Study On Some Matrix Equations Involving The Weighted Geometric Mean and Their Application

In this paper we consider two matrix equations that involve the weighted geometric mean. We use the fixed point theorem in the cone of positive definite matrices to prove the existence of a unique positive definite solution. In addition, we study the multi-step stationary iterative method for those equations and prove the corresponding convergence. A fidelity measure for quantum states based on the matrix geometric mean is introduced as an application of matrix equation.

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

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.

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

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.

Best Proximity Point Results for Generalized Θ-Contractions and Application to 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.