Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation $$A_1X_1B_1+A_2X_2B_2=C$$.(English)Zbl 1171.15310

Summary: A matrix $$P\in\mathbb{R}^{n\times n}$$ is called a generalized reflection matrix if $$P^T=P$$ and $$P^{2}=I$$. An $$n\times n$$ matrix $$A$$ is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix $$P$$ if $$A=PAP$$ $$(A=-PAP)$$. In this paper, three iterative algorithms are proposed for solving the linear matrix equation $$A_{1}X_{1}B_{1}+A_{2}X_{2}B_{2}=C$$ over reflexive (anti-reflexive) matrices $$X_{1}$$ and $$X_{2}$$. When this matrix equation is consistent over reflexive (anti-reflexive) matrices, for any reflexive (anti-reflexive) initial iterative matrices, the reflexive (anti-reflexive) solutions can be obtained within finite iterative steps in the absence of roundoff errors. By the proposed iterative algorithms, the least Frobenius norm reflexive (anti-reflexive) solutions can be derived when spacial initial reflexive (anti-reflexive) matrices are chosen. Furthermore, we also obtain the optimal approximation reflexive (anti-reflexive) solutions to the given reflexive (anti-reflexive) matrices in the solution set of the matrix equation. Finally, some numerical examples are presented to support the theoretical results of this paper.

MSC:

 15A24 Matrix equations and identities 65F10 Iterative numerical methods for linear systems
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References:

 [1] Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), The Johns Hopkins University Press: The Johns Hopkins University Press Baltimore, London · Zbl 0865.65009 [2] Horn, R. A.; Johnson, C. R., Topics in Matrix Analysis (1991), Cambridge University Press · Zbl 0729.15001 [3] Chen, H. C., Generalized reflexive matrices: Special properties and applications, SIAM J. Matrix Anal. Appl., 19, 140-153 (1998) · Zbl 0910.15005 [4] Chen, H. C.; Sameh, A., Numerical linear algebra algorithms on the ceder system, (Noor, A. K., Parallel Computations and Their Impact on Mechanics. Parallel Computations and Their Impact on Mechanics, AMD, vol. 86 (1987), The American Society of Mechanical Engineers), 101-125 [5] Zhou, F. Z., The solvability conditions for the inverse eigenvalue problems of reflexive matrices, J. Comput. Appl. Math., 188, 180-189 (2006) · Zbl 1088.15015 [7] 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 [8] Dehghan, M.; Hajarian, M., An iterative algorithm for solving a pair of matrix equations $$A Y B = E, C Y D = F$$ over generalized centro-symmetric matrices, Comput. Math. Appl., 56, 3246-3260 (2008) · Zbl 1165.15301 [9] Wang, Q. W., Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl., 49, 641-650 (2005) · Zbl 1138.15003 [10] Wang, Q. W., The general solution to a system of real quaternion matrix equations, Comput. Math. Appl., 49, 665-675 (2005) · Zbl 1138.15004 [11] Wang, Q. W.; Zhang, F., The reflexive re-nonnegative definite solution to a quaternion matrix equation, Electron. J. Linear Algebra, 17, 88-101 (2008) · Zbl 1147.15012 [12] Wang, Q. W.; Zhang, H. S.; Yu, S. W., On solutions to the quaternion matrix equation $$A X B + C Y D = E$$, Electron. J. Linear Algebra, 17, 343-358 (2008) [13] Zhou, B.; Duan, G. R., An explicit solution to the matrix equation $$A X - X F = B Y$$, Linear Algebra Appl., 402, 345-366 (2005) [14] Peng, Y. X.; Hu, X. Y., An iteration method for the symmetric solutions and optimal approximation solution of the matrix equation $$A X B = C$$, Appl. Math. Comput., 160, 163-167 (2005) [15] 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 [16] Peng, X. Y.; Hu, X. Y.; Zhang, L., The reflexive and anti-reflexive solutions of the matrix equation $$A^H X B = C$$, J. Comput. Appl. Math., 186, 638-645 (2007) [17] Xu, G.; Wei, M.; Zheng, D., On solutions of matrix equation $$A X B + C Y D = F$$, Linear Algebra Appl., 279, 93-109 (1980) [18] Chu, K. E., Singular value and generalized value decompositions and the solution of linear matrix equations, Linear Algebra Appl., 87, 83-98 (1987) · Zbl 0612.15003 [19] Wang, Q. W., A system of four matrix equations over von Neumann regular rings and its applications, Acta Math. Sinica, Ser. A, 21, 323-334 (2005) · Zbl 1083.15021 [20] 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 [21] Wu, A. G.; Fu, Y. M.; Duan, G. R., On solutions of matrix equations $$V - A V F = B W$$ and $$V - A \overline{V} F = B W$$, Math. Comput. Model., 47, 1181-1197 (2008) · Zbl 1145.15302 [22] Wu, A. G.; Duan, G. R.; Yu, H. H., On solutions of the matrix equations $$X F - A X = C$$ and $$X F - A \overline{X} = C$$, Appl. Math. Comput., 183, 932-941 (2006) · Zbl 1112.15018 [23] Zhou, B.; Duan, G. R., A new solution to the generalized Sylvester matrix equation $$A V - E V F = B W$$, Systems Control Lett., 55, 193-198 (2006) [24] Zhou, B.; Duan, G. R., Solutions to generalized Sylvester matrix equation by Schur decomposition, Internat. J. Systems Sci., 38, 369-375 (2007) · Zbl 1126.65034 [25] Zhou, B.; Li, Z. Y.; Duan, G. R.; Wang, Y., Weighted least squares solutions to general coupled Sylvester matrix equations, J. Comput. Appl. Math., 224, 759-776 (2009) · Zbl 1161.65034 [26] Zhou, B.; Duan, G. R., On the generalized Sylvester mapping and matrix equations, Systems Control Lett., 57, 200-208 (2008) · Zbl 1129.93018 [27] 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 [28] Ding, F.; Chen, T., Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica, 41, 315-325 (2005) · Zbl 1073.93012 [29] Ding, F.; Chen, T., Iterative least squares solutions of coupled Sylvester matrix equations, Systems Control Lett., 54, 95-107 (2005) · Zbl 1129.65306 [30] Ding, F.; Chen, T., On iterative solutions of general coupled matrix equations, SIAM J. Control Optim., 44, 2269-2284 (2006) · Zbl 1115.65035 [31] Ding, F.; Chen, T., Hierarchical least squares identification methods for multivariable systems, IEEE Trans. Automat. Control, 50, 397-402 (2005) · Zbl 1365.93551 [32] Cvetković-Ilić, D. S., The reflexive solutions of the matrix equation $$A X B = C$$, Comput. Math. Appl., 51, 879-902 (2006) · Zbl 1136.15011 [33] 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 [36] Duan, G. R.; Zhou, B., Solution to the second-order Sylvester matrix equation $$M V F^2 - D V F - K V = B W$$, IEEE Trans. Automat. Control, 51, 805-809 (2006) [37] Zhou, B.; Yan, Z. B., Solutions to right coprime factorizations and generalized Sylvester matrix equations, Trans. Inst. Meas. Control, 30, 397-426 (2008)
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