×

An iterative algorithm for solving a pair of matrix equations \(AYB=E\), \(CYD=F\) over generalized centro-symmetric matrices. (English) Zbl 1165.15301

Summary: A matrix \(P\in\mathbb{R}^{n\times n}\) is said to be a symmetric orthogonal matrix if \(P=P^T=P^{-1}\). A matrix \(A\in\mathbb{R}^{n\times n}\) is said to be generalized centro-symmetric (generalized central anti-symmetric) with respect to \(P\), if \(A=PAP\) (\(A= - PAP\)). The generalized centro-symmetric matrices have wide applications in information theory, linear estimate theory and numerical analysis. In this paper, we propose a new iterative algorithm to compute a generalized centro-symmetric solution of the linear matrix equations \(AYB=E\), \(CYD=F\). We show, when the matrix equations are consistent over generalized centro-symmetric matrix \(Y\), for any initial generalized centro-symmetric matrix \(Y_{1}\), the sequence \(\{Y_k\}\) generated by the introduced algorithm converges to a generalized centro-symmetric solution of matrix equations \(AYB=E\), \(CYD=F\). The least Frobenius norm generalized centro-symmetric solution can be derived when a special initial generalized centro-symmetric matrix is chosen. Furthermore, the optimal approximation generalized centro-symmetric solution to a given generalized centro-symmetric matrix can be derived. Several numerical examples are given to show the efficiency of the presented method.

MSC:

