Systems of quaternionic linear matrix equations: solution, computation, algorithm, and applications

Abdur Rehman; Muhammad Zia Ur Rahman; Asim Ghaffar; Carlos Martin-Barreiro; Cecilia Castro; Víctor Leiva; Xavier Cabezas; Abdur Rehman; Muhammad Zia Ur Rahman; Asim Ghaffar; Carlos Martin-Barreiro; Cecilia Castro; Víctor Leiva; Xavier Cabezas

doi:10.3934/math.20241284

AIMS Mathematics

2024, Volume 9, Issue 10: 26371-26402. doi: 10.3934/math.20241284

Previous Article Next Article

Research article

Systems of quaternionic linear matrix equations: solution, computation, algorithm, and applications

1.
Department of Basic Sciences and Humanities, University of Engineering and Technology, Lahore, Faisalabad Campus, Faisalabad, Pakistan
2.
Department of Mechanical, Mechatronics and Manufacturing Engineering, University of Engineering and Technology, Lahore, Faisalabad Campus, Faisalabad, Pakistan
3.
Facultad de Ingeniería, Universidad Espíritu Santo, Samborondón, Ecuador
4.
Centre of Mathematics, Universidade do Minho, Braga, Portugal
5.
Escuela de Ingeniería Industrial, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile
6.
Facultad de Ciencias Naturales y Matemáticas, Escuela Superior Politécnica del Litoral ESPOL, Guayaquil, Ecuador
7.
Centro de Estudios e Investigaciones Estadísticas, Escuela Superior Politécnica del Litoral ESPOL, Guayaquil, Ecuador

Received: 08 July 2024 Revised: 02 August 2024 Accepted: 21 August 2024 Published: 11 September 2024
MSC : 62H25, 62H30

In applied and computational mathematics, quaternions are fundamental in representing three-dimensional rotations. However, specific types of quaternionic linear matrix equations remain few explored. This study introduces new quaternionic linear matrix equations and their necessary and sufficient conditions for solvability. We employ a methodology involving lemmas and ranks of coefficient matrices to develop a novel algorithm. This algorithm is validated through numerical examples, showing its applications in advanced fields. In control theory, these equations are used for analyzing control systems, particularly for spacecraft attitude control in aerospace engineering and for control of arms in robotics. In quantum computing, quaternionic equations model quantum gates and transformations, which are important for algorithms and error correction, contributing to the development of fault-tolerant quantum computers. In signal processing, these equations enhance multidimensional signal filtering and noise reduction, with applications in color image processing and radar signal analysis. We extend our study to include cases of $ \eta $-Hermitian and i-Hermitian solutions. Our work represents an advancement in applied mathematics, providing computational methods for solving quaternionic matrix equations and expanding their practical applications.
- control theory,
- Hermitian solutions,
- quaternionic equations,
- solvability conditions
Citation: Abdur Rehman, Muhammad Zia Ur Rahman, Asim Ghaffar, Carlos Martin-Barreiro, Cecilia Castro, Víctor Leiva, Xavier Cabezas. Systems of quaternionic linear matrix equations: solution, computation, algorithm, and applications[J]. AIMS Mathematics, 2024, 9(10): 26371-26402. doi: 10.3934/math.20241284

Related Papers:

Abstract

In applied and computational mathematics, quaternions are fundamental in representing three-dimensional rotations. However, specific types of quaternionic linear matrix equations remain few explored. This study introduces new quaternionic linear matrix equations and their necessary and sufficient conditions for solvability. We employ a methodology involving lemmas and ranks of coefficient matrices to develop a novel algorithm. This algorithm is validated through numerical examples, showing its applications in advanced fields. In control theory, these equations are used for analyzing control systems, particularly for spacecraft attitude control in aerospace engineering and for control of arms in robotics. In quantum computing, quaternionic equations model quantum gates and transformations, which are important for algorithms and error correction, contributing to the development of fault-tolerant quantum computers. In signal processing, these equations enhance multidimensional signal filtering and noise reduction, with applications in color image processing and radar signal analysis. We extend our study to include cases of $ \eta $-Hermitian and i-Hermitian solutions. Our work represents an advancement in applied mathematics, providing computational methods for solving quaternionic matrix equations and expanding their practical applications.

