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Positive Definite Solutions for the System of Nonlinear Matrix Equations <i>X</i> + <i>A<sup>*</sup>Y<sup>-n</sup>A</i> = <i>I</i>, <i>Y</i> + <i>B<sup>*</sup>X<sup>-m</sup>B</i> = <i>I</i>
Department of Scientific Computing, Faculty of Computers and Informatics, Benha University, Benha, Egypt
Faculty of Science and Arts, Qassim University, Qassim, KSA
- 1 Department of Scientific Computing, Faculty of Computers and Informatics, Benha University, Benha, Egypt
- 2 Faculty of Science and Arts, Qassim University, Qassim, KSA
Applied Mathematics·Volume 05 (2014)·Pages 1977–1987·Published 7 July 2014·DOI10.4236/am.2014.513190
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Abstract
In this paper, some properties of the positive definite solutions for the nonlinear system of matrix equations X + A * Y -n A = I , Y + B * X -m B = I are derived. As a matter of fact, an effective iterative method to obtain the positive definite solutions of the system is established. These solutions are based on the convergence of monotone sequences of positive definite matrices. Moreover, the necessary and sufficient conditions for the existence of the positive definite solutions are obtained. Finally, some numerical results are given.
KeywordsSystem of Nonlinear Matrix EquationsIterative MethodsMonotonic SequencePositive Definite Matrices
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