An Iterative Method for Solving the Matrix Equation X-A^{*}XA-B^{*}X^{-1}B=I

Aynur Ali, Vejdi Hasanov


In this paper we study iterative computing a positive definite solution of the matrix
equation X-A^{*}XA-B^{*}X^{-1}B=I. We propose an iterative method for finding
a positive definite solution of the considered equation. The theoretical results are
illustrated by numerical examples


Matrix equation; Positive definite solution; Iterative methods

Full Text:



A. Ali, For the matrix equation X − A∗XA − B∗X−1B = I, In: MATTEX 2018, Conference proceeding, Vol. 1, 161-166, (2018). (in Bulgarian)

V. Hasanov, Necessary and sufficient condition for the existence of a positive definite solution of a matrix equation, Annual of Konstantin Preslavsky University of Shumen, vol. XX C, pp. 13-19, (2019).

D. Gao, Iterative methods for solving the nonlinear matrix equation X − A∗XpA − B∗X−qB = I (0 < p, q < 1), Advances in Linear Algebra and Matrix Theory, 7, 72-78, (2017).

V. Hasanov, Positive definite solutions of a linearly perturbed matrix equation, submitted

A. Ferrante, B. Levy, Hermitian solutions of the equation X = Q + NX−1N∗, Linear Algebra Appl., 247, 359-373, (1996).

C.-H. Guo, P. Lancaster, Iterative Solution of Two Matrix Equations, Math. Comput., 68, 1589-1603, (1999).

W. N. Anderson, T. D. Morley, and G. E. Trapp, Positive Solution to X = A−BX−1B∗,Linear Algebra Appl., 134, 53-62, (1990).

J.C. Engwerda, A.C.M. Ran And A.L.Rijkeboer, Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution of the Matrix Equation X + A∗X−1A = Q,Linear Algebra Appl., 186, 255-275, (1993).

XZ. Zhan Computing the extreme positive definite solutions of a matrix equation, SIAM J. Sci. Comput., 17, 632-645, (1996).

B. Meini, Efficient computation of the extreme solutions of X + A∗X−1A = Q and X − A∗X−1A = Q, Math. Comput., 71, 1189-1204, (2001).

M. Berzig, X. Duan, B. Samet, Positive definite solution of the matrix equation X = Q−A∗X−1A + B∗X−1B via Bhaskar-Lakshmikantham fixed point theorem, Mathematical Sciences, 6, 27, (2012).

X. Duan, Q. Wang, C. Li, Positive definite solution of a class of nonlinear matrix equation, Linear Multilinear A, 62, 839-852 (2014).

A.A. Ali, V.I. Hasanov, On some sufficient conditions for the existence of a positive definite solution of the matrix equation X + A∗X−1A − B∗X−1B = I, In: Pasheva V, Popivanov N, Venkov G, editors, 41st International Conference Applications of Mathematics in Engineering and Economics AMEE 2015. AIP Conf. Proc. 1690, 060001 (2015), doi:10.1063/1.4936739.

V. Hasanov, On the matrix equation X + A∗X−1A − B∗X−1B = I, Linear Multilinear A., 66, 1783-1798, (2018).

A.C.R. Ran, M.C.B. Reurings, The symmetric linear matrix equation, Electron. J. Linear Al., 9, 93-107, (2002).

P. Lancaster, M. Tismenetsky, The Theory of Matrices, 2nd ed. Academic Press, San Diego (CA), (1985).

K. Deimling, Nonlinear functional analysis, Springer-Verlag, Berlin, (1985).


  • There are currently no refbacks.
We use cookies.