This paper addresses the n-quaternion matrix equation $AX - YB = C$. We investigate the solutions to this equation, including the general solution, the Hermitian solution, and the anti-Hermitian solution. Firstly, we present the left real element representation and the right real element representation of the quaternion. Secondly, by combining the semi-tensor product of matrices, the n-quaternion matrix equation is transformed into a real linear system. Subsequently, the $\mathcal {H}$-representation is utilized to obtain its special solutions. Thirdly, numerical experiments are conducted to verify the effectiveness of the proposed method. Finally, the solving method of the n-quaternion matrix equation is applied to the encryption and decryption process of color images.
| [1] |
Asari, S.S., Amirfakhrian, M.: Numerical solution of Sylvester matrix equations: application to dynamical systems. J. Interpolat. Approx. Sci. Comput. 2016, 1–13 (2016)
|
| [2] |
Cayley, A.: A memoir on the theory of matrices. Proceedings of the Royal Society of London 9, 100-101(1859)
|
| [3] |
Cheng, D.Z., Ji, Z.P.: On universal eigenvalues and eigenvectors of hypermatrices. Journal of the Franklin Institute 362, 108126 (2025)
|
| [4] |
Cheng DZ, Qi HS, Liu ZQ. From STP to game-based control. Sci. China Inf. Sci., 2018, 611010201
|
| [5] |
Cheng DZ, Qi HS, Xue AC. A survey on semi-tensor product of matrices. J. Syst. Sci. Complex., 2007, 202304322.
|
| [6] |
Cheng, D.Z., Qi, H.S., Zhao, Y.: An Introduction to Semi-tensor Product of Matrices and Its Applications. World Scientific, Singapore (2012)
|
| [7] |
Clifford W. Preliminary sketch of bi-quaternions. Proc. Lond. Math. Soc., 1873, 4: 381-395
|
| [8] |
Ding WX, Li Y, Wang D. 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, 2021, 31578.
|
| [9] |
Gajic, Z., Qureshi, M.T.J.: Lyapunov Matrix Equation in System Stability and Control. Academic Press, San Diego (1995)
|
| [10] |
Golub, G.H., van Loan, C.F.: Matrix Computation. 4th ed. The Johns Hopkins University Press, Baltimore (2013)
|
| [11] |
Hamilton, W.R.: Theory of quaternions. Proceedings of the Royal Irish Academy 3, 1–16 (1844)
|
| [12] |
Li Y, Zhang FX, Guo WB, Zhao JL. Solutions with special structure to the linear matrix equation $AX=B$. Comput. Math. Appl., 2011, 612374383.
|
| [13] |
Liu XC, Li Y, Ding WX, Tao RY. A real method for solving octonion matrix equation $AXB= C$ based on semi-tensor product of matrices. Adv. Appl. Clifford Algebras, 2024, 34212.
|
| [14] |
Roth, W. E.: The equations $AX-YB=C$ and $AX-XB=C$ in matrices. Proceedings of the American Mathematical Society 3(3), 392–396 (1952)
|
| [15] |
Sadek, L.: Control theory for fractional differential Sylvester matrix equations with Caputo fractional derivative. J. Vib. Control 31(9/10), 1586–1602 (2025)
|
| [16] |
Shafiei, S.G., Hajarian, M.: Developing Kaczmarz method for solving Sylvester matrix equations. Journal of the Franklin Institute 359(16), 8991–9005 (2022)
|
| [17] |
Sun YP, Fu SH, Xia LY, Xu JY. Minimal observability of switching Boolean networks. Math. Methods Appl. Sci., 2025, 48332913303.
|
| [18] |
Tian YG. Matrix representations of octonions and their applications. Adv. Appl. Clifford Algebras, 2000, 1016190.
|
| [19] |
Wang D, Li Y, Ding WX. Several kinds of special least squares solutions to quaternion matrix equation $AXB= C$. J. Appl. Math. Comput., 2022, 68318811899.
|
| [20] |
Wang ZR, Fu SH, Pan JF, Zhao JL, Wang ZY. Nash equilibrium, dynamics and control of congestion games with resource failures. Nonlinear Dyn., 2024, 112181658716599.
|
| [21] |
Wang ZR, Fu SH, Wang JJ, Zhao XY. Strategy consensus of networked evolutionary games with time invariant delays. Dyn. Games Appl., 2024, 144981996.
|
| [22] |
Zhang WH, Chen BS. $\cal{H} $-representation and applications to generalized Lyapunov equations and linear stochastic systems. IEEE Trans. Autom. Control, 2012, 571230093022.
|
| [23] |
Zhu XY. Higher quaternion matrix equations with applications. Comput. Appl. Math., 2026, 45268.
|
Funding
National Natural Science Foundation of China(62176112)
Natural Science Foundation of Shandong Province(ZR2022MA030)
Discipline with Strong Characteristic of Liaocheng University Intelligent Science and Technology(319462208)
Rights & permissions
Shanghai University
PDF
