Abstract
In this paper, we put forward two efficient methods for solving the commutative quaternion Stein matrix equation \(X+AXB=C\). By combining real representation of a commutative quaternion matrix and \({\mathcal {H}}\)-representation or generalized \({\mathcal {H}}\)-representation of special matrices, we investigate the minimal norm least squares L-structured and generalized L-structured solutions of the previous commutative quaternion matrix equation and derive their expressions. In this way, we first present a theoretical study on extending L-structured real matrices to generalized L-structured matrices, and introduce some generalized L-structured matrices. Based on them, we then discuss their applications in commutative quaternion Stein matrix equation. The algorithms only involve real operations. Consequently, it is very simple and convenient, and it can be used for all kinds of commutative quaternion matrix equation with similar problems. Furthermore, an illustrative example is provided to show the feasibility of the given methods.

Similar content being viewed by others
References
Antoulas AC (2005) Approximation of large-scale dynamical systems. Society for Industrial and Applied Mathematics
Bouhamidi A, Jbilou K (2007) Sylvester Tikhonov-regularization methods in image restoration. J Comput Appl Math 206:86–98
Chen J, Chen X (2001) Special matrices. Tsinghua University Press, Tsinghua
Chen H, Hou CP, Wang Q et al (2014) Cumulants-based Toeplitz matrices reconstruction method for 2-D coherent DOA estimation. IEEE Sens J 14(8):2824–2832
de Souza E, Bhattacharyya SP (1981) Controllability, observability and the solution of \(AX-XB=C\). Linear Algebra Appl 39:167–188
Ding WX, Li Y, Wang D (2021) Special least squares solutions of the reduced biquaternion matrix equation \(AX=B\) with applications. Comput Appl Math 40(8):1–15
Golub GH, Van Loan CF (2013) Matrix Computations, 4th edn. The Johns Hopkins University Press, Baltimore, MD
Hyland D, Bernstein D (1984) The optimal projection equations for fixed-order dynamic compensation. IEEE Trans Autom Control 29(11):1034–1037
Jia ZG, Ng MK (2021) Structure preserving quaternion generalized minimal residual method. SIAM J Matrix Anal Appl 42(2):616–634
Jia ZG, Wei MS, Ling ST (2013) A new structure-preserving method for quaternion Hermitian eigenvalue problems. J Comput Appl Math 239:12–24
Jia ZG, Wei MS, Zhao MX et al (2018) A new real structure-preserving quaternion QR algorithm. J Comput Appl Math 343:26–48
Jia ZG, Ng MK, Wang W (2019) Color image restoration by saturation-value total variation. SIAM J Imag Sci 12(2):972–1000
Jia ZG, Ng MK, Song GJ (2019) Robust quaternion matrix completion with applications to image inpainting. Numer Linear Algebra Appl 26(4):e2245
Jia ZG, Ng MK, Song GJ (2019) Lanczos method for large-scale quaternion singular value decomposition. Numer Algorithms 82(2):699–717
Khatri CG, Mitra SK (1976) Hermitian and nonnegative definite solutions of linear matrix equations. SIAM J Appl Math 31(4):579–585
Laub A, Heath M, Paige C et al (1987) Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans Autom Control 32(2):115–122
Li YT, Wu WJ (2008) Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations. Comput Math Appl 55(6):1142–1147
Liao AP, Bai ZZ (2005) Least squares symmetric and skew-symmetric solutions of the matrix equation \(AXA^T+BYB^T=C\) with the least norm. Math Numer Sin 27(2):81–95
Liu LS, Wang QW, Chen JF et al (2022) An exact solution to a quaternion matrix equation with an application. Symmetry 14(2):375
Magnus JR (1983) L-structured matrices and linear matrix equations. Linear Multilinear A 14(1):67–88
Mehany MS, Wang QW (2022) Three symmetrical systems of coupled Sylvester-like quaternion matrix equations. Symmetry 14(3):550
Narendra KS, Shorten R (2010) Hurwitz stability of Metzler matrices. IEEE Trans Autom Control 55(6):1484–1487
Roberts JD (1980) Linear model reduction and solution of the algebraic Riccati equation by use of the sign function. Int J Control 32(4):677–687
Segre C (1892) The real representations of complex elements and extension to bicomplex systems. Math Ann 40:413–467
Song CQ, Chen GL, Liu QB (2012) Explicit solutions to the quaternion matrix equations \(X-AXF=C\) and \(X-A\tilde{X}F=C\). Int J Comput Math 89(7):890–900
Yuan SF, Wang QW (2016) L-structured quaternion matrices and quaternion linear matrix equations. Linear Multilinear A 64(2):321–339
Yuan WJ, Wang QW (2022) The common solution of twelve matrix equations over the quaternions. Filomat 36(3):887–903
Yuan SF, Liao AP, Lei Y (2007) Least squares symmetric solution of the matrix equation \(AXB+CYD=E\) with the least norm. Math Numer Sin 29:203–216
Yuan SF, Liao AP, Lei Y (2008) Least squares Hermitian solution of the matrix equation \((AXB, CXD)=(E, F)\) with the least norm over the skew field of quaternions. Math Comput Model 48(1–2):91–100
Zhang WH, Chen BS (2012) \({\cal{H}}\)-Representation and applications to generalized Lyapunov equations and linear stochastic systems. IEEE Trans Autom Control 57(12):3009–3022
Zhang YN, Jiang DC, Wang J (2002) A recurrent neural network for solving Sylvester equation with time-varying coefficients. IEEE Trans Neural Netw 13(5):1053–1063
Zhang W, Shen SQ, Han ZZ (2008) Sufficient conditions for Hurwitz stability of matrices. Lat Am Appl Res 38(3):253–258
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jinyun Yuan.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supported by the National Natural Science Foundation of China under Grant 62176112, the Natural Science Foundation of Shandong under Grant ZR2020MA053.
Rights and permissions
About this article
Cite this article
Wei, A., Li, Y., Ding, W. et al. Two algebraic methods for least squares L-structured and generalized L-structured problems of the commutative quaternion Stein matrix equation. Comp. Appl. Math. 41, 251 (2022). https://doi.org/10.1007/s40314-022-01943-x
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s40314-022-01943-x
Keywords
- Commutative quaternion
- Matrix equation
- Real representation matrix
- Generalized \({\mathcal {H}}\)-representation
- Least squares solution