{"title":"Iterative solutions to the linear matrix equation AXB + CXTD = E","authors":"Yongxin Yuan, Jiashang Jiang","volume":55,"journal":"International Journal of Mathematical and Computational Sciences","pagesStart":1046,"pagesEnd":1050,"ISSN":"1307-6892","URL":"https:\/\/publications.waset.org\/pdf\/12577","abstract":"In this paper the gradient based iterative algorithm is\r\npresented to solve the linear matrix equation AXB +CXTD = E,\r\nwhere X is unknown matrix, A,B,C,D,E are the given constant\r\nmatrices. It is proved that if the equation has a solution, then the\r\nunique minimum norm solution can be obtained by choosing a special\r\nkind of initial matrices. Two numerical examples show that the\r\nintroduced iterative algorithm is quite efficient.","references":"[1] G. R. Duan. Solutions to matrix equation AV + BW = V F and\r\ntheir application to eigenstructure assignment in linear systems. IEEE\r\nTransactions on Automatic Control, 38 (1993) 276-280.\r\n[2] S. P. Bhattacharyya, E. De Souza. Pole assignment via Sylvester-s\r\nequation. Systems and Control Letters, 1 (1972) 261-263.\r\n[3] G. R. Duan. On the solution to Sylvester matrix equation AV +BW =\r\nEV F. IEEE Transactions on Automatic Control, 41 (1996) 612-614.\r\n[4] K. Zhou, J. Doyle, K. Glover. Robust and optimal control. Prentice-Hall,\r\n1996.\r\n[5] B. Zhou, G. R. Duan. An explicit solution to the matrix equation AX\u2212 XF = BY . Linear Algebra and its Applications, 402 (2005) 345-366.\r\n[6] B. Zhou, G. R. Duan. A new solution to the generalized Sylvester matrix\r\nequation AV \u2212 EV F = BW. Systems & Control Letters, 55 (2006)\r\n193-198.\r\n[7] F. Ding, T. Chen. Iterative least squares solutions of coupled Sylvester\r\nmatrix equations. Systems & Control Letters, 54 (2005) 95-107.\r\n[8] H. Mukaidani, H. Xu, K. Mizukami. New iterative algorithm for\r\nalgebraic Riccati equation related to H\u221e control problem of singularly\r\nperturbed systems. IEEE Transactions on Automatic Control, 46 (2001)\r\n1659-1666.\r\n[9] I. Borno, Z. Gajic. Parallel algorithm for solving coupled algebraic\r\nLyapunov equations of discrete-time jump linear systems. Computers\r\n& Mathematics with Applications, 30 (1995) 1-4.\r\n[10] C. T. Kelley. Iterative Methods for Linear and Nonlinear Equations.\r\nSIAM, Philadelphia, 1995.\r\n[11] T. Mori, A. Derese. A brief summary of the bounds on the solution of\r\nthe algebraic matrix equations in control theory. International Journal of\r\nControl, 39 (1984) 247-256.\r\n[12] M. Mrabti, M. Benseddik. Unified type non-stationary Lyapunov matrix\r\nequations-simultaneous eigenvalue bounds. Systems & Control Letters,\r\n18 (1995) 73-81.\r\n[13] L. Xie, Y. Liu, H. Yang. Gradient based and least squares based iterative\r\nalgorithms for matrix equations AXB + CXTD = F. Applied\r\nMathematics and Computation, 217 (2010) 2191-2199.\r\n[14] L. Xie, J. Ding, F. Ding. Gradient based iterative solutions for general\r\nlinear matrix equations. Computers and Mathematics with Applications,\r\n58 (2009) 1441-1448.\r\n[15] F. Ding, T. Chen. On iterative solutions of general coupled matrix\r\nequations. SIAM Journal on Control and Optimization, 44 (2006) 2269-\r\n2284.\r\n[16] F. Ding, P. X. Liu, J. Ding. Iterative solutions of the generalized Sylvester\r\nmatrix equations by using the hierarchical identification principle. Applied\r\nMathematics and Computation, 197 (2008) 41-50.\r\n[17] J. Ding, Y. Liu, F. Ting. Iterative solutions to matrix equations of the\r\nform AiXBi = Fi. Computers and Mathematics with Applications, 59\r\n(2010) 3500-3507.\r\n[18] P. Lancaster, M. Tismenetsky. The Theory of Matrices. 2rd Edition.\r\nLondon: Academic Press, 1985.\r\n[19] A. Ben-Israel, T. N. E. Greville. Generalized Inverses. Theory and\r\nApplications (second ed). New York: Springer, 2003.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 55, 2011"}