Journal of Applied Mathematics

Volume 2013 (2013), Article ID 697947, 11 pages

http://dx.doi.org/10.1155/2013/697947

Research Article

## An Iterative Method for the Least-Squares Problems of a General Matrix Equation Subjects to Submatrix Constraints

School of Mathematics and Statistics, Tianshui Normal University, Tianshui, Gansu 741001, China

Received 26 July 2013; Accepted 22 October 2013

Academic Editor: Debasish Roy

Copyright © 2013 Li-fang Dai et al. 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

- G. H. Golub and C. F. Van Loan,
*Matrix Computations*, John Hopkins University Press, Baltimore, Md, USA, 1996. - 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 - J. Respondek, “Controllability of dynamical systems with constraints,”
*Systems & Control Letters*, vol. 54, no. 4, pp. 293–314, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. S. Pressman, “Matrices with multiple symmetry properties: applications of centro-Hermitian and per-Hermitian matrices,”
*Linear Algebra and its Applications*, vol. 284, no. 1–3, pp. 239–258, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F.-Z. Zhou, X.-Y. Hu, and L. Zhang, “The solvability conditions for the inverse eigenvalue problems of centro-symmetric matrices,”
*Linear Algebra and Its Applications*, vol. 364, pp. 147–160, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Boley and G. H. Golub, “A survey of matrix inverse eigenvalue problems,”
*Inverse Problems*, vol. 3, no. 4, pp. 595–622, 1987. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-J. Bai, “The inverse eigenproblem of centrosymmetric matrices with a submatrix constraint and its approximation,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 26, no. 4, pp. 1100–1114, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. C. Paige, “Computing the generalized singular value decomposition,”
*Society for Industrial and Applied Mathematics*, vol. 7, no. 4, pp. 1126–1146, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X.-P. Pan, X.-Y. Hu, and L. Zhang, “A class of constrained inverse eigenproblem and associated approximation problem for skew symmetric and centrosymmetric matrices,”
*Linear Algebra and its Applications*, vol. 408, pp. 66–77, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F.-Z. Zhou, L. Zhang, and X.-Y. Hu, “Least-square solutions for inverse problems of centrosymmetric matrices,”
*Computers & Mathematics with Applications*, vol. 45, no. 10-11, pp. 1581–1589, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F.-L. Li, X.-Y. Hu, and L. Zhang, “Left and right inverse eigenpairs problem of skew-centrosymmetric matrices,”
*Applied Mathematics and Computation*, vol. 177, no. 1, pp. 105–110, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - O. Rojo and H. Rojo, “Some results on symmetric circulant matrices and on symmetric centrosymmetric matrices,”
*Linear Algebra and its Applications*, vol. 392, pp. 211–233, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. X. Xie, X. Y. Hu, and Y.-P. Sheng, “The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations,”
*Linear Algebra and Its Applications*, vol. 418, no. 1, pp. 142–152, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Y. Liu and H. Faßbender, “Some properties of generalized $K$-centrosymmetric $H$-matrices,”
*Journal of Computational and Applied Mathematics*, vol. 215, no. 1, pp. 38–48, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - F.-L. Li, X.-Y. Hu, and L. Zhang, “Left and right inverse eigenpairs problem of generalized centrosymmetric matrices and its optimal approximation problem,”
*Applied Mathematics and Computation*, vol. 212, no. 2, pp. 481–487, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M.-L. Liang, C.-H. You, and L.-F. Dai, “An efficient algorithm for the generalized centro-symmetric solution of matrix equation $AXB=C$,”
*Numerical Algorithms*, vol. 44, no. 2, pp. 173–184, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Q. X. Yin, “Construction of real antisymmetric and bi-antisymmetric matrices with prescribed spectrum data,”
*Linear Algebra and Its Applications*, vol. 389, pp. 95–106, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Deift and T. Nanda, “On the determination of a tridiagonal matrix from its spectrum and a submatrix,”
*Linear Algebra and its Applications*, vol. 60, pp. 43–55, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. S. Gong, X. Y. Hu, and L. Zhang, “The expansion problem of anti-symmetric matrix under a linear constraint and the optimal approximation,”
*Journal of Computational and Applied Mathematics*, vol. 197, no. 1, pp. 44–52, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. X. Yuan and H. Dai, “Inverse problems for symmetric matrices with a submatrix constraint,”
*Applied Numerical Mathematics*, vol. 57, no. 5-7, pp. 646–656, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. J. Zhao, X. Y. Hu, and L. Zhang, “Least squares solutions to $AX=B$ for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation,”
*Linear Algebra and its Applications*, vol. 428, no. 4, pp. 871–880, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A.-P. Liao and Y. Lei, “Least-squares solutions of matrix inverse problem for bi-symmetric matrices with a submatrix constraint,”
*Numerical Linear Algebra with Applications*, vol. 14, no. 5, pp. 425–444, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. P. Xu, M. S. Wei, and D. S. Zheng, “On solutions of matrix equation $AXB+CYD=F$,”
*Linear Algebra and Its Applications*, vol. 279, no. 1–3, pp. 93–109, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Zhou and G.-R. Duan, “On the generalized Sylvester mapping and matrix equations,”
*Systems & Control Letters*, vol. 57, no. 3, pp. 200–208, 2008. 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, “Finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns,”
*Mathematical and Computer Modelling*, vol. 52, no. 9-10, pp. 1463–1478, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Ding and T. Chen, “Iterative least-squares solutions of coupled Sylvester matrix equations,”
*Systems & Control Letters*, vol. 54, no. 2, pp. 95–107, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-H. Peng, X.-Y. Hu, and L. Zhang, “The bisymmetric solutions of the matrix equation ${A}_{1}{X}_{1}{B}_{1}+{A}_{2}{X}_{2}{B}_{2}+\cdots +{A}_{l}{X}_{l}{B}_{l}=C$,”
*Linear Algebra and its Applications*, vol. 426, no. 2-3, pp. 583–595, 2007. 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 - J.-F. Li, X.-Y. Hu, and L. Zhang, “The submatrix constraint problem of matrix equation $AXB+CYD=E$,”
*Applied Mathematics and Computation*, vol. 215, no. 7, pp. 2578–2590, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Meng, “Experimental design and decision support,” in
*Expert Systems, the Technology of Knowledge Management and Decision Making-Forthe 21st Century*, C. Leondes, Ed., Academic Press, 2001. View at Google Scholar - M. Baruch, “Optimization procedure to correct stiffness and flexibility matrices using vibration tests,”
*AIAA Journal*, vol. 16, no. 11, pp. 1208–1210, 1978. View at Publisher · View at Google Scholar - N. J. Higham, “Computing a nearest symmetric positive semidefinite matrix,”
*Linear Algebra and its Applications*, vol. 103, pp. 103–118, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-Y. Peng, X.-Y. Hu, and L. Zhang, “The nearest bisymmetric solutions of linear matrix equations,”
*Journal of Computational Mathematics*, vol. 22, no. 6, pp. 873–880, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A.-P. Liao, Z.-Z. Bai, and Y. Lei, “Best approximate solution of matrix equation $AXB+CYD=E$,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 27, no. 3, pp. 675–688, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z.-Y. Peng and Y.-X. Peng, “An efficient iterative method for solving the matrix equation $AXB+CYD=E$,”
*Numerical Linear Algebra with Applications*, vol. 13, no. 6, pp. 473–485, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y.-B. Deng, Z.-Z. Bai, and Y.-H. Gao, “Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations,”
*Numerical Linear Algebra with Applications*, vol. 13, no. 10, pp. 801–823, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet