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The general coupled matrix equations over generalized bisymmetric matrices. (English) Zbl 1187.65042
Authors’ abstract: By extending the idea of the conjugate gradient (CG) method, we construct an iterative method to solve the general coupled matrix equations
$\sum _{j=1}^p A_{ij} X_j B_{ij} = M_i, \quad i = 1,2, \dots, p,$
(including the generalized (coupled) Lyapunov and Sylvester matrix equations as special cases) over generalized bisymmetric matrix group $$(X_1,X_2,\dots,X_p)$$. By using the iterative method, the solvability of the general coupled matrix equations over the generalized bisymmetric matrix group can be determined in the absence of roundoff errors. When the general coupled matrix equations are consistent over generalized bisymmetric matrices, a generalized bisymmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors.
The least Frobenius norm generalized bisymmetric solution group of the general coupled matrix equations can be derived when an appropriate initial iterative matrix group is chosen. In addition, the optimal approximation generalized bisymmetric solution group to a given matrix group $$(\widehat {X_1},\widehat {X_2},\dots,\widehat {X_p})$$ in Frobenius norm can be obtained by finding the least Frobenius norm generalized bisymmetric solution group of new general coupled matrix equations. The numerical results indicate that the iterative method works quite well in practice.

##### MSC:
 65F30 Other matrix algorithms (MSC2010) 15A24 Matrix equations and identities 65F10 Iterative numerical methods for linear systems
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##### References:
  Chang, X.W.; Wang, J.S., The symmetric solution of the matrix equations $$\mathit{AX} + \mathit{YA} = C$$, $$\mathit{AXA}^T + \mathit{BYB}^T + C$$ and $$(A^T \mathit{XA} . B^T \mathit{XB}) = (C, D)$$, Linear algebra appl., 179, 171-189, (1993)  Dai, H., On the symmetric solutions of a linear matrix equations, Linear algebra appl., 131, 1-7, (1990) · Zbl 0712.15009  Jameson, A.; Kreindler, E.; Lancaster, P., Symmetric, positive semidefinite, and positive definite real solutions or $$\mathit{AX} = \mathit{XA}^T$$ and $$\mathit{AX} = \mathit{YB}$$, Linear algebra appl., 160, 189-215, (1992)  Chu, K.W.E., Symmetric solutions of linear matrix equations by matrix decompositions, Linear algebra appl., 119, 35-50, (1989) · Zbl 0688.15003  Don, F.J.H., On the symmetric solutions of a linear matrix equation, Linear algebra appl., 93, 1-7, (1987) · Zbl 0622.15001  Baksalary, J.K.; Kala, R., The matrix equation $$\mathit{AXB} + \mathit{CYD} = E$$, Linear algebra appl., 30, 141-147, (1980) · Zbl 0437.15005  Dehghan, M.; Hajarian, M., An iterative algorithm for solving a pair of matrix equations $$\mathit{AYB} = E$$, $$\mathit{CYD} = F$$ over generalized centro-symmetric matrices, Comput. math. appl., 56, 3246-3260, (2008) · Zbl 1165.15301  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  Dehghan, M.; Hajarian, M., Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation $$A_1 X_1 B_1 + A_2 X_2 B_2 = C$$, Math. comput. model., 49, 1937-1959, (2009) · Zbl 1171.15310  Dehghan, M.; Hajarian, M., On the reflexive solutions of the matrix equation $$\mathit{AXB} + \mathit{CYD} = E$$, Bull. Korean math. soc., 46, 511-519, (2009) · Zbl 1170.15004  Dehghan, M.; Hajarian, M., Efficient iterative method for solving the second-order Sylvester matrix equation $$\mathit{EVF}^2 - \mathit{AVF} - \mathit{CV} = \mathit{BW}$$, IET control theory appl., 3, 1401-1408, (2009)  Dehghan, M.; Hajarian, M., A lower bound for the product of eigenvalues of solutions to matrix equations, Appl. math. lett., 22, 1786-1788, (2009) · Zbl 1190.15022  M. Dehghan, M. Hajarian, On the reflexive and anti-reflexive solutions of the generalized coupled Sylvester matrix equations, Int. J. Systems Sci., in press. · Zbl 1196.65081  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  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  