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Finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns. (English) Zbl 1205.15027

Summary: This paper is concerned with a class of complex matrix equations, in which there exist the conjugate and the transpose of the unknown matrices. The considered matrix equation includes some previously investigated matrix equations as its special cases. An iterative algorithm is presented for solving this class of matrix equations. When the matrix equation is consistent, a solution can be obtained within finite iteration steps for any initial values in the absence of round-off errors. A numerical example is given to illustrate the effectiveness of the proposed method.

MSC:

15A24 Matrix equations and identities
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[1] Bevis, J. H.; Hall, F. J.; Hartwing, R. E., Consimilarity and the matrix equation \(A \overline{X} - X B = C\), (Current Trends in Matrix Theory (Auburn, Ala., 1986) (1987), North-Holland: North-Holland New York), 51-64
[2] Horn, R. A.; Johnson, C. R., Matrix Analysis (1990), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0704.15002
[3] Huang, L., Consimilarity of quaternion matrices and complex matrices, Linear Algebra and its Applications, 331, 21-30 (2001) · Zbl 0982.15019
[4] Jiang, T.; Cheng, X.; Chen, L., An algebraic relation between consimilarity and similarity of complex matrices and its applications, Journal of Physics A (Mathematical and General), 39, 9215-9222 (2006) · Zbl 1106.15008
[5] Bevis, J. H.; Hall, F. J.; Hartwig, R. E., The matrix equation \(A \overline{X} - X B = C\) and its special cases, SIAM Journal on Matrix Analysis and Applications, 9, 3, 348-359 (1988) · Zbl 0655.15013
[6] Jiang, T.; Wei, M., On solutions of the matrix equations \(X - A X B = C\) and \(X - A \overline{X} B = C\), Linear Algebra and its Application, 367, 225-233 (2003)
[7] Wu, A. G.; Duan, G. R.; Yu, H. H., On solutions of \(X F - A X = C\) and \(X F - A \overline{X} = C\), Applied Mathematics and Computation, 182, 2, 932-941 (2006) · Zbl 1112.15018
[8] Wu, A. G.; Wang, H. Q.; Duan, G. R., On matrix equations \(X - A X F = C\) and \(X - A \overline{X} F = C\), Journal of Computational and Applied Mathematics, 230, 690-698 (2009) · Zbl 1390.15055
[9] Wu, A. G.; Fu, Y. M.; Duan, G. R., On solutions of matrix equations \(V - A V F = B W\) and \(V - A \overline{V} F = B W\), Mathematical and Computer Modelling, 47, 11-12, 1181-1197 (2008) · Zbl 1145.15302
[10] Wu, A. G.; Feng, G.; Hu, J.; Duan, G. R., Closed-form solutions to the nonhomogeneous Yakubovich-conjugate matrix equation, Applied Mathematics and Computation, 214, 442-450 (2009) · Zbl 1176.15021
[11] Wu, A. G.; Feng, G.; Duan, G. R.; Wu, W. J., Closed-form solutions to Sylvester-conjugate matrix equations, Computers and Mathematics with Applications, 60, 1, 95-111 (2010) · Zbl 1198.15013
[12] Ding, F.; Chen, T., Gradient based iterative algorithms for solving a class of matrix equations, IEEE Transactions on Automatic Control, 50, 8, 1216-1221 (2005) · Zbl 1365.65083
[13] Ding, F.; Chen, T., Performance analysis of multi-innovation gradient type identification methods, Automatica, 43, 1-14 (2007) · Zbl 1140.93488
[14] Ding, F.; Liu, P. X.; Ding, J., Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle, Applied Mathematics and Computation, 197, 41-50 (2008) · Zbl 1143.65035
[15] Ding, F.; Chen, T., Hierarchical least squares identification methods for multivariable systems, IEEE Transactions on Automatic Control, 50, 3, 397-402 (2005) · Zbl 1365.93551
[16] Wu, A. G.; Zeng, X.; Duan, G. R.; Wu, W. J., Iterative solutions to the extended Sylvester-conjugate matrix equations, Applied Mathematics and Computation (2010)
[17] Zhou, B.; Duan, G. R.; Li, Z. Y., Gradient based iterative algorithm for solving coupled matrix equations, Systems & Control Letters, 58, 327-333 (2009) · Zbl 1159.93323
[18] Wang, M.; Feng, Y., An iterative algorithm for solving a class of matrix equations, Journal of Control Theory and Applications, 7, 1, 68-72 (2009)
[19] Hou, J. J.; Peng, Z. Y.; Zhang, X. L., An iterative method for the least squares symmetric solution of matrix equation \(A X B = C\), Numerical Algorithms, 42, 181-192 (2006)
[20] Peng, Z. Y., An iterative method for the least squares symmetric solution of the linear matrix equation \(A X B = C\), Applied Mathematics and Computation, 170, 711-723 (2005) · Zbl 1081.65039
[21] Wang, M.; Cheng, X.; Wei, M., Iterative algorithms for solving the matrix equation \(A X B + C X^T D = E\), Applied Mathematics and Computation, 187, 2, 622-629 (2007)
[22] Piao, F.; Zhang, Q.; Wang, Z., The solution to matrix equation \(A X + X^T C = B\), Journal of the Franklin Institute, 344, 8, 1056-1062 (2007) · Zbl 1171.15015
[23] Zhang, X., Matrix Analysis and Applications (2004), Tsinghua University Press: Tsinghua University Press Beijing
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