References

[1]	A. Rehman, Q. W. Wang, A system of matrix equations with five variables, Appl. Math. Comput., 271 (2015), 805–819. https://doi.org/10.1016/j.amc.2015.09.066 doi: 10.1016/j.amc.2015.09.066
[2]	W. R. Hamilton, LXXVIII. On quaternions; or on a new system of imaginaries in Algebra: to the editors of the Philosophical Magazine and Journal, London Edinburgh Dublin Philos. Mag. J. Sci., 25 (1844), 489–495. https://doi.org/10.1080/14786444408645047 doi: 10.1080/14786444408645047
[3]	S. L. Adler, Quaternionic quantum mechanics and quantum fields, New York, USA: Oxford University Press, 1995.
[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]	C. C. Took, D. P. Mandic, Augmented second-order statistics of quaternion random signals, Signal Process., 91 (2011), 214–224. https://doi.org/10.1016/j.sigpro.2010.06.024 doi: 10.1016/j.sigpro.2010.06.024
[6]	J. B. Kuipers, Quaternions and rotation sequences: a primer with applications to orbits, aerospace, and virtual reality, Princeton: Princeton University Press, 1999. https://doi.org/10.2307/j.ctvx5wc3k
[7]	C. C. Took, D. P. Mandic, The quaternion LMS algorithm for adaptive filtering of hypercomplex processes, IEEE Trans. Signal Proces., 57 (2009), 1316–1327. https://doi.org/10.1109/TSP.2008.2010600
[8]	C. C. Took, D. P. Mandic, A quaternion widely linear adaptive filter, IEEE Trans. Signal Proces., 58 (2010), 4427–4431. https://doi.org/10.1109/TSP.2010.2048323 doi: 10.1109/TSP.2010.2048323
[9]	C. C. Took, D. P. Mandic, F. Zhang, On the unitary diagonalisation of a special class of quaternion matrices, Appl. Math. Lett., 24 (2011), 1806–1809. https://doi.org/10.1016/j.aml.2011.04.038 doi: 10.1016/j.aml.2011.04.038
[10]	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
[11]	A. Rehman, I. Kyrchei, I. Ali, M. Akram, A. Shakoor, Constraint solution of a classical system of quaternion matrix equations and its Cramer's rule, Iran. J. Sci. Technol. Trans. A: Sci., 45 (2021), 1015–1024. https://doi.org/10.1007/s40995-021-01083-7 doi: 10.1007/s40995-021-01083-7
[12]	R. K. Cavin, S. P. Bhattacharyya, Robust and well-conditioned eigenstructure assignment via Sylvester's equation, Optimal Control Appl. Methods, 4 (1983), 205–212. https://doi.org/10.1002/oca.4660040302 doi: 10.1002/oca.4660040302
[13]	V. L. Syrmos, F. L. Lewis, Coupled and constrained Sylvester equations in system design, Circuits, System Signal Process., 13 (1994), 663–694. https://doi.org/10.1007/BF02523122 doi: 10.1007/BF02523122
[14]	M. Darouach, Solution to Sylvester equation associated to linear descriptor systems, Syst. Control Lett., 55 (2006), 835–838. https://doi.org/10.1016/j.sysconle.2006.04.004 doi: 10.1016/j.sysconle.2006.04.004
[15]	A. Rehman, I. 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
[16]	S. Gupta, Linear quaternion equations with application to spacecraft attitude propagation, 1998 IEEE Aerospace Conference Proceedings, Snowmass at Aspen, CO, 1 (1998), 69–76. https://doi.org/10.1109/AERO.1998.686806
[17]	V. N. Kovalnogov, R. V. Fedorov, D. A. Demidov, M. A. Malyoshina1, T. E. Simos, V. N. Katsikis, et al., Zeroing neural networks for computing quaternion linear matrix equation with application to color restoration of images, AIMS Math., 8 (2023), 14321–14339. https://doi.org/10.3934/math.2023733 doi: 10.3934/math.2023733
[18]	A. Shakoor, I. Ali, S. Wali, A. Rehman, Some formulas on the Drazin inverse for the sum of two matrices and block matrices, Bull. Iran. Math. Soc., 48 (2022), 351–366. https://doi.org/10.1007/s41980-020-00521-3 doi: 10.1007/s41980-020-00521-3
