Solving two-sided Sylvester quaternionic matrix equations: Theoretical insights, computational implementation, and practical applications

Abdur Rehman; Cecilia Castro; Víctor Leiva; Muhammad Zia Ur Rahman; Carlos Martin-Barreiro; Abdur Rehman; Cecilia Castro; Víctor Leiva; Muhammad Zia Ur Rahman; Carlos Martin-Barreiro

doi:10.3934/math.2025702

AIMS Mathematics

2025, Volume 10, Issue 7: 15663-15697. doi: 10.3934/math.2025702

Previous Article Next Article

Research article Topical Sections

Solving two-sided Sylvester quaternionic matrix equations: Theoretical insights, computational implementation, and practical applications

1.
Department of Basic Sciences and Humanities, University of Engineering and Technology, Lahore, Faisalabad Campus, Faisalabad, Pakistan
2.
Centre of Mathematics, Universidade do Minho, Braga, Portugal
3.
School of Industrial Engineering, Universidad Catolica de Valparaíso, Valparaíso, Chile
4.
Department of Electrical Electronics and Telecommunication Engineering, University of Engineering and Technology, Lahore, Faisalabad Campus, Faisalabad, Pakistan
5.
Facultad de Ingeniería, Universidad Espíritu Santo, Samborondón, Ecuador

Received: 12 March 2025 Revised: 28 May 2025 Accepted: 06 June 2025 Published: 09 July 2025
MSC : 15A33, 15A09, 65F05

In this article, we investigate a class of quaternionic functional equations of the two-sided Sylvester type, which arise in areas such as control theory, robotics, signal processing, and image analysis. Although quaternionic matrix equations have been extensively studied, two-sided Sylvester systems remain particularly challenging due to the noncommutativity of quaternion multiplication and the increased structural complexity they entail. We derive general solutions to these systems by establishing the necessary and sufficient conditions for solvability, unifying and by extending previous theoretical results. We propose an efficient algorithm to compute the general solution in both full-rank and rank-deficient cases, using Moore-Penrose inverses and projection operators The practical interest of our method is demonstrated through applications in perturbation theory, image processing, and robust control. We present numerical examples to validate the proposed approach, including a case study involving a multi-joint robotic manipulator. These results highlight both the theoretical relevance and computational advantages of the proposed method.
Citation: Abdur Rehman, Cecilia Castro, Víctor Leiva, Muhammad Zia Ur Rahman, Carlos Martin-Barreiro. Solving two-sided Sylvester quaternionic matrix equations: Theoretical insights, computational implementation, and practical applications[J]. AIMS Mathematics, 2025, 10(7): 15663-15697. doi: 10.3934/math.2025702

Related Papers:

Abstract

In this article, we investigate a class of quaternionic functional equations of the two-sided Sylvester type, which arise in areas such as control theory, robotics, signal processing, and image analysis. Although quaternionic matrix equations have been extensively studied, two-sided Sylvester systems remain particularly challenging due to the noncommutativity of quaternion multiplication and the increased structural complexity they entail. We derive general solutions to these systems by establishing the necessary and sufficient conditions for solvability, unifying and by extending previous theoretical results. We propose an efficient algorithm to compute the general solution in both full-rank and rank-deficient cases, using Moore-Penrose inverses and projection operators The practical interest of our method is demonstrated through applications in perturbation theory, image processing, and robust control. We present numerical examples to validate the proposed approach, including a case study involving a multi-joint robotic manipulator. These results highlight both the theoretical relevance and computational advantages of the proposed method.

