Iterative Solutions to Some Linear Matrix Equations

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href="https://publications.waset.org/search?q=Yongxin%20Yuan"> Yongxin Yuan</a> </p> <p class="card-text"><strong>Abstract:</strong></p> <p>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.</p> <iframe src="https://publications.waset.org/8105.pdf" style="width:100%; height:400px;" frameborder="0"></iframe> <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/search?q=Matrix%20equation" title="Matrix equation">Matrix equation</a>, <a href="https://publications.waset.org/search?q=iterative%20algorithm" title=" iterative algorithm"> iterative algorithm</a>, <a href="https://publications.waset.org/search?q=parameter%20estimation" title=" parameter estimation"> parameter estimation</a>, <a href="https://publications.waset.org/search?q=minimum%20norm%20solution." title=" minimum norm solution."> minimum norm solution.</a> </p> <p class="card-text"><strong>Digital Object Identifier (DOI):</strong> <a href="https://doi.org/10.5281/zenodo.1070271" target="_blank">doi.org/10.5281/zenodo.1070271</a> </p> <a href="https://publications.waset.org/8105/iterative-solutions-to-some-linear-matrix-equations" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/8105/apa" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">APA</a> <a href="https://publications.waset.org/8105/bibtex" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">BibTeX</a> <a href="https://publications.waset.org/8105/chicago" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Chicago</a> <a href="https://publications.waset.org/8105/endnote" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">EndNote</a> <a href="https://publications.waset.org/8105/harvard" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Harvard</a> <a href="https://publications.waset.org/8105/json" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">JSON</a> <a href="https://publications.waset.org/8105/mla" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">MLA</a> <a href="https://publications.waset.org/8105/ris" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">RIS</a> <a href="https://publications.waset.org/8105/xml" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">XML</a> <a href="https://publications.waset.org/8105/iso690" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">ISO 690</a> <a href="https://publications.waset.org/8105.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">1856</span> </span> <p class="card-text"><strong>References:</strong></p> <br>[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. <br>[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. <br>[3] F. Ding, T. Chen, Performance analysis of multi-innovation gradient type identification methods, Automatica, 43 (2007) 1-14. <br>[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. <br>[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. <br>[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. <br>[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. <br>[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. <br>[9] B. Zhou, G. R. Duan, Solutions to generalized Sylvester matrix equation by Schur decomposition, International Journal of Systems Science, 38 (2007) 369-375. <br>[10] B. Zhou, G. R. Duan, On the generalized Sylvester mapping and matrix equations, Systems & Control Letters 57 (3) (2008) 200-208. <br>[11] F. Ding, T. Chen, Iterative least squares solutions of coupled Sylvester matrix equations, Systems & Control Letters, 54 (2005) 95-107. <br>[12] F. Ding, T. Chen, On iterative solutions of general coupled matrix equations, SIAM Journal on Control and Optimization, 44 (2006) 2269- 2284. <br>[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. <br>[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. <br>[15] P. Lancaster, M. Tismenetsky, The Theory of Matrices. 2rd Edition. London: Academic Press, 1985. <br>[16] A. Ben-Israel, T. N. E. Greville, Generalized Inverses. Theory and Applications (second ed). New York: Springer, 2003. <br>[17] H. Dai, The Theory of Matrices. 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Iterative Solutions to Some Linear Matrix Equations