[19]	D. Zhang, T. Jiang, G. Wang, V. I. Vasilev, On singular value decomposition and generalized inverse of a commutative quaternion matrix and applications, Appl. Math. Comput., 460 (2024), 128291. https://doi.org/10.1016/j.amc.2023.128291
[20]	V. N. Kovalnogov, R. V. Fedorov, I. I. Shepelev, V. V. Sherkunov, T. E. Simos, S. D. Mourtas, et al., A novel quaternion linear matrix equation solver through zeroing neural networks with applications to acoustic source tracking, AIMS Math., 8 (2023), 25966–25989. https://doi.org/10.3934/math.20231323 doi: 10.3934/math.20231323
[21]	X. Liu, The $\eta$-anti-Hermitian solution to some classic matrix equations, Appl. Math. Comput., 320 (2018), 264–270. https://doi.org/10.1016/j.amc.2017.09.033 doi: 10.1016/j.amc.2017.09.033
[22]	A. Rehman, Q. W. Wang, Z. H. He, Solution to a system of real quaternion matrix equations encompassing $\eta$-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
[23]	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
[24]	R. T. Farouki, Pythagorean-hodograph curves: algebra and geometry inseparable, New York: Springer, 2008.
[25]	A. Altavilla, C. de Fabritiis, Equivalence of slice semi-regular functions via Sylvester operators, Linear Algebra Appl., 607 (2020), 151–189. https://doi.org/10.1016/j.laa.2020.08.009 doi: 10.1016/j.laa.2020.08.009
[26]	J. A. Díaz-García, V. Leiva, M. Galea, Singular elliptic distribution: density and applications, Commun. Stat.: Theory Methods, 31 (2002), 665–681. https://doi.org/10.1081/STA-120003646 doi: 10.1081/STA-120003646
[27]	A. Barraud, S. Lesecq, N. Christov, From sensitivity analysis to random floating point arithmetics-application to Sylvester equations, Numer. Anal. Appl., 1988 (2001), 35–41. https://doi.org/10.1007/3-540-45262-1_5 doi: 10.1007/3-540-45262-1_5
[28]	R. C. Li, A bound on the solution to a structured Sylvester equation with an application to relative perturbation theory, SIAM J. Matrix Anal. Appl., 21 (1999), 440–445. https://doi.org/10.1137/S0895479898349586 doi: 10.1137/S0895479898349586
[29]	V. L. Syrmos, F. L. Lewis, Output feedback eigenstructure assignment using two Sylvester equations, IEEE Trans. Autom. Control, 38 (1993), 495–499. https://doi.org/10.1109/9.210155 doi: 10.1109/9.210155
[30]	Y. N. Zhang, D. C. Jiang, J. Wang, A recurrent neural network for solving Sylvester equation with time-varying coefficients, IEEE Trans. Neural Networks, 13 (2002), 1053–1063. https://doi.org/10.1109/TNN.2002.1031938 doi: 10.1109/TNN.2002.1031938
[31]	Z. Z. Bai, On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations, J. Comput. Math., 29 (2011), 185–198.
[32]	L. Rodman, Topics in quaternion linear algebra, Princeton: Princeton University Press, 2014.
[33]	W. E. Roth, The equations $AX-YB = C$ and $AX-XB = C$ in matrices, Proc. Amer. Math. Soc., 3 (1952), 392–396. https://doi.org/10.2307/2031890 doi: 10.2307/2031890
[34]	J. K. Baksalary, R. Kala, The matrix equation $AX-YB = C$, Linear Algebra Appl., 25 (1979), 41–43. https://doi.org/10.1016/0024-3795(79)90004-1 doi: 10.1016/0024-3795(79)90004-1
[35]	L. Wang, Q. Wang, Z. He, The common solution of some matrix equations, Algebra Colloq., 23 (2016), 71–81. https://doi.org/10.1142/S1005386716000092 doi: 10.1142/S1005386716000092
[36]	Q. W. Wang, Z. H. He, Solvability conditions and general solution for the mixed Sylvester equations, Automatica, 49 (2013), 2713–2719. https://doi.org/10.1016/j.automatica.2013.06.009 doi: 10.1016/j.automatica.2013.06.009
