Solvability, Unique Solvability, and Representation of Solutions for Systems of Coupled Linear Matrix Equations

Thitison Nuchniyom, Pattrawut Chansangiam


We investigate a system of coupled linear matrix equations of the form AXB + CYD = E, CXD + AYB = Fwhere A,B,C,D,E,F are rectangular complex matrices and X,Y are unknown complex matrices. We obtain several criterions for solvability and unique solvability of the system and tis special cases.These criterions rely on Kronecker product, vector operator, Moore Penrose inverses, and ranks.Moreover, explicit formulas of solutions are presented.


linear matrix equation; Kronecker product; vector operator, Moore-Penrose inverse

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Z. Al Zhour, A. Kilicman, New algebraic method for solving the axial N-index transportation problem based on the Kronecker product, Matematika 21(2), 113--123, (2005).

J. K. Baksalary and R. Kala, The matrix equation $AX - YB = C$, Linear Algebra Appl. 25, 41--43, (1979).

J. K. Baksalary and R. Kala, The matrix equation $AXB + CYD = E$, Linear Algebra Appl. 30, 141--147, (1980).

S. Barnett, Introduction to mathematical control theory, Oxford university Press, (1975).

R. Carlson, An inverse problem for the matrix Schrodinger equation, J. Math. Anal. Appl. 267, 564--575, (2002).

F. Ding and T. Chen, Gradient based iterative algorithms for solving a class of matrix equations, Appl. Math. Comput. 212, 327--336, (2009).

F. Ding, X. Liu and J. Ding, Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Appl. Math. Comput. 197, 41--50, (2008).

A. Kilicman and Z. Al Zhour, The general common exact solutions of coupled linear matrix and matrix differential equations, J. Anal. Comput. 1(1), 15--30, (2005).

A. Kilicman and Z. Al Zhour, Vector least-squares solutions for coupled singular matrix equations, J. Comput. Appl. Math. 206, 1051--1069, (2007).

J.R. Magnus and H. Neudecker, Matrix differential calculus with applications in statistics and econometrics, John Wiley and Sons Ltd, Baffins Lane, Chichester, England, (1999).

S. Tanimoto, Image processing in middle-school mathematics. Proceedings of the First IEEE International Conference on Image Processing, 501--505, (1994).

L. Tongxing, Solution of the matrix equation $AX-XB=C$, Computing 37, 351--355, (1986).

L. Xie, H. Yang, Y. Liu and F. Ding, Iterative solutions for general coupled matrix equations with real coefficients, American Control Conference on O'Farrell Street, San Francisco, CA, USA, (2011).

G. Xu, M. Wei and D. Zheng, On solutions of matrix equation $AXB + CYD = F$, Linear Algebra Appl. 279, 93--109, (1998).

B. Zhou, G.-R. Duan and Z.-Y. Li, Gradient based iterative algorithm for solving coupled matrix equations, System and Control Letter 58, 327--333, (2009).

B. Zhou, Z.-Y. Li, G.-R. Duan and Y. Wang, Solutions to a family of matrix equations by using the Kronecker matrix polynomials, Appl. Math. Comput. 212, 327--336, (2009).


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