References

[1]	W. R. Hamilton, On quaternions; or on a new system of imaginaries in algebra, Philosophical Magazine, 25 (1844), 489–495. https://doi.org/10.1080/14786444408645047
[2]	S. L. Adler, Quaternionic quantum mechanics and quantum fields, New York: Oxford University Press, 1995. https://doi.org/10.1093/oso/9780195066432.001.0001
[3]	C. C. Took, D. P. Mandic, Augmented second-order statistics of quaternion random signals, Signal Proce., 91 (2011), 214–224. https://doi.org/10.1016/j.sigpro.2010.06.024 doi: 10.1016/j.sigpro.2010.06.024
[4]	S. D. 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
[5]	F. Z. Zhang, Quaternions and matrices of quaternions, Linear Algebra Appl., 251 (1997), 21–57. https://doi.org/10.1016/0024-3795(95)00543-9 doi: 10.1016/0024-3795(95)00543-9
[6]	V. L. Syrmos, F. L. Lewis, Coupled and constrained Sylvester equations in system design, Circ. Syst. Signal Pr., 13 (1994), 663–694. https://doi.org/10.1007/BF02523122 doi: 10.1007/BF02523122
[7]	Q. W. Wang, Z. H. He, Systems of coupled generalized Sylvester matrix equations, Automatica, 50 (2014), 2840–2844. https://doi.org/10.1016/j.automatica.2014.10.033 doi: 10.1016/j.automatica.2014.10.033
[8]	Z. H. He, Q. W. Wang, The $\eta$-biHermitian solution to a system of real quaternion matrix equations, Linear Multilinear A., 62 (2013), 1509–1528. https://doi.org/10.1080/03081087.2013.839667 doi: 10.1080/03081087.2013.839667
[9]	Z. H. He, Q. W. Wang, Y. Zhang, The complete equivalence canonical form of four matrices over an arbitrary division ring, Linear Multilinear A., 66 (2018), 74–95. https://doi.org/10.1080/03081087.2017.1284740 doi: 10.1080/03081087.2017.1284740
[10]	A. Barraud, S. Lesecq, N. Christov, From sensitivity analysis to random floating point arithmetics-application to Sylvester equations, In: Numerical analysis and its applications, Springer, Berlin, 2000, 35–41. https://doi.org/10.1007/3-540-45262-1_5
[11]	H. K. Wimmer, Consistency of a pair of generalized Sylvester equations, IEEE Transact. Automat. Contr., 39 (1994), 1014–1016. https://doi.org/10.1109/9.284883 doi: 10.1109/9.284883
[12]	H. L. Wang, N. C. Wu, Y. F. Nie, Two accelerated gradient-based iteration methods for solving the Sylvester matrix equation $AX + XB = C$, AIMS Math., 9 (2024), 34734–34752. https://doi.org/10.3934/math.20241654 doi: 10.3934/math.20241654
[13]	S. B. Aoun, N. Derbel, H. Jerbi, T. E. Simos, S. D. Mourtas, V. N. Katsikis, A quaternion Sylvester equation solver through noise-resilient zeroing neural networks with application to control the SFM chaotic system, AIMS Math., 8 (2023), 27376–27395. https://doi.org/10.3934/math.20231401 doi: 10.3934/math.20231401
[14]	A. Rehman, Q. W. Wang, Z. H. He, Solution to a system of real quaternion matrix equations involving $\eta$-Hermicity, Appl. Math. Comput., 265 (2015), 945–957. https://doi.org/10.1016/j.amc.2015.05.104 doi: 10.1016/j.amc.2015.05.104
[15]	A. Rehman, Q. W. Wang, I. Ali, M. Akram, M. O. Ahmad, A constraint system of generalized Sylvester quaternion matrix equations, Adv. Appl. Clifford Algebras, 27 (2017), 3183–3196. https://doi.org/10.1007/s00006-017-0803-1 doi: 10.1007/s00006-017-0803-1
[16]	Z. H. He, Some new results on a system of Sylvester-type quaternion matrix equations, Linear Multilinear A., 69 (2021), 3069–3091. https://doi.org/10.1080/03081087.2019.1704213 doi: 10.1080/03081087.2019.1704213
[17]	A. Rehman, I. Kyrchei, M. Z. U. Rahman, V. Leiva, C. Castro, Solvability and algorithm for Sylvester-type quaternion matrix equations with potential applications, AIMS Math., 9 (2024), 19967–19996. https://doi.org/10.3934/math.2024974 doi: 10.3934/math.2024974
[18]	A. Rehman, M. Z. U. Rahman, A. Ghaffar, C. Martin-Barreiro, C. Castro, V. Leiva, et al., Systems of quaternionic linear matrix equations: solution, computation, algorithm, and applications, AIMS Math., 9 (2024), 26371–26402. https://doi.org/10.3934/math.20241284 doi: 10.3934/math.20241284
[19]	A. Rehman, I. Kyrchei, Solving and algorithm to system of quaternion Sylvester-type matrix equations with $\ast$-Hermicity, Adv. Appl. Clifford Algebras, 32 (2022), 49. https://doi.org/10.1007/s00006-022-01222-2 doi: 10.1007/s00006-022-01222-2
[20]	J. Qin, Q. W. Wang, Solving a system of two-sided Sylvester-like quaternion tensor equations, Comp. Appl. Math., 42 (2023), 232. https://doi.org/10.1007/s40314-023-02349-z doi: 10.1007/s40314-023-02349-z
[21]	J. S. Respondek, Fast matrix multiplication with applications, Switzerland: Springer, 2025. https://doi.org/10.1007/978-3-031-76930-6
