×

The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations. (English) Zbl 1198.15011

By using appropriate matrix decompositions, necessary and sufficient solvability conditions and explicit reflexive and anti-reflexive solutions for the following matrix equation: 9mm
(I)
\(A_1 XB_1 = D_1\), and systems of matrix equations:
(II)
\(A_1 X = C_1, XB_2 = C_2\), and
(III)
\(A_1 X = C_1, XB_2 = C_2, A_3 X = C_3, XB_4 = C_4\) are derived and formulated, respectively.

MSC:

15A24 Matrix equations and identities
15A06 Linear equations (linear algebraic aspects)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] H.C. Chen, Generalized reflexive matrices : Special properties and applications , SIAM J. Matrix Anal. Appl. 19 (1998), 140-153. · Zbl 0910.15005
[2] ——–, The SAS domain decomposition method for structural analysis , CSRD Teach, report 754, Center for Supercomputing Research and Development, University of Illinois, Urbana, IL, 1988.
[3] H.C. Chen and A. Sameh, Numerical linear algebra algorithms on the ceder system , in Parallel computations and their impact on mechanics , A.K. Noor, ed., AMD 86 , The American Society of Mechanical Engineers, 1987
[4] K.W.E. Chu, Singular value and generalized sigular value decompositions and the solution of linear matrix equations , Linear Algebra Appl. 88 /89 (1978), 83-89. · Zbl 0612.15003
[5] K.W.E. Chu, Symmetric solutions of linear matrix equations by matrix decompositions , Linear Algebra Appl. 119 (1989), 35-50. · Zbl 0688.15003
[6] D.S. Cvetković-Ilíc, The reflexive solutions of the matrix equations \(AXB=C\) , Comp. Math. Appl. 51 (2006), 879-902. · Zbl 1136.15011
[7] H. Dai, On the symmetric solutions of linear matrix equations , Linear Algebra Appl. 131 (1990), 1-7. · Zbl 0712.15009
[8] M. Dehghan and M. Hajarian, An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation , Appl. Math. Comput. 202 (2008), 571-588. · Zbl 1154.65023
[9] ——–, An iterative algorithm for solving a pair of matrix equations \(AY B = E\), \(CY D = F\) over generalized centro-symmetric matrices , Comput. Math. Appl. 56 (2008), 3246-3260. · Zbl 1165.15301
[10] ——–, On the reflexive solutions of the matrix equation \(AXB + CY D = E\) , Bull. Korean Math. Soc. 46 (2009), 511-519. · Zbl 1170.15004
[11] ——–, Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation \(A_1X_1B_1 + A_2X_2B_2 = C\) , Math. Comput. Model. 49 (2009), 1937-1959. · Zbl 1171.15310
[12] ——–, An efficient iterative method for solving the second-order Sylvester matrix equation \(EV F^2 - AV F - CV = BW\) , IET Control Theory Appl. 3 (2009), 1401-1408.
[13] ——–, The general coupled matrix equations over generalized bisymmetric matrices , Linear Algebra Appl. 432 (2010), 1531-1552. · Zbl 1187.65042
[14] ——–, An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices , Appl. Math. Modelling 34 (2010), 639-654. · Zbl 1185.65054
[15] ——–, An efficient algorithm for solving general coupled matrix equations and its application , Math. Comput. Model. 51 (2010), 1118-1134. · Zbl 1208.65054
[16] Y.-B Deng and X.-Y Hu, On solutions of matrix equation \(AXA^T + BYB^T = C\) , J. Comp. Math. 23 (2005), 17-26. · Zbl 1067.15008
[17] F. Ding and T. Chen, Gradient based iterative algorithms for solving a class of matrix equations , IEEE Trans. Autom. Contr. 50 (2005), 1216-1221. · Zbl 1365.65083
[18] ——–, Hierarchical gradient-based identification of multivariable discrete-time systems , Automatica 41 (2005), 315-325. · Zbl 1073.93012
[19] ——–, Iterative least squares solutions of coupled Sylvester matrix equations , Systems Contr. Lett. 54 (2005), 95-107. · Zbl 1129.65306
[20] ——–, On iterative solutions of general coupled matrix equations , SIAM J. Control Optim. 44 (2006), 2269-2284. · Zbl 1115.65035
[21] ——–, Hierarchical least squares identification methods for multivariable systems , IEEE Trans. Autom. Contr. 50 (2005), 397-402. · Zbl 1365.93551
