Matrix Inequalities

In this article, we introduce some refinements of the reverse young inequality for scalars. As applications of our results, we establish corresponding inequalities for matrices. The obtained inequalities in this article can be viewed as refinements of the derived inequalities by Burqan and Khandaqji [4].

The effects of porosity on the performance of gas bearings are investigated. A thin-wall approximation is developed to simplify the coupling between the Reynolds equation in the fluid film and the three dimensional Laplace equation in the porous matrix. The Newton-Raphson method is used to linearize the modified Reynolds equation and the column method is employed to solve the system of linear equations in matrix forms. The solution of a solid-wall bearing is included as a special case. Bearing characteristics are presented in tables and charts for varied permeabilities, slip coefficients, porous layer thicknesses, eccentricity ratios, length-to-diameter ratios, and for a wide range of compressibility number.

PurposeThis article proposes a relaxed gradient iterative (RGI) algorithm to solve coupled Sylvester-conjugate transpose matrix equations (CSCTME) with two unknowns.Design/methodology/approachThis article proposes a RGI algorithm to solve CSCTME with two unknowns.FindingsThe introduced (RGI) algorithm is more efficient than the gradient iterative (GI) algorithm presented in Bayoumi (2014), where the author's method exhibits quick convergence behavior.Research limitations/implicationsThe introduced (RGI) algorithm is more efficient than the GI algorithm presented in Bayoumi (2014), where the author's method exhibits quick convergence behavior.Practical implicationsIn systems and control, Lyapunov matrix equations, Sylvester matrix equations and other matrix equations are commonly encountered.Social implicationsIn systems and control, Lyapunov matrix equations, Sylvester matrix equations and other matrix equations are commonly encountered.Originality/valueThis article proposes a relaxed gradient iterative (RGI) algorithm to solve coupled Sylvester conjugate transpose matrix equations (CSCTME) with two unknowns. For any initial matrices, a sufficient condition is derived to determine whether the proposed algorithm converges to the exact solution. To demonstrate the effectiveness of the suggested method and to compare it with the gradient-based iterative algorithm proposed in [6] numerical examples are provided.

This paper is concerned with the numerical solutions of the general matrix equation $\sum ^{p}_{i=1}{\sum ^{s_{i}}_{j=1} }\,\,{A_{ij}X_{i}}{B_{ij}} = C$ , and the general discrete-time periodic matrix equations $\sum ^{p}_{i=1}\sum ^{s_{i}}_{j=1} (A_{i,j,k}X_{i,k}B_{i,j,k}+ C_{i,j, k}X_{i,k+1}D_{i,j,k}) = M_{k}$ , for $k = 1, 2, \ldots ,t$ , which include the well-known Lyapunov, Stein, and Sylvester matrix equations that appear in a wide range of applications in engineering and mechanical problems. Recently the generalized product-type BiCOR method, denoted as GPBiCOR, has been originally proposed to solve the nonsymmetric linear systems $Ax = b$ , and its significant convergence performance has been confirmed in many numerical results. By applying the Kronecker product and the vectorization operator, we develop a matrix form of the GPBiCOR method to approximate the solutions of the above-mentioned general matrix equation and general discrete-time periodic matrix equations. We present the theoretical background of the extended GPBiCOR method and its main computational aspects. Furthermore, several numerical examples of matrix equations arising in different applications are considered to exhibit the accuracy and the efficiency of the proposed method as compared with other popular methods in the literature.

The ultra slow light in a cholesteric liquid crystal given by similar Schrödinger equation

An efficient algorithm for solving general coupled matrix equations and its application

Finite iterative solutions to a class of complex matrix equations with conjugate and transpose of the unknowns

A lower bound for the product of eigenvalues of solutions to matrix equations

It is known that solving coupled matrix equations with complex matrices can be very difficult and it is sufficiently complicated. In this work, we propose two iterative algorithms based on the Conjugate Gradient method (CG) for finding the reflexive and Hermitian reflexive solutions of the coupled Sylvester-conjugate matrix equations(including Sylvester and Lyapunov matrix equations as special cases). The iterative algorithms can automatically judge the solvability of the matrix equations over the reflexive and Hermitian reflexive matrices, respectively. When the matrix equations are consistent over reflexive and Hermitian reflexive matrices, for any initial reflexive and Hermitian reflexive matrices, the iterative algorithms can obtain reflexive and Hermitian reflexive solutions within a finite number of iterations in the absence of roundoff errors, respectively. Finally, two numerical examples are presented to illustrate the proposed algorithms.

