Browse > Article
http://dx.doi.org/10.4134/JKMS.j160429

HERMITIAN POSITIVE DEFINITE SOLUTIONS OF THE MATRIX EQUATION Xs + A*X-tA = Q  

Masoudi, Mohsen (Department of Mathematics Shahid Bahonar University of Kerman)
Moghadam, Mahmoud Mohseni (Department of Mathematics Shahid Bahonar University of Kerman)
Salemi, Abbas (Department of Mathematics Shahid Bahonar University of Kerman)
Publication Information
Journal of the Korean Mathematical Society / v.54, no.6, 2017 , pp. 1667-1682 More about this Journal
Abstract
In this paper, the Hermitian positive definite solutions of the matrix equation $X^s+A^*X-^tA=Q$, where Q is an $n{\times}n$ Hermitian positive definite matrix, A is an $n{\times}n$ nonsingular complex matrix and $s,t{\in}[1,{\infty})$ are discussed. We find a matrix interval which contains all the Hermitian positive definite solutions of this equation. Also, a necessary and sufficient condition for the existence of these solutions is presented. Iterative methods for obtaining the maximal and minimal Hermitian positive definite solutions are proposed. The theoretical results are illustrated by numerical examples.
Keywords
iterative algoritheorem; nonlinear matrix equation; positive definite solution; fixed point theorem;
Citations & Related Records
연도 인용수 순위
  • Reference
1 R. P. Agarwal, M. Meehan, and D. O'Regan, Fixed Point Theory and Applications, Cambridge University, 2001.
2 W. N. Anderson, T. D. Morley, and G. E. Trapp, Ladder networks, fixed points, and the geometric mean, Circuits Systems Signal Process. 2 (1983), no. 3, 259-268.   DOI
3 T. Ando, Limit of iterates of cascade addition of matrices, Numer. Funct. Anal. Optim. 2 (1980), no. 7-8, 579-589.   DOI
4 R. Bhatia, Matrix Analysis, Graduate Text in Mathematics, Springer-Verlag New York, 1997.
5 R. S. Bucy, A priori bound for the Riccati equation, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 111, 645-656, Probability Theory, University of California Press, Berkeley, 1972.
6 J. Cai and G. Chen, Some investigation on Hermitian positive definite solutions of the matrix equation $X^s+A^*X^{-t}A=Q$, Linear Algebra Appl. 430 (2009), no. 8-9, 2448-2456.   DOI
7 Sh. Du and J. Hou, Positive definite solutions of operator equations $X^m+A^*X^{-n}A = I$, Linear Multilinear Algebra 51 (2003), no. 2, 163-173.   DOI
8 X. F. Duan and A. P. Liao, On the existence of Hermitian positive definite solutions of the matrix equation $X^s+A^*X^{-t}A=Q$, Linear Algebra Appl. 429 (2008), no. 4, 673-687.   DOI
9 X. F. Duan and A. P. Liao, On the nonlinear matrix equation $X+A^*X^{-q}A=Q$ (q $\geq$ 1), Math. Comput. Modelling 49 (2009), no. 5-6, 936-945.   DOI
10 S. M. El-Sayed and A. M. Al-Dbiban, A new inversion free iteration for solving the equation $X+A^TX^{-1}A=Q$, J. Comput. Appl. Math. 181 (2005), no. 1, 148-156.   DOI
11 J. C. Engwerda, On the existence of a positive definite solution of the matrix equation $X+A^TX^{-1}A=I$, Linear Algebra Appl. 194 (1993), 91-108.   DOI
12 T. Furuta, Operator inequalities associated with Holder-McCarthy and Kantorovich inequalities, J. Inequal. Appl. 2 (1998), no. 2, 137-148.
13 W. L. Green and E. W. Kamen, Stabilization of linear systems over a commutative normed algebra with applications to spatially distributed parameter dependent systems, SIAM J. Control Optim. 23 (1985), 1-18.   DOI
14 V. I. Hasanov and I. G. Ivanov, Solutions and perturbation estimates for the matrix equations $X{\pm}A^*X^{-n}A=Q$, Appl. Math. Comput. 156 (2004), no. 2, 513-525.   DOI
15 R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1991.
16 I. G. Ivanov, On positive definite solutions of the family of matrix equations $X+A^*X^{-n}A=Q$, J. Comput. Appl. Math. 193 (2006), no. 1, 277-301.   DOI
17 M. Parodi, La localisation des valeurs caracterisiques des matrices et ses applications, Gauthier-Villars, Paris, 1959.
18 Z. Y. Peng, S. M. El-Sayed, and X. L. Zhang, Iterative methods for the extremal positive definite solution of the matrix equation $X+A^*X^{-{\alpha}}A=Q$, J. Comput. Appl. Math. 200 (2007), no. 2, 520-527.   DOI
19 W. Pusz and S. L. Woronowitz, Functional calculus for sesquilinear forms and the puri cation map, Rep. Mathematical Phys. 8 (1975), no. 2,159-170.   DOI
20 X. T. Wang and Y. M. Li, On equations that are equivalent to the nonlinear matrix equation $X+A^*X^{-{\alpha}}A=Q$, J. Comput. Appl. Math. 234 (2010), no. 8, 2441-2449.   DOI
21 X. Y. Yin, S. Y. Liu, and L. Fang, Solutions and perturbation estimates for the matrix equation $X^s+A^*X^{-t}A=Q$, Linear Algebra Appl. 431 (2009), no. 9, 1409-1421.   DOI
22 Y. Yueting, The iterative method for solving nonlinear matrix equation $X^s+A^*X^{-t}A=Q$, Appl. Math. Comput. 188 (2007), no. 1, 46-53.   DOI
23 X. Zhan, Matrix Inequalities, Springer-Verlag Berlin Heidelberg, 2002.
24 D. Zhou, G. Chen, G. Wu, and X. Zhang, Some properties of the nonlinear matrix equation $X^s+A^*X^{-t}A=Q$, J. Math. Anal. Appl. 392 (2012), no. 1, 75-82.   DOI