[22] F. Ding, P.X. Liu and J. Ding, Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle , Appl. Math. Comput. 197 (2008), 41-50. · Zbl 1143.65035
[23] L. Huang and Q. Zeng, The solvability of matrix equation \(AXB + CYD = E\) over a simple Artinian ring , Linear and Multilinear Algebra 38 (1995), 225-232. · Zbl 0824.15015
[24] T. Jiang and M. Wei, On solutions of the matrix equations \(X-AXB=C\) and \(X-A\overlineXB=C\) , Linear Algebra Appl. 367 (2003), 429-436. · Zbl 1019.15002
[25] Y.-T. Li and W.-J. Wu, Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations , Comp. Math. with Applications (2007),
[26] S.K. Mitra, Common solutions to a pair of linear matrix equations \(A_1XB_1=C_1\) and \(A_2XB_2=C_2\) , Proc. Cambridge Philos. Soc. 74 (1973), 213-216. · Zbl 0262.15010
[27] ——–, A pair of simultaneous linear matrix equations and a matrix programming problem , Linear Algera Appl. 131 (1990), 97-123. · Zbl 0712.15010
[28] G.L. Morris and P. L. Odell, Common solutions for \(n\) matrix equations, J. Assn. Com. Mach. 15 (1968), 272–274. · Zbl 0157.22602
[29] A. Navarra, P.L. Odell and D.M. Young, A representation of the general common solution to the Matrix equations \(A_1XB_1=C_1\) and \(A_2XB_2=C_2\) with applications , Comp. Math. with Appl. 41 (2001), 929-935. · Zbl 0983.15016
[30] Z.Y. Peng, An iterative method for the least squares symmetric solution of the linear matrix equation \(AXB = C\) , Appl. Math. Comp. 170 (2005), 711-723. · Zbl 1081.65039
[31] Z.Y. Peng and X.Y. Hu, The reflexive and antireflexive solutions of the matrix equation \(AX=B\) , Linear Algebra Appl. 375 (2003), 147-155. · Zbl 1050.15016
[32] X.-Y. Peng, X.-Y. Hu and L. Zhang, The reflexive and anti-reflexive solutions of the matrix equation \(A^HXB=C\) , J. Comp. Appl. Math. 200 (2007), 749-760. · Zbl 1115.15014
[33] W.E. Roth, The equations \(AX - YB = C\) and \(AX - XB = C\) in matrices , Proc. Amer. Math. Soc. 3 (1952), 392-396. · Zbl 0047.01901
[34] N. Shinozaki and M. Sibuya, Consistency of a pair of matrix equations with an application , Keio Engineering Rep. 27 (1974), 141-146. · Zbl 0409.15010
[35] J. von der Woude, Feedback decoupling and stabilization for linear systems with multiple exogenous variables , Ph.D Thesis, Technical Univ. of Eindhoven, Netherland, 1997.
[36] D. von Rosen, Some results on homogeneous matrix equations , SIAM J. Matrix Anal. 14 (1993), 137-145. · Zbl 0768.15008
[37] Q.W. Wang, A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity , Linear Algebra Appl. 384 (2004), 43-54. · Zbl 1058.15015
[38] ——–, A system of four matrix equations over von Neumann regular rings and its applications , Acta Math. Sinica, English Series 21 (2005), 323-334. · Zbl 1083.15021
[39] ——–, The general solution to a system of real quaternion matrix equations , Comput. Math. Appl. 49 (2005), 665-675. · Zbl 1138.15004
[40] ——–, Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations , Comput. Math. Appl. 49 (2005), 641-650. · Zbl 1138.15003
[41] Q.W. Wang, H.X. Chang and Q. Ning, The common solution to six quaternion matrix equations with applications , Appl. Math. Comput. 198 (2008), 209-226. · Zbl 1141.15016
[42] Q.W. Wang and C.K. Li, Ranks and the least-norm of the general solution to a system of quaternion matrix equations , Linear Algebra Appl. 430 (2009), 1626-1640. · Zbl 1158.15010
[43] Q.W. Wang, H.S. Zhang and S.W. Yu, On solutions to the quaternion matrix equation \(AXB +CY D = E\) , Electron. J. Linear Algebra 17 (2008), 343-358. · Zbl 1154.15019
[44] J.R. Weaver, Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, eigenvectors , Amer. Math. Monthly 92 (1985), 711-717. JSTOR: · Zbl 0619.15021
[45] G. Xu, M. Wei and D. Zheng, On solutions of matrix equation \(AXB+CYD=E\) , Linear Algebra Appl. 279 (1998), 93-109. · Zbl 0933.15024
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.