Analysis of solvability and representation of general solutions for anti-Hermitian constrained quaternion matrix equations

Abdur Rehman; Ivan Kyrchei; Khuram Ali Khan; Tamador Alihia; Salwa El-Morsy; Abdur Rehman; Ivan Kyrchei; Khuram Ali Khan; Tamador Alihia; Salwa El-Morsy

doi:10.3934/math.20251154

AIMS Mathematics

2025, Volume 10, Issue 11: 26237-26259. doi: 10.3934/math.20251154

Previous Article Next Article

Research article

Analysis of solvability and representation of general solutions for anti-Hermitian constrained quaternion matrix equations

1.
University of Engineering & Technology, Lahore, Pakistan
2.
Pidstrygach Institute for Applied Problems of Mechanics and Mathematics of NASU, Lviv, Ukraine
3.
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
4.
Department of Mathematics, College of Science, Qassim University, Saudi Arabia

Received: 24 July 2025 Revised: 21 October 2025 Accepted: 28 October 2025 Published: 13 November 2025
MSC : 15A09, 15A15, 15A24, 15B33

This paper addresses the constrained system of quaternion matrix equations incorporating anti-Hermitian properties, driven by the significance of symmetric solutions in diverse applications. Solvability conditions are determined via rank equalities and relationships derived from Moore–Penrose inverses and induced projectors. Explicit solution representations are obtained, utilizing the Moore–Penrose inverse and projections. The originality of the results is established through a novel technique based on quaternion row-column determinant theory, supported by a numerical validation. This approach retains its innovative character even when extended to complex matrix equations using conventional determinants.
- quaternion matrix,
- matrix equation,
- generalized inverse,
- anti-Hermitian solution,
- symmetric solution,
- noncommutative determinant,
- Cramer's rule,
- mathematical operators
Citation: Abdur Rehman, Ivan Kyrchei, Khuram Ali Khan, Tamador Alihia, Salwa El-Morsy. Analysis of solvability and representation of general solutions for anti-Hermitian constrained quaternion matrix equations[J]. AIMS Mathematics, 2025, 10(11): 26237-26259. doi: 10.3934/math.20251154

Related Papers:

Abstract

This paper addresses the constrained system of quaternion matrix equations incorporating anti-Hermitian properties, driven by the significance of symmetric solutions in diverse applications. Solvability conditions are determined via rank equalities and relationships derived from Moore–Penrose inverses and induced projectors. Explicit solution representations are obtained, utilizing the Moore–Penrose inverse and projections. The originality of the results is established through a novel technique based on quaternion row-column determinant theory, supported by a numerical validation. This approach retains its innovative character even when extended to complex matrix equations using conventional determinants.

