Lower bounds of the minimal eigenvalue of a Hermitian positive-definite matrix

2000 ◽  
Vol 46 (7) ◽  
pp. 2760-2762 ◽  
Author(s):  
Weiwei Sun
Keyword(s):  
Lower Bounds ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Minimal Eigenvalue ◽  
Hermitian Positive Definite Matrix

scholarly journals Inequalities for the eigenvalues of the positive definite solutions of the nonlinear matrix equation

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2667-2671
Author(s):  
Guoxing Wu ◽  
Ting Xing ◽  
Duanmei Zhou
Keyword(s):  
Matrix Equation ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
The Matrix ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix ◽  
Positive Integers ◽  
Equivalent Conditions

In this paper, the Hermitian positive definite solutions of the matrix equation Xs + A*X-tA = Q are considered, where Q is a Hermitian positive definite matrix, s and t are positive integers. Bounds for the sum of eigenvalues of the solutions to the equation are given. The equivalent conditions for solutions of the equation are obtained. The eigenvalues of the solutions of the equation with the case AQ = QA are investigated.

scholarly journals Extensions of the Eneström-Kakeya theorem for matrix polynomials

2019 ◽  
Vol 7 (1) ◽  
pp. 304-315
Author(s):  
A. Melman
Keyword(s):  
Lower Bounds ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Upper And Lower Bounds ◽  
Unified Approach ◽  
Matrix Polynomials ◽  
Positive Coefficients

Abstract The classical Eneström-Kakeya theorem establishes explicit upper and lower bounds on the zeros of a polynomial with positive coefficients and has been generalized for positive definite matrix polynomials by several authors. Recently, extensions that improve the (scalar) Eneström-Kakeya theorem were obtained with a transparent and unified approach using just a few tools. Here, the same tools are used to generalize these extensions to positive definite matrix polynomials, while at the same time generalizing the tools themselves. In the process, a framework is developed that can naturally generate additional similar results.

scholarly journals Extended Branciari quasi-b-distance spaces, implicit relations and application to nonlinear matrix equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Reena Jain ◽  
Hemant Kumar Nashine ◽  
Vahid Parvaneh
Keyword(s):  
Positive Definite Matrix ◽  
Weak Limit ◽  
Positive Definite ◽  
Contractive Condition ◽  
Sufficient Condition ◽  
Nonlinear Matrix Equation ◽  
Property A ◽  
Underlying Space ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix

AbstractThis study introduces extended Branciari quasi-b-distance spaces, a novel implicit contractive condition in the underlying space, and basic fixed-point results, a weak well-posed property, a weak limit shadowing property and generalized Ulam–Hyers stability. The given notions and results are exemplified by suitable models. We apply these results to obtain a sufficient condition ensuring the existence of a unique positive-definite solution of a nonlinear matrix equation (NME) $\mathcal{X}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G(X)}\mathcal{A}_{i}$ X = Q + ∑ i = 1 k A i ∗ G ( X ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive-definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices, and $\mathcal{G}$ G is a nonlinear self-mapping of the set of all Hermitian matrices that are continuous in the trace norm. We demonstrate this sufficient condition for the NME $\mathcal{X}= \mathcal{Q} +\mathcal{A}_{1}^{*}\mathcal{X}^{1/3} \mathcal{A}_{1}+\mathcal{A}_{2}^{*}\mathcal{X}^{1/3} \mathcal{A}_{2}+ \mathcal{A}_{3}^{*}\mathcal{X}^{1/3}\mathcal{A}_{3}$ X = Q + A 1 ∗ X 1 / 3 A 1 + A 2 ∗ X 1 / 3 A 2 + A 3 ∗ X 1 / 3 A 3 , and visualize this through convergence analysis and a solution graph.

The Hermitian Positive Definite Solution of the Nonlinear Matrix Equation

2017 ◽  
Vol 18 (5) ◽  
pp. 293-301
Author(s):  
Xindong Zhang ◽  
Xinlong Feng
Keyword(s):  
Matrix Equation ◽  
Iterative Algorithms ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Nonlinear Matrix Equation ◽  
Positive Definite Solution ◽  
Hermitian Positive Definite Solution ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix ◽  
Definite Solution

Abstract:In this paper, we study the nonlinear matrix equation $X^{s}\pm\sum^{m}_{i=1}A^{T}_{i}X^{\delta_{i}}A_{i}=Q$, where $A_{i}\;(i=1,2,\ldots,m)$ is $n\times n$ nonsingular real matrix and $Q$ is $n\times n$ Hermitian positive definite matrix. It is shown that the equation has an unique Hermitian positive definite solution under some conditions. Iterative algorithms for obtaining the Hermitian positive definite solution of the equation are proposed. Finally, numerical examples are reported to illustrate the effectiveness of algorithms.

scholarly journals An implicit relation, relational theoretic approach under w-distance and application to nonlinear matrix equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Reena Jain ◽  
Hemant Kumar Nashine ◽  
Manuel De la Sen
Keyword(s):  
Metric Spaces ◽  
Sufficient Conditions ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Contractive Condition ◽  
Theoretic Approach ◽  
Nonlinear Matrix Equation ◽  
Surface Plot ◽  
Hermitian Positive Definite Matrix ◽  
Nonlinear Matrix

AbstractWe propose a new class of implicit relations and an implicit type contractive condition based on it in the relational metric spaces under w-distance functional. Further we derive fixed points results based on them. Useful examples illustrate the applicability and effectiveness of the presented results. We apply these results to discuss sufficient conditions ensuring the existence of a unique positive definite solution of the nonlinear matrix equation (NME) of the form $\mathcal{U}=\mathcal{Q} + \sum_{i=1}^{k}\mathcal{A}_{i}^{*} \mathcal{G}\mathcal{(U)}\mathcal{A}_{i}$ U = Q + ∑ i = 1 k A i ∗ G ( U ) A i , where $\mathcal{Q}$ Q is an $n\times n$ n × n Hermitian positive definite matrix, $\mathcal{A}_{1}$ A 1 , $\mathcal{A}_{2}$ A 2 , …, $\mathcal{A}_{m}$ A m are $n \times n$ n × n matrices and $\mathcal{G}$ G is a nonlinear self-mapping of the set of all Hermitian matrices which are continuous in the trace norm. In order to demonstrate the obtained conditions, we consider an example together with convergence and error analysis and visualisation of solutions in a surface plot.

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