[22]	R. G. Aykroyd, V. Leiva, F. Ruggeri, Recent developments of control charts, identification of big data sources and future trends of current research, Technol. Forecast. Soc. Change, 144 (2019), 221–232. https://doi.org/10.1016/j.techfore.2019.01.005 doi: 10.1016/j.techfore.2019.01.005
[23]	D. Calvetti, L. Reichel, Application of ADI iterative methods to the restoration of noisy images, SIAM J. Matrix Anal. Appl., 17 (1996), 165–186. https://doi.org/10.1137/S0895479894273687 doi: 10.1137/S0895479894273687
[24]	D. Rosen, Some results on homogeneous matrix equations, SIAM J. Matrix Anal. Appl., 14 (1993), 137–145. https://doi.org/10.1137/0614013 doi: 10.1137/0614013
[25]	Z. H. He, The general solution to a system of coupled Sylvester-type quaternion tensor equations involving $\eta$-Hermicity, Bull. Iran. Math. Soc., 45 (2019), 1407–1430. https://doi.org/10.1007/s41980-019-00205-7 doi: 10.1007/s41980-019-00205-7
[26]	Z. H. He, Q. W. Wang, Y. Zhang, A system of quaternary coupled Sylvester-type real quaternion matrix equations, Automatica, 87 (2018), 25–31. https://doi.org/10.1016/j.automatica.2017.09.008 doi: 10.1016/j.automatica.2017.09.008
[27]	H. Liping, Z. Qingguang, The matrix equation $AXB+CYD = E$ over a simple artinian ring, Linear Multilinear A., 38 (1995), 225–232. https://doi.org/10.1080/03081089508818358 doi: 10.1080/03081089508818358
[28]	Q. W. Wang, Z. H. He, Y. Zhang, Constrained two-sided coupled Sylvester-type quaternion matrix equations, Automatica, 101 (2019), 207–213. https://doi.org/10.1016/j.automatica.2018.12.001 doi: 10.1016/j.automatica.2018.12.001
[29]	Q. W. Wang, A. Rehman, Z. H. He, Y. Zhang, Constraint generalized Sylvester matrix equations, Automatica, 69 (2016), 60–64. https://doi.org/10.1016/j.automatica.2016.02.024 doi: 10.1016/j.automatica.2016.02.024
[30]	G. Marsaglia, G. P. H. Styan, Equalities and inequalities for ranks of matrices, Linear Multilinear A., 2 (1974), 269–292. https://doi.org/10.1080/03081087408817070 doi: 10.1080/03081087408817070
[31]	Q. W. Wang, Z. C. Wu, C. Y. Lin, Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications, Appl. Math. Comput., 182 (2006), 1755–1764. https://doi.org/10.1016/j.amc.2006.06.012 doi: 10.1016/j.amc.2006.06.012
[32]	Z. H. He, Q. W. Wang, The general solutions to some systems of matrix equations, Linear Multilinear A., 63 (2014), 2017–2032. https://doi.org/10.1080/03081087.2014.896361 doi: 10.1080/03081087.2014.896361
[33]	J. A. Díaz-García, V. Leiva-Sánchez, M. Galea, Singular elliptic distribution: Density and applications, Commun. Stat. Theor. M., 31 (2002), 665–681. https://doi.org/10.1081/STA-120003646 doi: 10.1081/STA-120003646
[34]	J. A. Ramírez-Figueroa, C. Martin-Barreiro, A. B. Nieto, V. Leiva, M. P. Galindo-Villardón, A new principal component analysis by particle swarm optimization with an environmental application for data science, Stoch. Environ. Res. Risk Assess., 35 (2021), 1969–1984. https://doi.org/10.1007/s00477-020-01961-3 doi: 10.1007/s00477-020-01961-3
[35]	A. Ghaffar, M. Z. U. Rahman, V. Leiva, C. Martin-Barreiro, X. Cabezas, C. Castro, Efficiency, optimality, and selection in a rigid actuation system with matching capabilities for an assistive robotic exoskeleton, Eng. Sci. Technol. Inter. J., 51 (2024), 101613. https://doi.org/10.1016/j.jestch.2023.101613 doi: 10.1016/j.jestch.2023.101613
[36]	A. Ghaffar, M. Z. U. Rahman, V. Leiva, C. Castro, Optimized design and analysis of cable-based parallel manipulators for enhanced subsea operations, Ocean Eng., 297 (2024), 117012. https://doi.org/10.1016/j.oceaneng.2024.117012 doi: 10.1016/j.oceaneng.2024.117012
[37]	A. Ghaffar, M. Z. U. Rahman, V. Leiva, C. Castro, C. Martin-Barreiro, Multi-factor optimization and failure-tolerant design of cable-driven parallel manipulators in deep-sea robotics, IEEE Access, 13 (2025), 79280–79290. https://doi.org/10.1109/ACCESS.2025.3561041 doi: 10.1109/ACCESS.2025.3561041
[38]	V. Leiva, C. Castro, Artificial intelligence and blockchain in clinical trials: Enhancing data governance efficiency, integrity, and transparency, Bioanalysis, 17 (2025), 161–176. https://doi.org/10.1080/17576180.2025.2452774 doi: 10.1080/17576180.2025.2452774
[39]	W. Alkady, K. ElBahnasy, V. Leiva, W. Gad, Classifying COVID-19 based on amino acids encoding with machine learning algorithms, Chemometr. Intell. Lab. Syst., 224 (2022), 104535. https://doi.org/10.1016/j.chemolab.2022.104535 doi: 10.1016/j.chemolab.2022.104535

Reader Comments

Your name:*

Email:*
© 2025 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)