15A24 Matrix equations and identities
65F30 Other matrix algorithms (MSC2010)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Datta, B. N., Numerical Linear Algebra and Applications (1995), Brooks/Cole Publishing Co.: Brooks/Cole Publishing Co. Pacific Grove, CA · Zbl 1182.65001
[2] Golub, G. H.; Van Loan, C. F., Matrix computations (1996), The Johns Hopkins University Press: The Johns Hopkins University Press Baltimore and London · Zbl 0865.65009
[3] Liang, M. L.; You, C. H.; Dai, L. F., An efficient algorithm for the generalized centro-symmetric solution of matrix equation \(A X B = C\), Numer. Algor., 4, 173-184 (2007) · Zbl 1129.65030
[4] van der Woude, W., On the existence of a common solution \(X\) to the matrix equations \(A_i X B_j = C_{i j},(i, j) \in \Gamma \), Linear Algebra Appl., 375, 135-145 (2003) · Zbl 1037.15014
[5] Chu, K. E., Symmetric solutions of linear matrix equations by matrix decompositions, Linear Algebra Appl., 119, 35-50 (1989) · Zbl 0688.15003
[6] Chang, X. W.; Wang, J. S., The symmetric solution of the matrix equation \(A Y + Z A = C, A Y A^T + B Z B^T = C\), and \((A^T Y A, B^T Y B) = (C, D)\), Linear Algebra Appl., 179, 171-189 (1993) · Zbl 0765.15002
[7] Bischof, C. H.; Datta, B. N.; Purkayastha, A., Parallel algorithm for the Sylvester-observer equation, SIAM J. Sci. Computat., 17, 686-698 (1996) · Zbl 0855.65043
[8] Dai, H., On the symmetric solutions of linear matrix equations, Linear Algebra Appl., 131, 1-7 (1990) · Zbl 0712.15009
[9] Hou, J. J.; Peng, Z. Y.; Zhang, X. L., An iterative method for the least squares solution of matrix equation \(A X B = C\), Numer. Algor., 42, 181-192 (2006) · Zbl 1122.65038
[10] Dajić, A.; Koliha, J. J., Positive solutions to the equations \(A X = C\) and \(X B = D\) for Hilbert space operators, J. Math. Anal. Appl., 333, 567-576 (2007) · Zbl 1120.47009
[11] Cvetković-Iliíc, D.; Dajić, A.; Koliha, J. J., Positive and real-positive solutions to the equation \(a x a^\ast = c\) in \(C^\ast \)-algebras, Linear Multilinear Algebra, 55, 535-543 (2007) · Zbl 1180.47014
[12] Cvetković-Iliíc, D., The reflexive solutions of the matrix equations \(A X B = C\), Comput. Math. Appl., 51, 879-902 (2006) · Zbl 1136.15011
[13] Navarra, A.; Odell, P. L.; Young, D. M., A representation of the general common solution to the matrix equations \(A_1 X B_1 = C_1\) and \(A_2 X B_2 = C_2\) with applications, Comput. Math. Appl., 41, 929-935 (2001) · Zbl 0983.15016
[14] Xu, G.; Wei, M.; Zheng, D., On solutions of matrix equation \(A X B + C Y D = F\), Linear Algebra Appl., 279, 93-109 (1998) · Zbl 0933.15024
[15] Wang, Q. W., A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity, Linear Algebra Appl., 384, 43-54 (2004) · Zbl 1058.15015
[16] Wang, Q. W., A system of four matrix equations over von Neumann regular rings and Its applications, Acta Math. Sin. Engl. Ser., 21, 323-334 (2005) · Zbl 1083.15021
[17] Peng, Y. X.; Hu, X. Y.; Zhang, L., An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation \(A X B = C\), Appl. Math. Comput., 160, 763-777 (2005) · Zbl 1068.65056
[18] Huang, G. X.; Yin, F.; Guo, K., An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation \(A X B = C\), J. Comput. Appl. Math., 212, 231-244 (2008) · Zbl 1146.65036
[19] Peng, Y. X.; Hu, X. Y.; Zhang, L., An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation \(A X B = C\), Appl. Math. Comput., 160, 763-777 (2005) · Zbl 1068.65056
[20] Li, Y. T.; Wu, W. J., Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl., 55, 1142-1147 (2008) · Zbl 1143.15012
[21] Wang, M.; Cheng, X.; Wei, M., Iterative algorithms for solving the matrix equation \(A X B + C X^T D = E\), Appl. Math. Comput., 187, 622-629 (2007) · Zbl 1121.65048
[22] Jiang, T.; Wei, M., On solutions of the matrix equations \(X - A X B = C\) and \(X - A \overline{X} B = C\), Linear Algebra Appl., 367, 429-436 (2003)
[23] Burde, D., On the matrix equation \(X A - A X = X^p\), Linear Algebra Appl., 404, 147-165 (2005) · Zbl 1079.15015
[24] Ding, F.; Chen, T., Gradient based iterative algorithms for solving a class of matrix equations, IEEE Trans. Automat. Control, 50, 1216-1221 (2005) · Zbl 1365.65083
[25] Ding, F.; Liu, P. X.; Ding, J., Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Appl. Math. Comput., 197, 41-50 (2008) · Zbl 1143.65035
[26] Ding, F.; Chen, T., Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica, 41, 315-325 (2005) · Zbl 1073.93012
[27] Ding, F.; Chen, T., Iterative least squares solutions of coupled Sylvester matrix equations, Systems Control Lett., 54, 95-107 (2005) · Zbl 1129.65306
[28] Ding, F.; Chen, T., On iterative solutions of general coupled matrix equations, SIAM J. Control Optim., 44, 2269-2284 (2006) · Zbl 1115.65035
[29] Ding, F.; Chen, T., Hierarchical least squares identification methods for multivariable systems, IEEE Trans. Automat. Control, 50, 397-402 (2005) · Zbl 1365.93551
[30] M. Dehghan, M. Hajarian, The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations, Rocky Mountain J. Math. (in press); M. Dehghan, M. Hajarian, The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations, Rocky Mountain J. Math. (in press) · Zbl 1198.15011
[31] Dehghan, M.; Hajarian, M., An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation, Appl. Math. Comput., 202, 571-588 (2008) · Zbl 1154.65023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.