Ding, F.; Chen, T., Iterative least squares solutions of coupled Sylvester matrix equations, Systems control lett., 54, 95-107, (2005) · Zbl 1129.65306  Ding, F.; Chen, T., Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica, 41, 315-325, (2005) · Zbl 1073.93012  Ding, F.; Chen, T., Hierarchical least squares identification methods for multivariable systems, IEEE trans. automat. control, 50, 397-402, (2005) · Zbl 1365.93551  Ding, F.; Chen, T., On iterative solutions of general coupled matrix equations, SIAM J. control optim., 44, 2269-2284, (2006) · Zbl 1115.65035  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  Hestenes, M.R., Conjugate direction methods in optimization, (1980), Springer-Verlag Berlin · Zbl 0421.49033  Peng, Y.X.; Hu, X.Y.; Zhang, L., An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation $$\mathit{AXB} = C$$, Appl. math. comput., 160, 763-777, (2005) · Zbl 1068.65056  Peng, Z.H.; Hu, X.Y.; Zhang, L., An efficient algorithm for the least-squares reflexive solution of the matrix equation $$A_1 \mathit{XB}_1 = C_1$$, $$A_2 \mathit{XB}_2 = C_2$$, Appl. math. comput., 181, 988-999, (2006) · Zbl 1115.65048  Peng, Z.Y.; Peng, Y.X., An efficient iterative method for solving the matrix equation $$\mathit{AXB} + \mathit{CYD} = E$$, Numer. linear algebra appl., 13, 473-485, (2006)  Reid, J.K., On the method of conjugate gradients for the solution of large sparse systems of linear equations, () · Zbl 0259.65037  Wang, Q.W.; Sun, J.H.; Li, S.Z., Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra, Linear algebra appl., 353, 169-182, (2002) · Zbl 1004.15017  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  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  Wang, Q.W.; Zhang, H.S.; Song, G.J., A new solvable condition for a pair of generalized Sylvester equations, Electron. J. linear algebra, 18, 289-301, (2009) · Zbl 1190.15019  Wang, Q.W.; Chang, H.X.; Ning, Q., The common solution to six quaternion matrix equations with applications, Appl. math. comput., 198, 209-226, (2008) · Zbl 1141.15016  Wang, Q.W., Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. math. appl., 49, 641-650, (2005) · Zbl 1138.15003  Wang, Q.W., The general solution to a system of real quaternion matrix equations, Comput. math. appl., 49, 665-675, (2005) · Zbl 1138.15004  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  Wang, Q.W.; Zhang, H.S.; Yu, S.W., On solutions to the quaternion matrix equation $$\mathit{AXB} + \mathit{CYD} = E$$, Electron. J. linear algebra, 17, 343-358, (2008)  Wang, Q.W.; Woude, J.W.; Chang, H.X., A system of real quaternion matrix equations with applications, Linear algebra appl., 431, 2291-2303, (2009) · Zbl 1180.15019  Wang, Q.W.; Li, C.K., Ranks and the least-norm of the general solution to a system of quaternion matrix equations, Linear algebra appl., 430, 1626-1640, (2009) · Zbl 1158.15010  Zhou, B.; Duan, G.R., An explicit solution to the matrix equation $$\mathit{AX} - \mathit{XF} = \mathit{BY}$$, Linear algebra appl., 402, 345-366, (2005)  Zhou, B.; Duan, G.R., A new solution to the generalized Sylvester matrix equation $$\mathit{AV} - \mathit{EVF} = \mathit{BW}$$, Systems control lett., 55, 193-198, (2006)  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  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  Zhou, B.; Yan, Z.B., Solutions to right coprime factorizations and generalized Sylvester matrix equations, Trans. inst. measure. control, 30, 397-426, (2008)  Zhou, B.; Duan, G.R., On the generalized Sylvester mapping and matrix equations, Systems control lett., 57, 200-208, (2008) · Zbl 1129.93018  Zhou, B.; Duan, G.R.; Li, Z.Y., Gradient based iterative algorithm for solving coupled matrix equations, Systems control lett., 58, 327-333, (2009) · Zbl 1159.93323  Zhou, B.; Li, Z.Y.; Duan, G.R.; Wang, Y., Solutions to a family of matrix equations by using the Kronecker matrix polynomials, Appl. math. comput., 212, 327-336, (2009) · Zbl 1181.15020
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