**Commenced**in January 2007

**Frequency:**Monthly

**Edition:**International

**Paper Count:**31515

##### Iterative Solutions to Some Linear Matrix Equations

**Authors:**
Jiashang Jiang,
Hao Liu,
Yongxin Yuan

**Abstract:**

In this paper the gradient based iterative algorithms are presented to solve the following four types linear matrix equations: (a) AXB = F; (b) AXB = F, CXD = G; (c) AXB = F s. t. X = XT ; (d) AXB+CYD = F, where X and Y are unknown matrices, A,B,C,D, F,G are the given constant matrices. It is proved that if the equation considered has a solution, then the unique minimum norm solution can be obtained by choosing a special kind of initial matrices. The numerical results show that the proposed method is reliable and attractive.

**Keywords:**
Matrix equation,
iterative algorithm,
parameter estimation,
minimum norm solution.

**Digital Object Identifier (DOI):**
doi.org/10.5281/zenodo.1070271

**References:**

[1] A. Kilicman, Z. Al Zhour, Vector least-squares solutions for coupled singular matrix equations, Journal of Computational and Applied Mathematics, 206 (2007) 1051-1069.

[2] F. Ding, L. Qiu, T. Chen, Reconstruction of continuous-time systems from their non-uniformly sampled discrete-time systems, Automatica, 45 (2009) 324-332.

[3] F. Ding, T. Chen, Performance analysis of multi-innovation gradient type identification methods, Automatica, 43 (2007) 1-14.

[4] F. Ding, P. X. Liu, G. Liu, Auxiliary model based multi-innovation extended stochastic gradient parameter estimation with colored measurement noises, Signal Processing, 89 (2009) 1883-1890.

[5] F. Ding, P. X. Liu, H. Z. Yang, Parameter identification and intersample output estimation for dual-rate systems, IEEE Transactions on Systems, Man, and Cybernetics, Part A: Systems and Humans, 38 (2008) 966-975.

[6] F. Ding, H. Z. Yang, F. Liu, Performance analysis of stochastic gradient algorithms under weak conditions, Science in China Series FInformation Sciences, 51 (2008) 1269-1280.

[7] M. Dehghan, M. Hajarian, An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation, Applied Mathematics and Computation, 202 (2008) 571-588.

[8] H. Mukaidani, S. Yamamoto, T. Yamamoto, A numerical algorithm for finding solution of cross-coupled algebraic Riccati equations, IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences, E91A (2008) 682-685.

[9] B. Zhou, G. R. Duan, Solutions to generalized Sylvester matrix equation by Schur decomposition, International Journal of Systems Science, 38 (2007) 369-375.

[10] B. Zhou, G. R. Duan, On the generalized Sylvester mapping and matrix equations, Systems & Control Letters 57 (3) (2008) 200-208.

[11] F. Ding, T. Chen, Iterative least squares solutions of coupled Sylvester matrix equations, Systems & Control Letters, 54 (2005) 95-107.

[12] F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM Journal on Control and Optimization, 44 (2006) 2269- 2284.

[13] F. Ding, P. X. Liu, J. Ding. Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle. Applied Mathematics and Computation, 197 (2008) 41-50.

[14] J. Ding, Y. Liu, F. Ting, Iterative solutions to matrix equations of the form AiXBi = Fi, Computers and Mathematics with Applications, 59 (2010) 3500-3507.

[15] P. Lancaster, M. Tismenetsky, The Theory of Matrices. 2rd Edition. London: Academic Press, 1985.

[16] A. Ben-Israel, T. N. E. Greville, Generalized Inverses. Theory and Applications (second ed). New York: Springer, 2003.

[17] H. Dai, The Theory of Matrices. Beijing: Science Press, 2002.