## Abstract and Applied Analysis

### Iterative Algorithm for Solving a Class of Quaternion Matrix Equation over the Generalized $\left(P,Q\right)$-Reflexive Matrices

#### Abstract

The matrix equation ${\sum }_{l=1}^{u}{A}_{l}X{B}_{l}+{\sum }_{s=1}^{v}{C}_{s}{X}^{T}{D}_{s}=F,$ which includes some frequently investigated matrix equations as its special cases, plays important roles in the system theory. In this paper, we propose an iterative algorithm for solving the quaternion matrix equation ${\sum }_{l=1}^{u}{A}_{l}X{B}_{l}+{\sum }_{s=1}^{v}{C}_{s}{X}^{T}{D}_{s}=F$ over generalized $\left(P,Q\right)$-reflexive matrices. The proposed iterative algorithm automatically determines the solvability of the quaternion matrix equation over generalized $\left(P,Q\right)$-reflexive matrices. When the matrix equation is consistent over generalized $\left(P,Q\right)$-reflexive matrices, the sequence $\left\{X\left(k\right)\right\}$ generated by the introduced algorithm converges to a generalized $\left(P,Q\right)$-reflexive solution of the quaternion matrix equation. And the sequence $\left\{X\left(k\right)\right\}$ converges to the least Frobenius norm generalized $\left(P,Q\right)$-reflexive solution of the quaternion matrix equation when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate generalized $\left(P,Q\right)$-reflexive solution for a given generalized $\left(P,Q\right)$-reflexive matrix ${X}_{\mathrm{0}}$ can be derived. The numerical results indicate that the iterative algorithm is quite efficient.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 831656, 15 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393512084

Digital Object Identifier
doi:10.1155/2013/831656

Mathematical Reviews number (MathSciNet)
MR3121402

#### Citation

Li, Ning; Wang, Qing-Wen. Iterative Algorithm for Solving a Class of Quaternion Matrix Equation over the Generalized $\left(P,Q\right)$ -Reflexive Matrices. Abstr. Appl. Anal. 2013 (2013), Article ID 831656, 15 pages. doi:10.1155/2013/831656. https://projecteuclid.org/euclid.aaa/1393512084

