Abstract and Applied Analysis

Volume 2013 (2013), Article ID 831656, 15 pages

http://dx.doi.org/10.1155/2013/831656

Research Article

## Iterative Algorithm for Solving a Class of Quaternion Matrix Equation over the Generalized -Reflexive Matrices

^{1}School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan,
Shandong Province 250002, China^{2}Department of Mathematics, Shanghai University, Shanghai 200444, China

Received 18 April 2013; Accepted 12 August 2013

Academic Editor: Douglas Anderson

Copyright © 2013 Ning Li and Qing-Wen Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Song, G. Chen, and X. Wang, “On solutions of quaternion matrix equations $XF-AX=BY$ and $XF-A\tilde{X}=BY$,”
*Acta Mathematica Scientia B*, vol. 32, no. 5, pp. 1967–1982, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - Z.-h. Peng, X.-y. Hu, and L. Zhang, “An efficient algorithm for the 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 Computation*, vol. 181, no. 2, pp. 988–999, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Ding and T. Chen, “Hierarchical gradient-based identification of multivariable discrete-time systems,”
*Automatica*, vol. 41, no. 2, pp. 315–325, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-y. Peng, “New matrix iterative methods for constraint solutions of the matrix equation $AXB=C$,”
*Journal of Computational and Applied Mathematics*, vol. 235, no. 3, pp. 726–735, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-Y. Peng, “Solutions of symmetry-constrained least-squares problems,”
*Numerical Linear Algebra with Applications*, vol. 15, no. 4, pp. 373–389, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. C. Paige, “Bidiagonalization of matrices and solutions of the linear equations,”
*SIAM Journal on Numerical Analysis*, vol. 11, pp. 197–209, 1974. View at Publisher · View at Google Scholar · View at MathSciNet - M. Wang, X. Cheng, and M. Wei, “Iterative algorithms for solving the matrix equation $AXB+C{X}^{T}D=E$,”
*Applied Mathematics and Computation*, vol. 187, no. 2, pp. 622–629, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Duan, A. Liao, and B. Tang, “On the nonlinear matrix equation $X-{\sum}_{i=1}^{m}{A}^{*}\mathrm{}{}_{i}{X}^{{\delta}_{i}}{A}_{i}=Q$,”
*Linear Algebra and its Applications*, vol. 429, no. 1, pp. 110–121, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Duan and A. Liao, “On the nonlinear matrix equation $X+{A}^{*}\mathrm{}{X}^{-q}A=Q(q\ge 1)$,”
*Mathematical and Computer Modelling*, vol. 49, no. 5-6, pp. 936–945, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Duan and A. Liao, “On Hermitian positive definite solution of the matrix equation $X-{\sum}_{i=1}^{m}{A}^{*}\mathrm{}{}_{i}{X}^{r}{A}_{i}=Q$,”
*Journal of Computational and Applied Mathematics*, vol. 229, no. 1, pp. 27–36, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Duan, C. Li, and A. Liao, “Solutions and perturbation analsis for the nonlinear matrix equation $X+{\sum}_{i=1}^{m}{A}_{i}^{*}{X}^{-1}{A}_{i}=I$,”
*Applied Mathematics and Computation*, vol. 218, no. 8, pp. 4458–4466, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - X. Duan and A. Liao, “On the existence of Hermitian positive definite solutions of the matrix equation ${X}^{s}+{A}^{*}\mathrm{}{X}^{-t}A=Q$,”
*Linear Algebra and its Applications*, vol. 429, no. 4, pp. 673–687, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Dehghan and M. Hajarian, “Finite iterative algorithms for the 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Dehghan and M. Hajarian, “Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices,”
*Computational & Applied Mathematics*, vol. 31, no. 2, pp. 353–371, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A.-G. Wu, X. Zeng, G.-R. Duan, and W.-J. Wu, “Iterative solutions to the extended Sylvester-conjugate matrix equations,”
*Applied Mathematics and Computation*, vol. 217, no. 1, pp. 130–142, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A.-G. Wu, G. Feng, G.-R. Duan, and W.-J. Wu, “Iterative solutions to coupled Sylvester-conjugate matrix equations,”
*Computers & Mathematics with Applications*, vol. 60, no. 1, pp. 54–66, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Zhang, “Quaternions and matrices of quaternions,”
*Linear Algebra and its Applications*, vol. 251, pp. 21–57, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Le Bihan and J. Mars, “Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing,”
*Signal Processing*, vol. 84, no. 7, pp. 1177–1199, 2004. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - S. J. Sangwine and N. Le Bihan, “Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations,”
*Applied Mathematics and Computation*, vol. 182, no. 1, pp. 727–738, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Google Scholar · View at Scopus - L. Dai,
*Singular Control Systems*, vol. 118 of*Lecture Notes in Control and Information Sciences*, Springer, Berlin, Germany, 1989. View at Publisher · View at Google Scholar · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Scopus - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at Publisher · View at Google Scholar · View at MathSciNet