zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Iterative solutions to coupled Sylvester-conjugate matrix equations. (English) Zbl 1198.65083
Summary: This paper is concerned with iterative solutions to the coupled Sylvester-conjugate matrix equation with a unique solution. By applying a hierarchical identification principle, an iterative algorithm is established to solve this class of complex matrix equations. With a real representation of a complex matrix as a tool, a sufficient condition is given to guarantee that the iterative solutions given by the proposed algorithm converge to the exact solution for any initial matrices. In addition, a sufficient convergence condition that is easier to compute is also given by the original coefficient matrices. Finally, a numerical example is given to verify the effectiveness of the proposed algorithm.
MSC:
65F30Other matrix algorithms
15A24Matrix equations and identities
References:
[1]Kagstrom, B.: A direct method for reording eigenvalues in the generalized real Schur form of a regular matrix pair (A,B), Linear algebra for large scale and real-time application, 195-218 (1993)
[2]Kagstrom, B.; Van Dooren, P.: A generalized state-space approach for the additive decomposition of a transfer matrix, International journal of numerical linear algebra with applications 1, No. 2, 165-181 (1992)
[3]Kagstrom, B.; Westin, L.: Generalized Schur methods with condition estimators for solving the generalized Sylvester equation, IEEE transactions on automatic control 34, No. 7, 745-751 (1989) · Zbl 0687.93025 · doi:10.1109/9.29404
[4]Ding, F.; Chen, T.: Iterative least-squares solutions of coupled Sylvester matrix equations, Systems control letters 54, 95-107 (2005) · Zbl 1129.65306 · doi:10.1016/j.sysconle.2004.06.008
[5]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 · doi:10.1016/j.amc.2007.07.040
[6]Ding, F.; Chen, T.: Gradient based iterative algorithms for solving a class of matrix equations, IEEE transactions on automatic control 50, No. 8, 1216-1221 (2005)
[7]Kilicman, A.; Zhour, Z. A.: Vector least-squares solutions for coupled singular matrix equations, Journal of computational and applied mathematics 206, 1051-1069 (2007) · Zbl 1132.65034 · doi:10.1016/j.cam.2006.09.009
[8]Zhou, B.; Lam, J.; Duan, G. R.: Convergence of gradient-based iterative solution of coupled Markovian jump Lyapunov equations, Computers and mathematics with applications 56, 3070-3078 (2008) · Zbl 1165.15304 · doi:10.1016/j.camwa.2008.07.037
[9]Peng, Z.; Hu, X.; Zhang, L.: An efficient algorithm for the least-squares reflexive solution of the matrix equation A1XB1=C1,A2XB2=C2, Applied mathematics and computation 181, 988-999 (2006) · Zbl 1115.65048 · doi:10.1016/j.amc.2006.01.071
[10]Dehghan, M.; Hajarian, M.: An iterative algorithm for solving a pair of matrix equations AYB=E,CYD=F over generalized centro-symmetric matrices, Computers and mathematics with applications 56, 3246-3260 (2008) · Zbl 1165.15301 · doi:10.1016/j.camwa.2008.07.031
[11]Dehghan, M.; Hajarian, M.: An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation, Applied mathematics and computation 202, 571-588 (2008) · Zbl 1154.65023 · doi:10.1016/j.amc.2008.02.035
[12]M. Dehghan, M. Hajarian, An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices, Applied Mathematical Modelling (2009). doi:10.1016/j.apm.2009.06.018.
[13]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 · doi:10.1016/j.sysconle.2008.12.004
[14]Zhou, B.; Li, Z. Y.; Duan, G. R.; Wang, Y.: Weighted least squares solutions to general coupled Sylvester matrix equations, Journal of computational and applied mathematics 224, 759-776 (2009) · Zbl 1161.65034 · doi:10.1016/j.cam.2008.06.014
[15]Wu, A. G.; Duan, G. R.; Xue, Y.: Kronecker maps and Sylvester-polynomial matrix equations, IEEE transactions on automatic control 52, No. 5, 905-910 (2007)
[16]Bevis, J. H.; Hall, F. J.; Hartwing, R. E.: Consimilarity and the matrix equation AX-XB=C, in: current trends in matrix theory (Auburn, ala., 1986), pp. 51–64, (1987) · Zbl 0655.15012
[17]Horn, R. A.; Johnson, C. R.: Matrix analysis, (1990)
[18]Huang, L.: Consimilarity of quaternion matrices and complex matrices, Linear algebra and its applications 331, 21-30 (2001) · Zbl 0982.15019 · doi:10.1016/S0024-3795(01)00266-X
[19]Jiang, T.; Cheng, X.; Chen, L.: An algebraic relation between consimilarity and similarity of complex matrices and its applications, J. phys. A: math. Gen. 39, 9215-9222 (2006) · Zbl 1106.15008 · doi:10.1088/0305-4470/39/29/014
[20]Bevis, J. H.; Hall, F. J.; Hartwig, R. E.: The matrix equation AX-XB=C and its special cases, SIAM journal on matrix analysis and applications 9, No. 3, 348-359 (1988) · Zbl 0655.15013 · doi:10.1137/0609029
[21]Wu, A. G.; Duan, G. R.; Yu, H. H.: On solutions of XF-AX=C and XF-AX=C, Applied mathematics and applications 182, No. 2, 932-941 (2006)
[22]Jiang, T.; Wei, M.: On solutions of the matrix equations X-AXB=C and X-AXB=C, Linear algebra and its application 367, 225-233 (2003) · Zbl 1019.15002 · doi:10.1016/S0024-3795(02)00633-X
[23]Wu, A. G.; Feng, G.; Hu, J.; Duan, G. R.: Closed-form solutions to the nonhomogeneous yakubovich-conjugate matrix equation, Applied mathematics and application 214, 442-450 (2009) · Zbl 1176.15021 · doi:10.1016/j.amc.2009.04.011
[24]Ding, F.; Chen, T.: On iterative solutions of general coupled matrix equations, SIAM journal on control and optimization 44, No. 6, 2269-2284 (2006) · Zbl 1115.65035 · doi:10.1137/S0363012904441350
[25]Xie, L.; Ding, J.; Ding, F.: Gradient based iterative solutions for general linear matrix equations, Computers mathematics with applications 58, No. 7, 1441-1448 (2009) · Zbl 1189.65083 · doi:10.1016/j.camwa.2009.06.047
[26]Ding, F.; Chen, T.: Hierarchical gradient-based identification of multivariable discrete-time systems, Automatica 41, No. 2, 315-325 (2005) · Zbl 1073.93012 · doi:10.1016/j.automatica.2004.10.010
[27]Zhou, K.; Doyle, J.; Glover, K.: Robust and optimal control, (1996) · Zbl 0999.49500