Abstract
In this paper, we give a new necessary and sufficient condition for the solvability of the system of generalized Sylvester real quaternion matrix equations \(A_{i}X_{i}+Y_{i}B_{i}+C_{i}ZD_{i} =E_{i}\), (\(i=1,2\)). Moreover, using the purely algebraic technique, we consider the solvability of the system of generalized Sylvester equations in a unital ring.
Similar content being viewed by others
References
Duan, G.R.: Eigen structure assignment and response analysis in descriptor linear systems with state feed back control. Int. J. Control 69, 663–694 (1998)
Kagstrom, B., Westin, L.: Generalized Schur methods with condition estimators for solving the generalized Sylvester equation. IEEE. Trans. Autom. Control 34, 745–751 (1989)
Wu, A.G., Duan, G.R., Zhou, B.: Solution to generalized Sylvester matrix equations. IEEE. Trans. Autom. Control. 53, 811–815 (2008)
Zhou, B., Duan, G.R.: A new solution to the generalized Sylvester matrix equation \(AV-EVF=BW\). Syst. Control Lett. 55, 193–198 (2006)
Zhou, B., Li, Z.Y., Duan, G.R., Wang, Y.: Weighted leasts quares solutions to general coupled Sylvester matrix equations. J. Comput. Appl. Math. 224, 759–776 (2009)
Cavin, R.K., Bhattacharyya, S.P.: Robust and well-condition Edeigen structure assignment via Sylvester sequation. Optim. Control Appl. Methods 4, 205–212 (1983)
Fletcher, L.R., Kautsky, J., Nichols, N.K.: Eigen structure assignment indescriptor systems. IEEE. Trans. Autom. Control 31, 1138–1141 (1986)
Robust, V.A.: Pole assignment via Sylvester equation based state feedback parametrization. In: Computer-aided control system design, 2000,CACSD2000, IEEE international symposium, vol. 57, p. 13C18 (2000)
Zhang, Y., Jiang, D., Wang, J.: A recurrent neural network for solving Sylvester equation with time-varying coefficients. IEEE. Trans. Neural Netw. 13, 1053–1063 (2002)
Roth, W.E.: The equations \(AX-YB=C\) and \(AX-XB=C\) in matrices. Proc. Am. Math. Soc. 3, 392–396 (1952)
Lin, Y.Q., Wei, Y.M.: Condition numbers of the generalized Sylvester equation. IEEE. Trans. Autom. Control 52, 2380–2385 (2007)
Axelsson, O., Bai, Z.Z., Qiu, S.X.: A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part. Numer. Algorithms 35, 351–372 (2004)
Deng, Y.B., Bai, Z.Z., Gao, Y.H.: Iterative orthogonal direction methods for Hermitian minimumnorm solutions of two consistent matrix equations. Numer. Linear Algebra Appl. 13, 801–823 (2006)
Liao, A.P., Bai, Z.Z.: Least-squares solution of \(AXB=D\) over symmetric positive semidefinite matrices \(X\). J. Comput. Math. 21, 175–182 (2003)
Liao, A.P., Bai, Z.Z.: Least squares symmetric and skew-symmetric solutions of the matrix equation \(AXA^{T}+BYB^{T}=C\) with the least norm. Math. Numer. Sin. 27, 81–95 (2005). (In Chinese)
Liao, A.P., Bai, Z.Z., Lei, Y.: Best approximate solution of matrix equation \(AXB+CYD=E\). SIAM. J. Matrix Anal. Appl. 27, 675–688 (2005)
Niu, Q., Wang, X., Lu, L.Z.: A relaxed gradient based algorithm for solving Sylvester equations. Asian J. Control 13, 461–464 (2011)
Wang, X., Li, W.W., Mao, L.Z.: On positive-definite and skew-Hermitian splitting iteration methods for continuous Sylvester equation \(AX+XB=C\). Comput. Math. Appl. 66, 2352–2361 (2013)
Wang, X., Dai, L., Liao, D.: A modified gradient based algorithm for solving Sylvester equations. Appl. Math. Comput. 218, 5620–5628 (2012)
Lee, S.G., Vu, Q.P.: Simultaneous solutions of matrix equations and simultaneous equivalence of matrices. Linear Algebra Appl. 437, 2325–2339 (2012)
Wang, Q.W., He, Z.H.: Solvability conditions and general solution for the mixed Sylvester equations. Automatica 49, 2713–2719 (2013)
He, Z.H., Wang, Q.W.: A pair of mixed generalized Sylvester matrix equations. J. Shanghai Univ. Nat. Sci. 20, 138–156 (2014)
Zhang, X.: A system of generalized Sylvester quaternion matrix equations and its applications. Appl. Math. Comput. 273, 74–81 (2016)
Cvetković-Ilić, D.S., Nikolov Radenković, J., Wang, Q.W.: Algebraic conditions for the solvability to some systems of matrix equations. Linear Multilinear Algebra (2019). https://doi.org/10.1080/03081087.2019.1633993
He, Z.H., Agudelo, O.M., Wang, Q.W., De Moor, B.: Two-sided coupled generalized Sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices. Linear Algebra Appl. 496, 549–593 (2016)
He, Z.H., Wang, Q.W.: A system of periodic discrete-time coupled Sylvester quaternion matrix equations. Algebra Colloq. 24, 169–180 (2017)
Wang, Q.W., He, Z.H.: Systems of coupled generalized Sylvester matrix equations. Automatica 50, 2840–2844 (2014)
Wang, Q.W., Sun, J.H., Li, S.Z.: Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra. Linear Algebra Appl. 353, 169–182 (2002)
Dmytryshyn, A., Kågström, B.: Coupled Sylvester-type matrix equations and block diagonalization. SIAM. J. Matrix Anal. Appl. 36, 580–593 (2015)
De Terán, F., Iannazzo, B.: Uniqueness of solution of a generalized \(\ast \)-Sylvester matrix equation. Linear Multilinear Algebra 493, 323–335 (2016)
Dajić, A., Koliha, J.J.: Equations \(ax =c\) and \(xb =d\) in rings and rings with involution with applications to Hilbert space operators. Linear Algebra Appl. 429, 1779–1809 (2008)
Dajić, A.: Common solutions of linear equations in a ring, with applications. Electron. J. Linear Algebra 30, 66–79 (2015)
Chen, H.X., Wang, L. Wang, Q.: Solvability conditions for mixed Sylvester equations in rings, Filomat (2020) (in press)
Funding
The author is highly grateful to the referee for his/her valuable comments and suggestions which led to improvements of this paper. The research is supported by the National Natural Science Foundation of China (11901510), the National Natural Science Foundation of Jiangsu Province (BK20170589), China Postdoctoral Science Foundation Funded Project (2017M611920).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Chen, H., Wang, L. & Li, T. A note on the solvability for generalized Sylvester equations. RACSAM 115, 64 (2021). https://doi.org/10.1007/s13398-020-00957-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-020-00957-6