#### References

• I. Kyrchei, “Explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations,” Linear Algebra and its Applications, vol. 438, no. 1, pp. 136–152, 2013.
• S. Yuan and A. Liao, “Least squares solution of the quaternion matrix equation $X-A\widehat{X}B=C$ with the least norm,” Linear and Multilinear Algebra, vol. 59, no. 9, pp. 985–998, 2011.
• C. Song, G. Chen, and X. Wang, “On solutions of quaternion matrix equations $XF-AX=BY$ and $XF-A\widetilde{X}=BY$,” Acta Mathematica Scientia B, vol. 32, no. 5, pp. 1967–1982, 2012.
• Z.-H. He and Q.-W. Wang, “A real quaternion matrix equation with applications,” Linear and Multilinear Algebra, vol. 61, no. 6, pp. 725–740, 2013.
• H.-C. Chen and A. H. Sameh, “A matrix decomposition method for orthotropic elasticity problems,” SIAM Journal on Matrix Analysis and Applications, vol. 10, no. 1, pp. 39–64, 1989.
• H.-C. Chen, “Generalized reflexive matrices: special properties and applications,” SIAM Journal on Matrix Analysis and Applications, vol. 19, no. 1, pp. 140–153, 1998.
• L. Datta and S. D. Morgera, “On the reducibility of centrosymmetric matrices–-applications in engineering problems,” Circuits, Systems, and Signal Processing, vol. 8, no. 1, pp. 71–96, 1989.
• Z.-h. Peng, X.-y. Hu, and L. Zhang, “An efficient algorithm forthe least-squares reflexive solution of the matrix equation ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,” Applied Mathematics and Com-putation, vol. 181, no. 2, pp. 988–999, 2006.
• Y.-X. Peng, X.-Y. Hu, and L. Zhang, “An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations ${A}_{1}X{B}_{1}={C}_{1}$, ${A}_{2}X{B}_{2}={C}_{2}$,” Applied Mathematics and Computation, vol. 183, no. 2, pp. 1127–1137, 2006.
• F. Ding and T. Chen, “Gradient based iterative algorithms for solving a class of matrix equations,” IEEE Transactions on Automatic Control, vol. 50, no. 8, pp. 1216–1221, 2005.
• F. Ding and T. Chen, “On iterative solutions of general coupled matrix equations,” SIAM Journal on Control and Optimization, vol. 44, no. 6, pp. 2269–2284, 2006.
• F. Ding and T. Chen, “Hierarchical gradient-based identification of multivariable discrete-time systems,” Automatica, vol. 41, no. 2, pp. 315–325, 2005.
• Z.-Y. Peng, “A matrix LSQR iterative method to solve matrix equation $AXB=C$,” International Journal of Computer Mathematics, vol. 87, no. 8, pp. 1820–1830, 2010.
• Z.-y. Peng, “New matrix iterative methods for constraint sol-utions of the matrix equation $AXB=C$,” Journal of Computational and Applied Mathematics, vol. 235, no. 3, pp. 726–735, 2010.
• Z.-Y. Peng, “Solutions of symmetry-constrained least-squares problems,” Numerical Linear Algebra with Applications, vol. 15, no. 4, pp. 373–389, 2008.
• C. C. Paige, “Bidiagonalization of matrices and solutions of the linear equations,” SIAM Journal on Numerical Analysis, vol. 11, pp. 197–209, 1974.
• M. Wang, X. Cheng, and M. Wei, “Iterative algorithms for solv-ing the matrix equation $AXB+C{X}^{T}D=E$,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 622–629, 2007.
• L. Xie, J. Ding, and F. Ding, “Gradient based iterative solutions for general linear matrix equations,” Computers & Mathematics with Applications, vol. 58, no. 7, pp. 1441–1448, 2009.
• Z.-Y. Li, Y. Wang, B. Zhou, and G.-R. Duan, “Least squares solution with the minimum-norm to general matrix equations via iteration,” Applied Mathematics and Computation, vol. 215, no. 10, pp. 3547–3562, 2010.
• X. Duan, A. Liao, and B. Tang, “On the nonlinear matrix equation $X-\sum _{i=1}^{m}{A}^{\ast\,\!} _{i}{X}^{{\delta }_{i}}{A}_{i}=Q$,” Linear Algebra and its Applications, vol. 429, no. 1, pp. 110–121, 2008.
• X. Duan and A. Liao, “On the nonlinear matrix equation $X+{A}^{\ast\,\!}{X}^{-q}A=Q(q\geq 1)$,” Mathematical and Computer Modelling, vol. 49, no. 5-6, pp. 936–945, 2009.
• X. Duan and A. Liao, “On Hermitian positive definite solution of the matrix equation $X-\sum _{i=1}^{m}{A}^{\ast\,\!} _{i}{X}^{r}{A}_{i}=Q$,” Journal of Computational and Applied Mathematics, vol. 229, no. 1, pp. 27–36, 2009.
• X. Duan, C. Li, and A. Liao, “Solutions and perturbation analsis for the nonlinear matrix equation $X+\sum _{i=1}^{m}{A}_{i}^{\ast\,\!}{X}^{-1}{A}_{i}=I$,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4458–4466, 2011.
• X. Duan and A. Liao, “On the existence of Hermitian positive definite solutions of the matrix equation ${X}^{s}+{A}^{\ast\,\!}{X}^{-t}A=Q$,” Linear Algebra and its Applications, vol. 429, no. 4, pp. 673–687, 2008.
• M. Dehghan and M. Hajarian, “Finite iterative algorithms forthe reflexive and anti-reflexive solutions of the matrix equation ${A}_{1}{X}_{1}{B}_{1}+{A}_{2}{X}_{2}{B}_{2}=C$,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1937–1959, 2009.
• M. Dehghan and M. Hajarian, “Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflex-ive matrices,” Computational & Applied Mathematics, vol. 31, no. 2, pp. 353–371, 2012.
• M. Dehghan and M. Hajarian, “Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations,” Applied Mathematical Modelling, vol. 35, no. 7, pp. 3285–3300, 2011.