Matrix equation problem is one of the topics of active research in the context of computational mathematics. The Hermitian positive definite solutions of a matrix equation play an important role in real applications. In this paper, we present the sufficient conditions for the existence of the positive definite solution to the nonlinear matrix equation X - A* (X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/2</sup> A = I and propose a natural and stable iteration algorithm for obtaining a positive definite solution of this matrix equation. Finally, two numerical examples for the convergence behavior of the proposed algorithm are conducted to demonstrate the effectiveness.

This note presents an iterative algorithm to solve the coupled Sylvester-transpose matrix equations (including the generalized coupled Sylvester matrix equations and Lyapunov matrix equations as special cases) over generalized centro-symmetric matrices. When the considered matrix equations are consistent, for any initial generalized centro-symmetric matrix group, a generalized centro-symmetric solution group can be obtained within finite iteration steps in the absence of roundoff errors. The least Frobenius norm generalized centro-symmetric solution group of the coupled Sylvester-transpose matrix equations can be derived when a suitable initial generalized centro-symmetric matrix group is chosen. In addition, for a given generalized centro-symmetric matrix group, the optimal approximation generalized centro-symmetric solution group can be obtained by finding the least Frobenius norm generalized centro-symmetric solution group of new coupled Sylvester-transpose matrix equations. Finally, a numerical example is given to demonstrate the efficiency of the introduced iterative algorithm.

In this article, we examine an efficient solution of the stochastic Galerkin (SG) matrix equations coming from the Darcy flow problem with uncertain material parameters on given interfaces. The solution of the SG system of equations, here represented as matrix equations, is usually a very challenging task. A relatively new approach to the solution of the SG matrix equations is the reduced basis (RB) solver, which looks for the low-rank representation of the solution. The construction of the RB is usually done iteratively and consists of multiple solutions of systems of equations. We aim to speed up the process using the deflated conjugate gradients (DCG). Other contributions of this work are a modified specific construction of the RB without the need of Cholesky factor and an adaptive choice of the candidate vectors for the expansion of the RB. The proposed approach allows an efficient parallel implementation.

Matrix equations involving a family of unknown matrices $$X_1, \ldots , X_k$$ can be written in a general form $$f(X_1, \ldots , X_k) = 0$$ . One of the fundamental problems in matrix equation theory is to establish identifying conditions for the matrix equation to always hold for all matrices $$X_1, \ldots , X_k$$ . In this article, we select some nonlinear matrix equations with two, three, and four unknown matrices as illustrative examples, and show how to establish a family of simple and well-organized necessary and sufficient conditions for these matrix equations to always hold with respect to all unknown matrices in the equations. The methods, results, and facts in this article can be used to efficiently establish various nonlinear identities with variables in different noncommutative algebraic structures.

In this paper, a modal expansion method is proposed to compute the field distribution inside a reverberation chamber excited by a monopole. The electromagnetic fields are expanded using modal functions derived from Helmholtz equation. The continuity conditions of tangential field components at regional interfaces are transformed into four matrix equations through the mode matching (MM) process. Another two matrix equations are obtained using the point matching (PM) method by imposing the boundary conditions on the four side walls. With those six matrix equations, all the unknown expansion coefficients are determined. This technique takes advantage of the orthogonality of the modal functions, and thus the obtained matrix is block-diagonal, which is easy to be inverted. Furthermore, discretization of the whole chamber is avoided so that the problem scale is greatly reduced. Therefore, the proposed method appears to provide a rigorous and efficient analysis of RC excited by a monopole antenna.

In a previous article by Aihua Li and Duane Randall, the existence of solutions to certain matrix equations is demonstrated via nonconstructive methods. A recurring example appears in that work, namely the matrix equation AXA = X 2 AX, where A is a fixed, square matrix with real entries and X is an unknown square matrix. In this paper, the solution space is explicitly constructed for all 2×2 complex matrices using Grobner basis techniques. When A is a 2×2 matrix, the equation AXA = X 2 AX is equivalent to a system of four polynomial equations. The solution space then is the variety defined by the polynomials involved. The ideal of the underlying polynomial ring generated by the defining polynomials plays an important role in solving the system. In our procedure for solving these equations, Grobner bases are used to transform the polynomial system into a simpler one, which makes it possible to classify all the solutions. In addition to classifying all solutions for 2 × 2 matrices, certain explicit solutions are produced in arbitrary dimensions when A is nonsingular. In higher dimensions, Grobner bases are extraordinarily computationally demanding, and so a different approach is taken. This technique can be applied to more general matrix equations, and the focus here is placed on solutions coming from a particular class of matrices.