References

[1]	S. L. Adler, Quaternionic quantum mechanics and quantum fields, New York: Oxford University Press, 1995. https://doi.org/10.1093/oso/9780195066432.001.0001
[2]	H. Aslaksen, Quaternionic determinants, Math. Intell., 18 (1996), 57–65. https://doi.org/10.1007/BF03024312
[3]	A. Bacciotti, L. Rosier, Liapunov functions and stability in control theory, 2 Eds., Berlin, Heidelberg: Springer, 2005. https://doi.org/10.1007/b139028
[4]	Z. Z. Bai, On Hermitian and skew-Hermitian splitting iteration methods for continuous Sylvester equations, J. Comput. Math., 29 (2011), 185–198. https://doi.org/10.4208/jcm.1009-m3152 doi: 10.4208/jcm.1009-m3152
[5]	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
[6]	J. K. Baksalary, R. Kala, The matrix equation AXB + CYD = E, Linear Algebra Appl., 30 (1980), 141–147. https://doi.org/10.1016/0024-3795(80)90189-5 doi: 10.1016/0024-3795(80)90189-5
[7]	R. K. Cavin, S. P. Bhattacharyya, Robust and well-conditioned eigenstructure assignment via Sylvester's equation, Optim. Control Appl. Methods, 4 (1983), 205–212. https://doi.org/10.1002/OCA.4660040302 doi: 10.1002/OCA.4660040302
[8]	N. Cohen, S. De Leo, The quaternionic determinant, Electron. J. Linear Algebra, 7 (2000), 100–111. https://doi.org/10.13001/1081-3810.1050
[9]	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
[10]	Y.-B. Deng, X.-Y. Hu, On solutions of matrix equation $AXA^T+BYB^T = C$, J. Comput. Math., 23 (2005), 17–26.
[11]	F. Ding, T. Chen, Iterative least squares solutions of coupled Sylvester matrix equations, Syst. Control Lett., 54 (2005), 95–107. http://doi.org/10.1016/j.sysconle.2004.06.008 doi: 10.1016/j.sysconle.2004.06.008
[12]	F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM J. Control Optim., 44 (2006), 2269–2284. https://doi.org/10.1137/S0363012904441350 doi: 10.1137/S0363012904441350
[13]	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
[14]	M. Hajarian, Developing CGNE algorithm for the periodic discrete-time generalized coupled Sylvester matrix equations, Comp. Appl. Math., 34 (2015), 755–771. http://doi.org/10.1007/s40314-014-0138-7 doi: 10.1007/s40314-014-0138-7
[15]	W. R. Hamilton, On quaternions, or on a new system of imaginaries in algebra, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 25 (1844), 10–13. https://doi.org/10.1080/14786444408644923 doi: 10.1080/14786444408644923
[16]	Z.-H. He, W.-L. Qin, J. Tian, X.-X. Wang, Y. Zhang, A new Sylvester-type quaternion matrix equation model for color image data transmission, Comp. Appl. Math., 43 (2024), 227. https://doi.org/10.1007/s40314-024-02732-4 doi: 10.1007/s40314-024-02732-4
[17]	Z.-H. He, Q.-W. Wang, A pair of mixed generalized Sylvester matrix equations, J. Shanghai Univ. Nat. Sci., 20 (2014), 138–156.
[18]	J. B. Kuipers, Quaternions and rotation sequences, Princeton: Princeton University Press, 1999.
[19]	I. I. Kyrchei, Determinantal representations of general and (skew-)Hermitian solutions to the generalized Sylvester-type quaternion matrix equation, Abstr. Appl. Anal., 2019 (2019), 5926832. https://doi.org/10.1155/2019/5926832 doi: 10.1155/2019/5926832
[20]	I. I. Kyrchei, Determinantal representations of the Moore-Penrose inverse over the quaternion skew field and corresponding Cramer's rules, Linear Multilinear Algebra, 59 (2011), 413–431. https://doi.org/10.1080/03081081003586860 doi: 10.1080/03081081003586860
[21]	I. I. Kyrchei, The theory of the column and row determinants in a quaternion linear algebra, In: Advances in Mathematics Research, New York: Nova Sci. Publ., 2012,301–358.
[22]	S.-G. Lee, Q.-P. Vu, Simultaneous solutions of matrix equations and simultaneous equivalence of matrices, Linear Algebra Appl., 437 (2012), 2325–2339. http://doi.org/10.1016/j.laa.2012.06.004 doi: 10.1016/j.laa.2012.06.004
[23]	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 (2000), 440–445. https://doi.org/10.1137/S0895479898349586 doi: 10.1137/S0895479898349586