• M. Hajarian and M. Dehghan, “The generalized centro-symmetric and least squares generalized centro-symmetric solutions of the matrix equation $AYB+C{Y}^{T}D=E$,” Mathematical Methods in the Applied Sciences, vol. 34, no. 13, pp. 1562–1579, 2011.
• M. Dehghan and M. Hajarian, “The general coupled matrix equations over generalized bisymmetric matrices,” Linear Algebra and its Applications, vol. 432, no. 6, pp. 1531–1552, 2010.
• M. Dehghan and M. Hajarian, “An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices,” Applied Mathematical Modelling. Simulation and Computation for Engineering and Environmental Systems, vol. 34, no. 3, pp. 639–654, 2010.
• A.-G. Wu, X. Zeng, G.-R. Duan, and W.-J. Wu, “Iterative sol-utions to the extended Sylvester-conjugate matrix equations,” Applied Mathematics and Computation, vol. 217, no. 1, pp. 130–142, 2010.
• A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, “Iterative sol-utions to coupled Sylvester-conjugate matrix equations,” Computers & Mathematics with Applications, vol. 60, no. 1, pp. 54–66, 2010.
• A.-G. Wu, B. Li, Y. Zhang, and G.-R. Duan, “Finite iterative solutions to coupled Sylvester-conjugate matrix equations,” Applied Mathematical Modelling, vol. 35, no. 3, pp. 1065–1080, 2011.
• A.-G. Wu, L. Lv, and M.-Z. Hou, “Finite iterative algorithms for a common solution to a group of complex matrix equations,” Applied Mathematics and Computation, vol. 218, no. 4, pp. 1191–1202, 2011.
• A.-G. Wu, L. Lv, and M.-Z. Hou, “Finite iterative algorithms for extended Sylvester-conjugate matrix equations,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2363–2384, 2011.
• M. Wang, M. Wei, and Y. Feng, “An iterative algorithm for least squares problem in quaternionic quantum theory,” Computer Physics Communications, vol. 179, no. 4, pp. 203–207, 2008.
• F. Zhang, “Quaternions and matrices of quaternions,” Linear Algebra and its Applications, vol. 251, pp. 21–57, 1997.
• N. Le Bihan and J. Mars, “Singular value decomposition of qua-ternion matrices: a new tool for vector-sensor signal processing,” Signal Processing, vol. 84, no. 7, pp. 1177–1199, 2004.
• F. O. Farid, Q.-W. Wang, and F. Zhang, “On the eigenvalues of quaternion matrices,” Linear and Multilinear Algebra, vol. 59, no. 4, pp. 451–473, 2011.
• C. C. Took, D. P. Mandic, and F. Zhang, “On the unitary diagonalisation of a special class of quaternion matrices,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1806–1809, 2011.
• S. J. Sangwine and N. Le Bihan, “Quaternion singular value decomposition based on bidiagonalization to a real or complexmatrix using quaternion Householder transformations,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 727–738, 2006.
• S. De Leo and G. Scolarici, “Right eigenvalue equation in quaternionic quantum mechanics,” Journal of Physics A, vol. 33, no. 15, pp. 2971–2995, 2000.
• Q.-W. Wang, X. Liu, and S.-W. Yu, “The common bisymmetric nonnegative definite solutions with extreme ranks and inertias to a pair of matrix equations,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2761–2771, 2011.
• Q.-W. Wang, Y. Zhou, and Q. Zhang, “Ranks of the common solution to six quaternion matrix equations,” Acta Mathematicae Applicatae Sinica, vol. 27, no. 3, pp. 443–462, 2011.
• Q.-W. Wang, H.-X. Chang, and C.-Y. Lin, “$P$-(skew)symmetric common solutions to a pair of quaternion matrix equations,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 721–732, 2008.
• Q.-W. Wang, J. W. van der Woude, and H.-X. Chang, “A system of real quaternion matrix equations with applications,” Linear Algebra and its Applications, vol. 431, no. 12, pp. 2291–2303, 2009.
• Q. Wang, S. Yu, and W. Xie, “Extreme ranks of real matrices in solution of the quaternion matrix equation $AXB=C$ with applications,” Algebra Colloquium, vol. 17, no. 2, pp. 345–360, 2010.
• L. R. Fletcher, J. Kautsky, and N. K. Nichols, “Eigenstructure assignment in descriptor systems,” IEEE Transactions on Automatic Control, vol. AC-31, no. 12, pp. 1138–1141, 1986.
• L. Dai, Singular Control Systems, vol. 118 of Lecture Notes in Control and Information Sciences, Springer, Berlin, Germany, 1989.
• P. M. Frank, “Fault diagnosis in dynamic systems using analytical and knowledge-based redundancy: a survey and some new results,” Automatica, vol. 26, no. 3, pp. 459–474, 1990.
• Y.-X. Peng, X.-Y. Hu, and L. Zhang, “An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation $AXB=C$,” Applied Mathematics and Computation, vol. 160, no. 3, pp. 763–777, 2005.
• L. Xie, Y. Liu, and H. Yang, “Gradient based and least squares based iterative algorithms for matrix equations $AXB+C{X}^{T}D=F$,” Applied Mathematics and Computation, vol. 217, no. 5, pp. 2191–2199, 2010.
• F. Piao, Q. Zhang, and Z. Wang, “The solution to matrix equation $AX+{X}^{T}C=B$,” Journal of the Franklin Institute, vol. 344, no. 8, pp. 1056–1062, 2007.
• X. D. Zhang, Matrix Analysis and Applications, Tsinghua University Press, Beijing, China, 2004.
• R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991.