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On the Hermitian positive definite solutions of nonlinear matrix equation $X^s + A^*X^{-t}A = Q$. (English) Zbl 1201.15005
Authors’ abstract: The nonlinear matrix equation $X^s + A^*X^{-t}A = Q$, where $A, Q$ are $n \times n$ complex matrices with $Q$ Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: $s \geqslant 1, 0 < t \leqslant 1$ and $0 < s \leqslant 1, t \geqslant 1$. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.

15A24Matrix equations and identities
65F30Other matrix algorithms
15B48Positive matrices and their generalizations; cones of matrices
15B57Hermitian, skew-Hermitian, and related matrices
Full Text: DOI
[1] Anderson, W. N.; Morley, T. D.; Trapp, G. E.: Positive solutions to X=A - BX - 1B$\ast $, Linear algebra appl. 134, 53-62 (1990) · Zbl 0702.15009 · doi:10.1016/0024-3795(90)90005-W
[2] Chen, X.; Li, W.: On the matrix equation X+A$\ast X - 1A=P$: solution and perturbation analysis, Math. num. Sin. 27, 303-310 (2005) · Zbl 1106.15301
[3] Engwerda, J. C.; Ran, A. C. M.; Rijkeboer, A. L.: Necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation X+A$\ast X - 1A=Q$, Linear algebra appl. 186, 255-275 (1993) · Zbl 0778.15008 · doi:10.1016/0024-3795(93)90295-Y
[4] Engwerda, J. C.: On the existence of a positive definite solution of the matrix equation X+A$\ast X - 1A=I$, Linear algebra appl. 194, 91-108 (1993) · Zbl 0798.15013 · doi:10.1016/0024-3795(93)90115-5
[5] El-Sayed, S. M.; Ran, A. C. M.: On an iterative methods for solving a class of nonlinear matrix equations, SIAM J. Matrix anal. Appl. 23, 632-645 (2001) · Zbl 1002.65061 · doi:10.1137/S0895479899345571
[6] Furuta, T.: Operator inequalities associated with holder -- McCarthy and Kantorovich inequalities, J. inequal. Appl. 6, 137-148 (1998) · Zbl 0910.47014 · doi:10.1155/S1025583498000083
[7] Guo, C.; Lancaster, P.: Iterative solution of two matrix equations, Math. comput. 68, 1589-1603 (1999) · Zbl 0940.65036 · doi:10.1090/S0025-5718-99-01122-9
[8] Hasanov, V. I.; Ivanov, I. G.: Solutions and perturbation estimates for the matrix equations X$\pm a\ast $X - na=Q, Appl. math. Comput. 156, 513-525 (2004) · Zbl 1063.15012 · doi:10.1016/j.amc.2003.08.007
[9] Hasanov, V. I.: Positive definite solutions of the matrix equations X$\pm a\ast $X - qa=Q, Linear algebra appl. 404, 166-182 (2005) · Zbl 1078.15012 · doi:10.1016/j.laa.2005.02.024
[10] Hasanov, V. I.; El-Sayed, S. M.: On the positive definite solutions of nonlinear matrix equation X+A$\ast X - \delta $A=Q, Linear algebra appl. 412, 154-160 (2006) · Zbl 1083.15018 · doi:10.1016/j.laa.2005.06.026
[11] Ivanov, I. G.; Uhlig, F.: Improved methods and starting values to solve the matrix equations X$\pm a\ast $X - 1A=I iteratively, Math. comput. 74, 263-278 (2005) · Zbl 1058.65051 · doi:10.1090/S0025-5718-04-01636-9
[12] Ivanov, I. G.; El-Sayed, S. M.: Properties of positive definite solutions of the equation X+A$\ast X - 2A=I$, Linear algebra appl. 279, 303-316 (1998) · Zbl 0935.65041 · doi:10.1016/S0024-3795(98)00023-8
[13] Ivanov, I. G.; Hasanov, V. I.; Minchev, B. V.: On matrix equations X$\pm a\ast $X - 2A=I, Linear algebra appl. 326, 27-44 (2001) · Zbl 0979.15007 · doi:10.1016/S0024-3795(00)00302-5
[14] Liu, X. G.; Gao, H.: On the positive definite solutions of the equation xs$\pm $ATX - ta=In, Linear algebra appl. 368, 83-97 (2003)
[15] Parodi, M.: La localisation des valeurs caracterisiques desmatrices etses applications, (1959) · Zbl 0087.01602
[16] Peng, Z.; El-Sayed, S. M.; Zhang, X.: Iterative methods for the extremal positive solution of the matrix equation X+A$\ast X - \alpha $A=Q, J. comput. Appl. math. 200, 520-527 (2007) · Zbl 1118.65029 · doi:10.1016/j.cam.2006.01.033
[17] Yang, Y.: The iterative method for solving nonlinear matrix equation xs+A$\ast X$ - ta=Q, Appl. math. Comput. 188, 46-53 (2007) · Zbl 1131.65039 · doi:10.1016/j.amc.2006.09.085
[18] Zhan, X.: Computing the extremal positive definite solution of a matrix equation, SIAM J. Sci. comput. 17, 1167-1174 (1996) · Zbl 0856.65044 · doi:10.1137/S1064827594277041
[19] Zhan, X.; Xie, J.: On the matrix equation X+ATX - 1A=I, Linear algebra appl. 247, 337-345 (1996) · Zbl 0863.15005 · doi:10.1016/0024-3795(95)00120-4