Z. Cheng, H. S. Qi, Z. Q. Li, Analysis and control of Boolean networks: A semi-tensor product approach, London: Springer, 2011. https://doi.org/10.1007/978-0-85729-097-7
[30]	D. Cheng, H. Qi, Z. Li, J. B. Liu, Stability and stabilization of Boolean networks, Int. J. Robust Nonlin., 21 (2011), 134–156. https://doi.org/10.1002/rnc.1581 doi: 10.1002/rnc.1581
[31]	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. https://doi.org/10.1049/iet-cta.2016.1659 doi: 10.1049/iet-cta.2016.1659
[32]	D. Z. Cheng, H. S. Qi, Controllability and observability of Boolean control networks, Automatica, 45 (2009), 1659–1667. https://doi.org/10.1016/j.automatica.2009.03.006 doi: 10.1016/j.automatica.2009.03.006
[33]	W. Ding, Y. Li, D. Wang, A real method for solving quaternion matrix equation $X-A\widehat{X}B = C$ based on semi-tensor product of matrices, Adv. Appl. Clifford Algebras, 31 (2021), 78. https://doi.org/10.1007/s00006-021-01180-1 doi: 10.1007/s00006-021-01180-1
[34]	W. Ding, Y. Li, A. L. Wei, Z. H. Liu, Solving reduced biquaternion matrices equation $\sum\limits_{i = 1}^nA_iXB_i = C$ with special structure based on semi tensor product of matrices, AIMS Mathematics, 7 (2022), 3258–3276. https://doi.org/10.3934/math.2022181 doi: 10.3934/math.2022181
[35]	D. Wang, Y. Li, W. Ding, Several kinds of special least squares solutions to quaternion matrix equation $AXB = C$, J. Appl. Math. Comput., 68 (2022), 1881–1899. https://doi.org/10.1007/s12190-021-01591-0 doi: 10.1007/s12190-021-01591-0
[36]	W. H. Zhang, B. S. Chen, $\mathcal{H}$-representation and applications to generalized Lyapunov equations and linear Stochastic systems, IEEE T. Automat Contr., 57 (2012), 3009–3022. https://doi.org/10.1109/TAC.2012.2197074 doi: 10.1109/TAC.2012.2197074
[37]	D. Z. Cheng, From dimension-free matrix theory to cross-dimensional dynamic systems, Academic Press, 2019. https://doi.org/10.1016/C2018-0-02653-5
[38]	G. H. Golub, C. F. Van Loan, Matrix computation, 4th edn, Baltimore: The Johns Hopkins University Press, 2013.
[39]	S. Gai, G. W. Yang, M. H. Wan, L. Wang, Denoising color images by reduced quaternion matrix singular value decomposition, Multidim. Syst. Sign. Process., 26 (2015), 307–320. https://doi.org/10.1007/s11045-013-0268-x doi: 10.1007/s11045-013-0268-x
[40]	S. Gai, M. H. Wan, L. Wang, C. H. Yang, Reduced quaternion matrix for color texture classification, Neural Comput. Appl., 25 (2014), 945–954. https://doi.org/10.1007/s00521-014-1578-0 doi: 10.1007/s00521-014-1578-0

Reader Comments

Your name:*

Email:*
© 2023 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)