The Hermitian Positive Definite Solution of Nonlinear Matrix Equations

2009 ◽  
Author(s):  
Guangyuan Zhao
Keyword(s):  
Positive Definite ◽  
Matrix Equations ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

scholarly journals Best Proximity Point Results for Generalized Θ-Contractions and Application to Matrix Equations

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 93
Author(s):  
Zhenhua Ma ◽  
Azhar Hussain ◽  
Muhammad Adeel ◽  
Nawab Hussain ◽  
Ekrem Savas
Keyword(s):  
Metric Spaces ◽  
Best Proximity Point ◽  
Positive Definite ◽  
Matrix Equations ◽  
Nonlinear Matrix Equation ◽  
Complete Metric Spaces ◽  
Partially Ordered ◽  
Positive Definite Solution ◽  
Nonlinear Matrix ◽  
Definite Solution

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.

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

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
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.

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

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.

scholarly journals Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2199
Author(s):  
Hemant Kumar Nashine ◽  
Rajendra Pant ◽  
Reny George
Keyword(s):  
Fixed Point ◽  
Positive Definite ◽  
Matrix Equations ◽  
Trace Norm ◽  
Time Analysis ◽  
Positive Definite Solution ◽  
Cpu Time ◽  
Nonlinear Matrix ◽  
The Given ◽  
Definite Solution

We discuss a pair of nonlinear matrix equations (NMEs) of the form X=R1+∑i=1kAi*F(X)Ai, X=R2+∑i=1kBi*G(X)Bi, where R1,R2∈P(n), Ai,Bi∈M(n), i=1,⋯,k, and the operators F,G:P(n)→P(n) are continuous in the trace norm. We go through the necessary criteria for a common positive definite solution of the given NME to exist. We develop the concept of a joint Suzuki-implicit type pair of mappings to meet the requirement and achieve certain existence findings under weaker assumptions. Some concrete instances are provided to show the validity of our findings. An example is provided that contains a randomly generated matrix as well as convergence and error analysis. Furthermore, we offer graphical representations of average CPU time analysis for various initializations.

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