Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation X+A* X-2 A+B* X-2B=I

ijmtt-v52p504

doi:https://doi.org/10.14445/22315373/IJMTT-V52P504

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 52 | Number 1 | Year 2017 | Article Id. IJMTT-V52P504 | DOI : https://doi.org/10.14445/22315373/IJMTT-V52P504

Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation X+A* X-2 A+B* X-2B=I

ijmtt-v52p504

Abstract

In this paper we derive necessary and sufficient conditions for the matrix equation X+A* X-2 A+B* X-2B=I to have a positive definite solution X, where , I is an n x n identity matrix and Aand B are n x n nonsingular complex matrices. We use these conditions to present some properties on the matrices A andB. Moreover, relations between the solution X and the matrices A and B are proposed.

Keywords

Nonlinear matrix equation, positive definite solution, necessary and sufficient conditions.

References

[1] C. H. Guo and P. Lancaster, “ Iterative Solution of two Matrix Equations, ” Math. of Comp. 228, 1589-1603 (1999).
[2] G. H. Golub and C. F. Van Loan, “ Matrix Computations, ” Johns Hopkins Univ. Press. Baltimore (1989).
[3] G. W. Stewart and J. G. Sun, “ Matrix Perturbation Theory, ” Academic Press, Boston, Mass, USA, 1990.

Citation :

ijmtt-v52p504, "Necessary and Sufficient Conditions for the Existence of a Positive Definite Solution for the Matrix Equation X+A* X-2 A+B* X-2B=I," International Journal of Mathematics Trends and Technology (IJMTT), vol. 52, no. 1, pp. 22-26, 2017. Crossref, https://doi.org/10.14445/22315373/IJMTT-V52P504