Atlas: On the Matrix Equation $X - A^* X^{-n} A = I$ by Vejdi Hasanov

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International Conference: 2004 - Dynamical Systems and Applications
July 5-10, 2004

Antalya, Turkey

Organizers
Akca Haydar, Basem S. Attili, Boucherif Abdelkader, Cho Yeol Je, Covachev Valery, Gyori Istvan, Maksimov Vyacheslav, Stavroulakis Ioannis P.

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On the Matrix Equation X - A^* X^-n A = I
by
Vejdi Hasanov
Laboratory of Mathematical Modelling, Shoumen University, Shoumen 9712, Bulgaria
Coauthors: Ivan Ivanov

We consider the nonlinear matrix equation

X - A^* X^-n A = I, (1)

where A is an m×m complex matrix, X is an unknown matrix, I is the identity matrix and n is a positive integer.

The equation (1) for n=1 has many applications [1]. Xin-Guo Lio and Hua Gao have considered more general equations X^s +/- A^T X^-t A = I . El-Sayed [2] has investigated the equation (1) and two iterative methods for computing a positive definite solution of (1). The sufficient conditions for the convergence of the considered methods for (1) with n=2^k are derived in [2]. The positive definite solutions of the equation X +/- A^* X^-2A = I and their properties have been discussed in [3].

In this paper we generalize El-Sayed's sufficient conditions for equation (1) with an arbitrary integer n . We compare some theorems which are proved in [3, 5]. We present some critical remarks on iterative methods which are considered in [4].

Numerical experiments to illustrate the performance of the methods are reported.

[1] A. Ferrante, B. C. Levy Hermitian Solution of the X = Q + N X^-1 N^*, Linear Algebra Appl., 247 (1996) 359-373.

[2] S. M. El-Sayed, Two Iteration Processes for Computing Positive Definite Solution of the Equation X - A^* X^-n A = Q Computers Mathematics with applications, 41 (2001) 579-588.

[3] I. G. Ivanov, V. I. Hasanov, B. V. Minchev, On Matrix Equation X +/- A^* X^-2 A = I, Linear Algebra Appl., 326 (2001) 27-44.

[4] X.-G. Lio, H. Gao, On the Positive Definite Solutions of the Matrix Equations X^s +/- A^T X^-t A = I, Linear Algebra Appl., 368 (2003) 83-97.

[5] M. Reurings, Symmetric Matrix Equations, PhD Thesis, Amsterdam 2003.

Date received: March 11, 2004