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An efficient algorithm for the reflexive solution of the quaternion matrix equation \(AXB + CX^HD = F\). (English) Zbl 1268.65060

Summary: We propose an iterative algorithm for solving the reflexive solution of the quaternion matrix equation \(AXB + CX^HD = F\). When the matrix equation is consistent over the reflexive matrix \(X\), a reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors. By the proposed iterative algorithm, the least Frobenius norm reflexive solution of the matrix equation can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate reflexive solution to a given reflexive matrix \(X_0\) can be derived by finding the least Frobenius norm reflexive solution of a new corresponding quaternion matrix equation. Finally, two numerical examples are given to illustrate the efficiency of the proposed methods.

MSC:

65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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[1] H. C. Chen and A. H. Sameh, “A matrix decomposition method for orthotropic elasticity problems,” SIAM Journal on Matrix Analysis and Applications, vol. 10, no. 1, pp. 39-64, 1989. · Zbl 0669.73010
[2] H. C. Chen, “Generalized reflexive matrices: special properties and applications,” SIAM Journal on Matrix Analysis and Applications, vol. 19, no. 1, pp. 140-153, 1998. · Zbl 0910.15005
[3] Q. W. Wang, H. X. Chang, and C. Y. Lin, “P-(skew)symmetric common solutions to a pair of quaternion matrix equations,” Applied Mathematics and Computation, vol. 195, no. 2, pp. 721-732, 2008. · Zbl 1149.15011
[4] S. F. Yuan and Q. W. Wang, “Two special kinds of least squares solutions for the quaternion matrix equation AXB+CXD=E,” Electronic Journal of Linear Algebra, vol. 23, pp. 257-274, 2012. · Zbl 1250.65051
[5] T. S. Jiang and M. S. Wei, “On a solution of the quaternion matrix equation X - AX\~B=C and its application,” Acta Mathematica Sinica (English Series), vol. 21, no. 3, pp. 483-490, 2005. · Zbl 1083.15019
[6] Y. T. Li and W. J. Wu, “Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations,” Computers & Mathematics with Applications, vol. 55, no. 6, pp. 1142-1147, 2008. · Zbl 1143.15012
[7] L. G. Feng and W. Cheng, “The solution set to the quaternion matrix equation AX-X\?B=0,” Algebra Colloquium, vol. 19, no. 1, pp. 175-180, 2012. · Zbl 1236.15032
[8] Z. Y. Peng, “An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C,” Applied Mathematics and Computation, vol. 170, no. 1, pp. 711-723, 2005. · Zbl 1081.65039
[9] Z. Y. Peng, “A matrix LSQR iterative method to solve matrix equation AXB=C,” International Journal of Computer Mathematics, vol. 87, no. 8, pp. 1820-1830, 2010. · Zbl 1195.65056
[10] Z. Y. Peng, “New matrix iterative methods for constraint solutions of the matrix equation AXB=C,” Journal of Computational and Applied Mathematics, vol. 235, no. 3, pp. 726-735, 2010. · Zbl 1206.65145
[11] Z. Y. Peng, “Solutions of symmetry-constrained least-squares problems,” Numerical Linear Algebra with Applications, vol. 15, no. 4, pp. 373-389, 2008. · Zbl 1212.65181
[12] C. C. Paige, “Bidiagonalization of matrices and solutions of the linear equations,” SIAM Journal on Numerical Analysis, vol. 11, pp. 197-209, 1974. · Zbl 0244.65023
[13] X. F. Duan, A. P. Liao, and B. Tang, “On the nonlinear matrix equation X-\sum i=1mAx2a;iX\delta iAi=Q,” Linear Algebra and Its Applications, vol. 429, no. 1, pp. 110-121, 2008. · Zbl 1148.15012
[14] X. F. Duan and A. P. Liao, “On the existence of Hermitian positive definite solutions of the matrix equation Xs+A*X - tA=Q,” Linear Algebra and Its Applications, vol. 429, no. 4, pp. 673-687, 2008. · Zbl 1143.15011
[15] X. F. Duan and A. P. Liao, “On the nonlinear matrix equation X+A*X - qA=Q(q\geq 1),” Mathematical and Computer Modelling, vol. 49, no. 5-6, pp. 936-945, 2009. · Zbl 1165.15302
[16] X. F. Duan and A. P. Liao, “On Hermitian positive definite solution of the matrix equation X - \sum i=1mAi*XrAi=Q,” Journal of Computational and Applied Mathematics, vol. 229, no. 1, pp. 27-36, 2009. · Zbl 1170.15005
[17] X. F. Duan, C. M. Li, and A. P. Liao, “Solutions and perturbation analysis for the nonlinear matrix equation X+\sum i=1mAi*X - 1Ai=I,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4458-4466, 2011. · Zbl 1250.15021
[18] 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. · Zbl 1143.65035
[19] F. Ding and T. Chen, “Iterative least-squares solutions of coupled Sylvester matrix equations,” Systems & Control Letters, vol. 54, no. 2, pp. 95-107, 2005. · Zbl 1129.65306
[20] F. Ding and T. Chen, “Hierarchical gradient-based identification of multivariable discrete-time systems,” Automatica, vol. 41, no. 2, pp. 315-325, 2005. · Zbl 1073.93012
[21] M. H. Wang, M. S. Wei, and S. H. Hu, “An iterative method for the least-squares minimum-norm symmetric solution,” CMES: Computer Modeling in Engineering & Sciences, vol. 77, no. 3-4, pp. 173-182, 2011. · Zbl 1356.65115