[37]	S. G. Lee, Q. P. Vu, Simultaneous solutions of matrix equations and simultaneous equivalence of matrices, Linear Algebra Appl., 437 (2012), 2325–2339. https://doi.org/10.1016/j.laa.2012.06.004 doi: 10.1016/j.laa.2012.06.004
[38]	Y. Q. Lin, Y. M. Wei, Condition numbers of the generalized Sylvester equation, IEEE Trans. Autom. Control, 52 (2007), 2380–2385. https://doi.org/10.1109/TAC.2007.910727 doi: 10.1109/TAC.2007.910727
[39]	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
[40]	X. Zhang, A system of generalized Sylvester quaternion matrix equations and its applications, Appl. Math. Comput., 273 (2016), 74–81. https://doi.org/10.1016/j.amc.2015.09.074
[41]	A. Dmytryshyn, B. Kåström, Coupled Sylvester-type matrix equations and block diagonalization, SIAM J. Matrix Anal. Appl., 38 (2015), 580–593. https://doi.org/10.1137/151005907 doi: 10.1137/151005907
[42]	F. O. Farid, Z. H. He, Q. W. Wang, The consistency and the exact solutions to a system of matrix equations, Linear Multilinear Algebra, 64 (2016), 2133–2158. https://doi.org/10.1080/03081087.2016.1140717 doi: 10.1080/03081087.2016.1140717
[43]	Z. H. He, M. Wang, X. Liu, On the general solutions to some systems of quaternion matrix equations, RACSAM, 114 (2020), 95.
[44]	H. K. Wimmer, Consistency of a pair of generalized Sylvester equations, IEEE Trans. Autom. Control, 39 (1994), 1014–1016. https://doi.org/10.1109/9.284883 doi: 10.1109/9.284883
[45]	B. Kågström, A perturbation analysis of the generalized Sylvester equation $(AR - LB, DR - LE) = (C, F)$, SIAM J. Matrix Anal. Appl., 15 (1994), 1045–1060. https://doi.org/10.1137/S0895479893246212 doi: 10.1137/S0895479893246212
[46]	Z. H. He, Q. W. Wang, A pair of mixed generalized Sylvester matrix equations, J. Shanghai Univ. Nat. Sci. Ed., 20 (2014), 138–156. https://doi.org/10.3969/j.issn.1007-2861.2014.01.021 doi: 10.3969/j.issn.1007-2861.2014.01.021
[47]	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
[48]	Z. H. He, Q. W. Wang, A system of periodic discrete-time coupled Sylvester quaternion matrix equations, Algebra Colloq., 24 (2017), 169–180. https://doi.org/10.1142/S1005386717000104 doi: 10.1142/S1005386717000104
[49]	R. Kristiansen, P. J. Nicklasson, Satellite attitude control by quaternion-based backstepping, Proceedings of the 2005 American Control Conference, 2005, 16–18.
[50]	D. Finkelstein, J. M. Jauch, S. Schiminovich, D. Speiser, Foundations of quaternion quantum mechanics, J. Math. Phys., 3 (1962), 207–220.
[51]	S. C. Pei, Y. Z. Hsiao, Colour image edge detection using quaternion quantized localized phase, Proc. 18th European Signal Processing Conference, 2010, 1766–1770.
[52]	G. Marsaglia, G. P. H. Styan, Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 2 (1974), 269–292. https://doi.org/10.1080/03081087408817070 doi: 10.1080/03081087408817070
[53]	J. N. Buxton, R. F. Churchouse, A. B. Tayler, Matrices methods and applications, Oxford, UK: Clarendon Press, 1990.
[54]	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
[55]	Z. H. He, Q. W. Wang, The general solutions to some systems of matrix equations, Linear Multilinear Algebra, 63 (2015), 2017–2032. https://doi.org/10.1080/03081087.2014.896361 doi: 10.1080/03081087.2014.896361
[56]	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
[57]	J. A. Ramirez-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
[58]	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., Int. J., 51 (2024), 101613. https://doi.org/10.1016/j.jestch.2023.101613
[59]	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

Reader Comments

Your name:*

Email:*
© 2024 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)