Finite iterative solutions to coupled Sylvester-conjugate matrix equations

Abstract This paper is concerned with solutions to the so-called coupled Sylveter-conjugate matrix equations, which include the generalized Sylvester matrix equation and coupled Lyapunov matrix equation as special cases. An iterative algorithm is constructed to solve this kind of matrix equations. By using the proposed algorithm, the existence of a solution to a coupled Sylvester-conjugate matrix equation can be determined automatically. When the considered matrix equation is consistent, it is proven by using a real inner product in complex matrix spaces as a tool that a solution can be obtained within finite iteration steps for any initial values in the absence of round-off errors. Another feature of the proposed algorithm is that it can be implemented by using original coefficient matrices, and does not require to transform the coefficient matrices into any canonical forms. The algorithm is also generalized to solve a more general case. Two numerical examples are given to illustrate the effectiveness of the proposed methods.

[1]  Zeyad Abdel Aziz Al Zhour,et al.  Vector least-squares solutions for coupled singular matrix equations , 2007 .

[2]  Musheng Wei,et al.  On solutions of the matrix equations X−AXB=C and X−AXB=C , 2003 .

[3]  James Lam,et al.  Iterative solutions of coupled discrete Markovian jump Lyapunov equations , 2008, Comput. Math. Appl..

[4]  Mehdi Dehghan,et al.  An iterative algorithm for solving a pair of matrix equations AYB=E, CYD=F over generalized centro-symmetric matrices , 2008, Comput. Math. Appl..

[5]  Ai-Guo Wu,et al.  On solutions of matrix equations V-AVF=BW and V-AV'F=BW , 2008, Math. Comput. Model..

[6]  Mehdi Dehghan,et al.  An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation , 2008, Appl. Math. Comput..

[7]  Lei Zhang,et al.  An efficient algorithm for the least-squares reflexive solution of the matrix equation A1XB1=C1, A2XB2=C2 , 2006, Appl. Math. Comput..

[8]  Guang-Ren Duan,et al.  Weighted least squares solutions to general coupled Sylvester matrix equations , 2009 .

[9]  M. Fragoso,et al.  Stability Results for Discrete-Time Linear Systems with Markovian Jumping Parameters , 1993 .

[10]  Ai-Guo Wu,et al.  Solution to Generalized Sylvester Matrix Equations , 2008, IEEE Transactions on Automatic Control.

[11]  Bing Zheng,et al.  The *congruence class of the solutions of some matrix equations , 2009, Comput. Math. Appl..

[12]  Ai-Guo Wu,et al.  Finite iterative algorithm for solving coupled Lyapunov equations appearing in discrete-time Markov jump linear systems [Brief Paper] , 2010 .

[13]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[14]  Ai-Guo Wu,et al.  Closed-form solutions to Sylvester-conjugate matrix equations , 2010, Comput. Math. Appl..

[15]  Feng Ding,et al.  Gradient Based Iterative Algorithms for Solving a Class of Matrix Equations , 2005, IEEE Trans. Autom. Control..

[16]  Ai-Guo Wu,et al.  Kronecker Maps and Sylvester-Polynomial Matrix Equations , 2007, IEEE Transactions on Automatic Control.

[17]  James Lam,et al.  Convergence of gradient-based iterative solution of coupled Markovian jump Lyapunov equations , 2008, Comput. Math. Appl..

[18]  I. Borno,et al.  Parallel algorithm for solving coupled algebraic Lyapunov equations of discrete-time jump linear systems , 1995 .

[19]  M. Mariton,et al.  Jump Linear Systems in Automatic Control , 1992 .

[20]  Ai-Guo Wu,et al.  Finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns , 2010, Math. Comput. Model..

[21]  Guang-Ren Duan,et al.  Gradient based iterative algorithm for solving coupled matrix equations , 2009, Syst. Control. Lett..

[22]  B. Kågström,et al.  Generalized Schur methods with condition estimators for solving the generalized Sylvester equation , 1989 .

[23]  Feng Ding,et al.  Iterative least-squares solutions of coupled Sylvester matrix equations , 2005, Syst. Control. Lett..

[24]  Mehdi Dehghan,et al.  An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices , 2010 .

[25]  I. Borno.  Parallel computation of the solutions of coupled algebraic Lyapunov equations , 1995, Autom..

[26]  Huang Liping.  Consimilarity of quaternion matrices and complex matrices , 2001 .

[27]  Richard H. Bartels,et al.  Algorithm 432 [C2]: Solution of the matrix equation AX + XB = C [F4] , 1972, Commun. ACM.

[28]  Junqiang Hu,et al.  Closed-form solutions to the nonhomogeneous Yakubovich-conjugate matrix equation , 2009, Appl. Math. Comput..

[29]  Gene H. Golub,et al.  Linear algebra for large scale and real-time applications , 1993 .

[30]  Yuan Lei,et al.  Least-squares solution with the minimum-norm for the matrix equation (AXB, GXH) = (C, D) , 2005 .

[31]  Ai-Guo Wu,et al.  Iterative solutions to coupled Sylvester-conjugate matrix equations , 2010, Comput. Math. Appl..

[32]  Paul Van Dooren,et al.  A generalized state-space approach for the additive decomposition of a transfer matrix , 1992 .

[33]  Ai-Guo Wu,et al.  On matrix equations X-AXF=C and X-AXF=C , 2009 .

[34]  Xuehan Cheng,et al.  An algebraic relation between consimilarity and similarity of complex matrices and its applications , 2006 .

[35]  B. Kågström,et al.  A Direct Method for Reordering Eigenvalues in the Generalized Real Schur form of a Regular Matrix Pair (A, B) , 1993 .

[36]  Feng Ding,et al.  Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle , 2008, Appl. Math. Comput..

[37]  Jean H. Bevis,et al.  The matrix equation A X – XB = C and its special cases , 1988 .

[38]  Ai-Guo Wu,et al.  Iterative solutions to the extended Sylvester-conjugate matrix equations , 2010, Appl. Math. Comput..