Journal of Discrete Mathematics

Volume 2013 (2013), Article ID 170263, 13 pages

http://dx.doi.org/10.1155/2013/170263

Research Article

## Finite Iterative Algorithm for Solving a Complex of Conjugate and Transpose Matrix Equation

^{1}Department of Mathematics, Faculty of Science, Menoufia University, Shebeen El-Koom, Egypt^{2}Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt

Received 4 August 2012; Accepted 4 November 2012

Academic Editor: Franck Petit

Copyright © 2013 Mohamed A. Ramadan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- F. Ding, P. X. Liu, and J. Ding, “Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle,”
*Applied Mathematics and Computation*, vol. 197, no. 1, pp. 41–50, 2008. View at Publisher · View at Google Scholar · View at Scopus - F. Ding and T. Chen, “Gradient based iterative algorithms for solving a class of matrix equations,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 8, pp. 1216–1221, 2005. View at Publisher · View at Google Scholar · View at Scopus - F. Ding and T. Chen, “Hierarchical gradient-based identification of multivariable discrete-time systems,”
*Automatica*, vol. 41, no. 2, pp. 315–325, 2005. View at Publisher · View at Google Scholar · View at Scopus - F. Ding and T. Chen, “Iterative least-squares solutions of coupled Sylvester matrix equations,”
*Systems and Control Letters*, vol. 54, no. 2, pp. 95–107, 2005. View at Publisher · View at Google Scholar · View at Scopus - F. Ding and T. Chen, “On iterative solutions of general coupled matrix equations,”
*SIAM Journal on Control and Optimization*, vol. 44, no. 6, pp. 2269–2284, 2006. View at Publisher · View at Google Scholar · View at Scopus - F. Ding and T. Chen, “Hierarchical least squares identification methods for multivariable systems,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 3, pp. 397–402, 2005. View at Publisher · View at Google Scholar · View at Scopus - F. Ding and T. Chen, “Performance analysis of multi-innovation gradient type identification methods,”
*Automatica*, vol. 43, no. 1, pp. 1–14, 2007. View at Publisher · View at Google Scholar · View at Scopus - A. G. Wu, X. Zeng, G. R. Duan, and W. J. Wu, “Iterative solutions to the extended Sylvester-conjugate matrix equations,”
*Applied Mathematics and Computation*, vol. 217, no. 1, pp. 130–142, 2010. View at Publisher · View at Google Scholar · View at Scopus - B. Zhou, G. R. Duan, and Z. Y. Li, “Gradient based iterative algorithm for solving coupled matrix equations,”
*Systems and Control Letters*, vol. 58, no. 5, pp. 327–333, 2009. View at Publisher · View at Google Scholar · View at Scopus - J. H. Bevis, F. J. Hall, and R. E. Hartwing, “Consimilarity and the matrix equation $A\overline{X}-XB=C$,” in
*Current Trends in Matrix Theory*, pp. 51–64, North-Holland, New York, NY, USA, 1987. View at Google Scholar - R. A. Horn and C. R. Johnson,
*Matrix Analysis*, Cambridge University Press, Cambridge, UK, 1990. - H. Liping, “Consimilarity of quaternion matrices and complex matrices,”
*Linear Algebra and Its Applications*, vol. 331, no. 1–3, pp. 21–30, 2001. View at Publisher · View at Google Scholar · View at Scopus - T. Jiang, X. Cheng, and L. Chen, “An algebraic relation between consimilarity and similarity of complex matrices and its applications,”
*Journal of Physics A*, vol. 39, no. 29, pp. 9215–9222, 2006. View at Publisher · View at Google Scholar · View at Scopus - J. H. Bevis, F. J. Hall, and R. E. Hartwig, “The matrix equation $A\overline{X}-XB=C$ and its special cases,”
*SIAM Journal on Matrix Analysis and Applications*, vol. 9, no. 3, pp. 348–359, 1988. View at Google Scholar - A. G. Wu, G. R. Duan, and H. H. Yu, “On solutions of the matrix equations $XF-AX=C$ and $XF-A\overline{X}=C$,”