[22] M. Dehghan and M. Hajarian, “An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation,” Applied Mathematics and Computation, vol. 202, no. 2, pp. 571-588, 2008. · Zbl 1154.65023
[23] M. Dehghan and M. Hajarian, “Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A1X1B1+A2X2B2=C,” Mathematical and Computer Modelling, vol. 49, no. 9-10, pp. 1937-1959, 2009. · Zbl 1171.15310
[24] M. Hajarian and M. Dehghan, “Solving the generalized Sylvester matrix equation \sum i=1pAiXBi+\sum j=1qCjYDj=E over reflexive and anti-reflexive matrices,” International Journal of Control and Automation, vol. 9, no. 1, pp. 118-124, 2011.
[25] M. Dehghan and M. Hajarian, “An iterative algorithm for solving a pair of matrix equations AYB=E, CYD=F over generalized centro-symmetric matrices,” Computers & Mathematics with Applications, vol. 56, no. 12, pp. 3246-3260, 2008. · Zbl 1165.15301
[26] M. Dehghan and M. Hajarian, “Analysis of an iterative algorithm to solve the generalized coupled Sylvester matrix equations,” Applied Mathematical Modelling, vol. 35, no. 7, pp. 3285-3300, 2011. · Zbl 1227.65037
[27] M. Dehghan and M. Hajarian, “The general coupled matrix equations over generalized bisymmetric matrices,” Linear Algebra and Its Applications, vol. 432, no. 6, pp. 1531-1552, 2010. · Zbl 1187.65042
[28] M. Dehghan and M. Hajarian, “An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices,” Applied Mathematical Modelling, vol. 34, no. 3, pp. 639-654, 2010. · Zbl 1185.65054
[29] 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. · Zbl 1211.15024
[30] A. G. Wu, L. 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. · Zbl 1222.65041
[31] A. G. Wu, L. L. Lv, and M. Z. Hou, “Finite iterative algorithms for extended Sylvester-conjugate matrix equations,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2363-2384, 2011. · Zbl 1235.65040
[32] N. L. Bihan and J. Mars, “Singular value decomposition of matrices of Quaternions matrices: a new tool for vector-sensor signal processing,” Signal Process, vol. 84, no. 7, pp. 1177-1199, 2004. · Zbl 1154.94331
[33] N. L. Bihan and S. J. Sangwine, “Jacobi method for quaternion matrix singular value decomposition,” Applied Mathematics and Computation, vol. 187, no. 2, pp. 1265-1271, 2007. · Zbl 1114.65321
[34] F. O. Farid, Q. W. Wang, and F. Zhang, “On the eigenvalues of quaternion matrices,” Linear and Multilinear Algebra, vol. 59, no. 4, pp. 451-473, 2011. · Zbl 1237.15016
[35] S. de Leo and G. Scolarici, “Right eigenvalue equation in quaternionic quantum mechanics,” Journal of Physics A, vol. 33, no. 15, pp. 2971-2995, 2000. · Zbl 0954.81008
[36] S. J. Sangwine and N. L. Bihan, “Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 727-738, 2006. · Zbl 1109.65037
[37] C. C. Took, D. P. Mandic, and F. Zhang, “On the unitary diagonalisation of a special class of quaternion matrices,” Applied Mathematics Letters, vol. 24, no. 11, pp. 1806-1809, 2011. · Zbl 1388.15009
[38] Q. W. Wang, H. X. Chang, and Q. Ning, “The common solution to six quaternion matrix equations with applications,” Applied Mathematics and Computation, vol. 198, no. 1, pp. 209-226, 2008. · Zbl 1141.15016
[39] Q. W. Wang, J. W. van der Woude, and H. X. Chang, “A system of real quaternion matrix equations with applications,” Linear Algebra and Its Applications, vol. 431, no. 12, pp. 2291-2303, 2009. · Zbl 1180.15019
[40] Q. W. Wang, S. W. Yu, and W. Xie, “Extreme ranks of real matrices in solution of the quaternion matrix equation AXB=C with applications,” Algebra Colloquium, vol. 17, no. 2, pp. 345-360, 2010. · Zbl 1188.15016
[41] Q. W. Wang and J. Jiang, “Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation,” Electronic Journal of Linear Algebra, vol. 20, pp. 552-573, 2010. · Zbl 1207.15016
[42] Q. W. Wang, X. Liu, and S. W. Yu, “The common bisymmetric nonnegative definite solutions with extreme ranks and inertias to a pair of matrix equations,” Applied Mathematics and Computation, vol. 218, no. 6, pp. 2761-2771, 2011. · Zbl 1267.15017
[43] Q. W. Wang, Y. Zhou, and Q. Zhang, “Ranks of the common solution to six quaternion matrix equations,” Acta Mathematicae Applicatae Sinica (English Series), vol. 27, no. 3, pp. 443-462, 2011. · Zbl 1304.15006
[44] F. Zhang, “Ger\vsgorin type theorems for quaternionic matrices,” Linear Algebra and Its Applications, vol. 424, no. 1, pp. 139-153, 2007. · Zbl 1117.15017
[45] F. Zhang, “Quaternions and matrices of quaternions,” Linear Algebra and Its Applications, vol. 251, pp. 21-57, 1997. · Zbl 0873.15008
[46] R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, UK, 1991. · Zbl 0729.15001
[47] Y. X. Peng, X. Y. Hu, and L. Zhang, “An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C,” Applied Mathematics and Computation, vol. 160, no. 3, pp. 763-777, 2005. · Zbl 1068.65056
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