The Hermitian Positive Definite Solution of Some Matrix Equations

In this paper, we introduce the notion of C ´ iri c ´ type α - ψ - Θ -contraction and prove best proximity point results in the context of complete metric spaces. Moreover, we prove some best proximity point results in partially ordered complete metric spaces through our main results. As a consequence, we obtain some fixed point results for such contraction in complete metric and partially ordered complete metric spaces. Examples are given to illustrate the results obtained. Moreover, we present the existence of a positive definite solution of nonlinear matrix equation X = Q + ∑ i = 1 m A i * γ ( X ) A i and give a numerical example.

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Necessary and Sufficient Conditions for the Existence of a Hermitian Positive Definite Solution of a Type of Nonlinear Matrix Equations

Mathematical Problems in Engineering ◽

10.1155/2009/672695 ◽

2009 ◽

Vol 2009 ◽

pp. 1-13 ◽

Cited By ~ 1

Author(s):

Wenling Zhao ◽

Hongkui Li ◽

Xueting Liu ◽

Fuyi Xu

Keyword(s):

Matrix Equation ◽

Sufficient Conditions ◽

Positive Definite ◽

Necessary And Sufficient Conditions ◽

Nonsingular Matrix ◽

Positive Definite Solution ◽

Hermitian Positive Definite Solution ◽

Nonlinear Matrix ◽

Necessary And Sufficient ◽

Definite Solution

We study the Hermitian positive definite solutions of the nonlinear matrix equationX+A∗X−2A=I, whereAis ann×nnonsingular matrix. Some necessary and sufficient conditions for the existence of a Hermitian positive definite solution of this equation are given. However, based on the necessary and sufficient conditions, some properties and the equivalent equations ofX+A∗X−2A=Iare presented while the matrix equation has a Hermitian positive definite solution.

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Notes on the Hermitian Positive Definite Solutions of a Matrix Equation

Journal of Applied Mathematics ◽

10.1155/2014/128249 ◽

2014 ◽

Vol 2014 ◽

pp. 1-8 ◽

Cited By ~ 2

Author(s):

Jing Li ◽

Yuhai Zhang

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Matrix Equation ◽

Positive Definite ◽

Nonlinear Matrix Equation ◽

Positive Definite Solution ◽

Hermitian Positive Definite Solution ◽

The Matrix ◽

Definite Solution

The nonlinear matrix equation,X-∑i=1mAi*XδiAi=Q,with-1≤δi<0is investigated. A fixed point theorem in partially ordered sets is proved. And then, by means of this fixed point theorem, the existence of a unique Hermitian positive definite solution for the matrix equation is derived. Some properties of the unique Hermitian positive definite solution are obtained. A residual bound of an approximate solution to the equation is evaluated. The theoretical results are illustrated by numerical examples.

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Latest Inversion-Free Iterative Scheme for Solving a Pair of Nonlinear Matrix Equations

Journal of Mathematics ◽

10.1155/2021/2667885 ◽

2021 ◽

Vol 2021 ◽

pp. 1-22

Author(s):

Sourav Shil ◽

Hemant Kumar Nashine

Keyword(s):

Convergence Rate ◽

Convergence Analysis ◽

Iterative Scheme ◽

Positive Definite ◽

Matrix Equations ◽

Identity Matrix ◽

Positive Definite Solution ◽

Nonlinear Matrix ◽

Definite Solution

In this work, the following system of nonlinear matrix equations is considered, X 1 + A ∗ X 1 − 1 A + B ∗ X 2 − 1 B = I and X 2 + C ∗ X 2 − 1 C + D ∗ X 1 − 1 D = I , where A , B , C , and D are arbitrary n × n matrices and I is the identity matrix of order n . Some conditions for the existence of a positive-definite solution as well as the convergence analysis of the newly developed algorithm for finding the maximal positive-definite solution and its convergence rate are discussed. Four examples are also provided herein to support our results.

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