*Applied Mathematics and Computation*, vol. 183, no. 2, pp. 932–941, 2006. View at Publisher · View at Google Scholar · View at Scopus - A. Navarra, P. L. Odell, and D. M. Young, “Representation of the general common solution to the matrix equations ${A}_{1}X{B}_{1}={C}_{1},{A}_{2}X{B}_{2}={C}_{2}$ with applications,”
*Computers and Mathematics with Applications*, vol. 41, no. 7-8, pp. 929–935, 2001. View at Publisher · View at Google Scholar · View at Scopus - J. W. Van der Woude, “On the existence of a common solution X to the matrix equations ${A}_{i}X{B}_{j}=C$ (i, j) E I,”
*Linear Algebra and Its Applications*, vol. 375, no. 1–3, pp. 135–145, 2003. View at Publisher · View at Google Scholar · View at Scopus - P. Bhimasankaram, “Common solutions to the linear matrix equations $AX=C,XB=D$, and $\text{EXF}=\text{G}$,”
*Sankhya Series A*, vol. 38, pp. 404–409, 1976. View at Google Scholar - S. Kumar Mitra, “The matrix equations $AX=C,XB=D$,”
*Linear Algebra and Its Applications*, vol. 59, pp. 171–181, 1984. View at Google Scholar · View at Scopus - S. K. Mitra, “A pair of simultaneous linear matrix equations ${A}_{1}X{B}_{1}={C}_{1},{A}_{2}X{B}_{2}={C}_{2}$ and a matrix programming problem,”
*Linear Algebra and Its Applications*, vol. 131, pp. 107–123, 1990. View at Google Scholar · View at Scopus - M. A. Ramadan, M. A. Abdel Naby, and A. M. E. Bayoumi, “On the explicit solutions of forms of the Sylvester and the Yakubovich matrix equations,”
*Mathematical and Computer Modelling*, vol. 50, no. 9-10, pp. 1400–1408, 2009. View at Publisher · View at Google Scholar · View at Scopus - K. W. E. Chu, “Singular value and generalized singular value decompositions and the solution of linear matrix equations,”
*Linear Algebra and Its Applications*, vol. 88-89, pp. 83–98, 1987. View at Google Scholar · View at Scopus - Y. X. Yuan, “The optimal solution of linear matrix equation by matrix decompositions, Math,”
*Numerica Sinica*, vol. 24, pp. 165–176, 2002. View at Google Scholar - A. P. Liao and Y. Lei, “Least-squares solution with the minimum-norm for the matrix equation $(AXB,GXH)=(C,D)$,”
*Computers and Mathematics with Applications*, vol. 50, no. 3-4, pp. 539–549, 2005. View at Publisher · View at Google Scholar · View at Scopus - Y. H. Liu, “Ranks of least squares solutions of the matrix equation $AXB=C$,”
*Computers and Mathematics with Applications*, vol. 55, no. 6, pp. 1270–1278, 2008. View at Publisher · View at Google Scholar · View at Scopus - X. Sheng and G. Chen, “A finite iterative method for solving a pair of linear matrix equations $(AXB,CXD)=(E,F)$,”
*Applied Mathematics and Computation*, vol. 189, no. 2, pp. 1350–1358, 2007. View at Publisher · View at Google Scholar · View at Scopus - A. G. Wu, W. Liu, and G. R. Duan, “On the conjugate product of complex polynomial matrices,”
*Mathematical and Computer Modelling*, vol. 53, no. 9-10, pp. 2031–2043, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. G. Wu, G. Feng, W. Liu, and G. R. Duan, “The complete solution to the Sylvester-polynomial-conjugate matrix equations,”
*Mathematical and Computer Modelling*, vol. 53, no. 9-10, pp. 2044–2056, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. G. Wu, B. Li, Y. Zhang, and G. R. Duan, “Finite iterative solutions to coupled Sylvester-conjugate matrix equations,”
*Applied Mathematical Modelling*, vol. 35, no. 3, pp. 1065–1080, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. G. Wu, L. Lv, and G. R. Duan, “Iterative algorithms for solving a class of complex conjugate and transpose matrix equations,”
*Applied Mathematics and Computation*, vol. 217, no. 21, pp. 8343–8353, 2011. View at Publisher · View at Google Scholar · View at Scopus - X. Zhang,
*Matrix Analysis and Application*, Tsinghua University Press, Beijing, China, 2004. - A. G. Wu, L. Lv, and M. Z. Hou, “Finite iterative algorithms for extended Sylvester-conjugate matrix equations,”
*Mathematical and Computer Modelling*, vol. 54, pp. 2363–2384, 2011. View at Publisher · View at Google Scholar · View at Scopus