Iterative Solutions to Some Linear Matrix Equations
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.
[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.
[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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:57186", author = "Jiashang Jiang and Hao Liu and Yongxin Yuan", title = "Iterative Solutions to Some Linear Matrix Equations", 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.", volume = "5", number = "4", pages = "602-6", }