[24]	Y. Q. Lin, Y. M. Wei, Condition numbers of the generalized Sylvester equation, IEEE Trans. Autom. Control, 52 (2007), 2380–2385. http://doi.org/10.1109/TAC.2007.910727 doi: 10.1109/TAC.2007.910727
[25]	X. Liu, The $\eta$-anti-Hermitian solution to some classic matrix equations, Appl. Math. Comput., 320 (2018), 264–270. http://doi.org/10.1016/j.amc.2017.09.033 doi: 10.1016/j.amc.2017.09.033
[26]	Y. H. Liu, Y. G. Tian, A simultaneous decomposition of a matrix triplet with applications, Numer. Linear Algebra Appl., 18 (2011), 69–85. http://doi.org/10.1002/nla.701 doi: 10.1002/nla.701
[27]	P. D. Mannheim, PT symmetry as a necessary and sufficient condition for unitary time evolution, Phil. Trans. R. Soc. A, 371 (2013), 20120060. https://doi.org/10.1098/rsta.2012.0060 doi: 10.1098/rsta.2012.0060
[28]	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
[29]	J. E. Marsden, T. S. Ratiu, Introduction to mechanics and symmetry, 2 Eds., New York: Springer, 1999. https://doi.org/10.1007/978-0-387-21792-5
[30]	A. Rehman, M. Akram, Optimization of a nonlinear Hermitian matrix expression with application, Filomat, 31 (2017), 2805–2819. https://doi.org/10.2298/FIL1709805R doi: 10.2298/FIL1709805R
[31]	A. Rehman, I. I. Kyrchei, Hermitian solution to constraint system of generalized Sylvester quaternion matrix equations, Arab. J. Math., 13 (2024), 595–610. https://doi.org/10.1007/s40065-024-00477-w doi: 10.1007/s40065-024-00477-w
[32]	A. Rehman, I. I. Kyrchei, Compact formula for skew-symmetric system of matrix equations, Arab. J. Math., 12 (2023), 587–600. https://doi.org/10.1007/s40065-023-00439-8 doi: 10.1007/s40065-023-00439-8
[33]	A. Rehman, I. I. Kyrchei, I. A. Khan, M. Nasir, I. Ali, Closed-form formula for a classical system of matrix equations, Arab J. Basic Appl. Sci., 29 (2022), 258–268. http://doi.org/10.1080/25765299.2022.2113497 doi: 10.1080/25765299.2022.2113497
[34]	A. Rehman, I. I. Kyrchei, I. Ali, M. Akram, A. Shakoor, The general solution of quaternion matrix equation having $\eta$-skew-Hermicity and its Cramer's rule, Math. Probl. Eng., 2019 (2019), 7939238. https://doi.org/10.1155/2019/7939238 doi: 10.1155/2019/7939238
[35]	A. Rehman, I. I. Kyrchei, I. Ali, M. Akram, A. Shakoor, Explicit formulas and determinantal representation for $\eta$-skew-Hermitian solution to a system of quaternion matrix equations, Filomat, 34 (2020), 2601–2627. https://doi.org/10.2298/FIL2008601R doi: 10.2298/FIL2008601R
[36]	A. Rehman, I. I. Kyrchei, Solving and algorithm to system of quaternion Sylvester-type matrix equations with $$-Hermicity, Adv. Appl. Clifford Algebras*, 32 (2022), 49. https://doi.org/10.1007/s00006-022-01222-2 doi: 10.1007/s00006-022-01222-2
[37]	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
[38]	V. L. Syrmos, F. L. Lewis, Coupled and constrained Sylvester equations in system design, Circ. Syst. Signal Process., 13 (1994), 663–694. https://doi.org/10.1007/BF02523122 doi: 10.1007/BF02523122
[39]	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
[40]	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
[41]	Q.-W. Wang, A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity, Linear Algebra Appl., 384 (2004), 43–54. https://doi.org/10.1016/j.laa.2003.12.039 doi: 10.1016/j.laa.2003.12.039
[42]	Q.-W. Wang, Z.-H. He, Solvability conditions and general solution for the mixed Sylvester equations, Automatica, 49 (2013), 2713–2719. http://doi.org/10.1016/j.automatica.2013.06.009 doi: 10.1016/j.automatica.2013.06.009
[43]	Q.-W. Wang, A. Rehman, Z.-H. He, Y. Zhang, Constraint generalized Sylvester matrix equations, Automatica, 69 (2016), 60–74. http://doi.org/10.1016/j.automatica.2016.02.024 doi: 10.1016/j.automatica.2016.02.024
[44]	Q.-W. Wang, J. W. Van der Woude, H.-X. Chang, A system of real quaternion matrix equations with applications, Linear Algebra Appl., 431 (2009), 2291–2303. https://doi.org/10.1016/j.laa.2009.02.010

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)