Class Of Matrix Equations Research Articles

Approximating Solutions of Matrix Equations via Fixed Point Techniques

The principal goal of this work is to investigate new sufficient conditions for the existence and convergence of positive definite solutions to certain classes of matrix equations. Under specific assumptions, the basic tool in our study is a monotone mapping, which admits a unique fixed point in the setting of a partially ordered Banach space. To estimate solutions to these matrix equations, we use the Krasnosel’skiĭ iterative technique. We also discuss some useful examples to illustrate our results.

Solving Bisymmetric Solution of a Class of Matrix Equations Based on Linear Saturated System Model Neural Network

In order to solve the complicated process and low efficiency and low accuracy of solving a class of matrix equations, this paper introduces the linear saturated system model neural network architecture to solve the bisymmetric solution of a class of matrix equations. Firstly, a class of matrix equations is constructed to determine the key problems of solving the equations. Secondly, the linear saturated system model neural network structure is constructed to determine the characteristic parameters in the process of bisymmetric solution. Then, the matrix equations is solved by using backpropagation neural network topology. Finally, the class normalization is realized by using the objective function of bisymmetric solution, and the bisymmetric solution of a class of matrix equations is realized. In order to verify the solving effect of the method in this paper, three indexes (accuracy, correction accuracy, and solving time) are designed in the experiment. The experimental results show that the proposed method can effectively reduce the solving time, can improve the accuracy and correction effect of the bisymmetric solution, and has high practicability.

Non-singular solutions of one class of matrix equations over a polynomial ring

Binomial two-sided homogeneous matrix equations with two unknowns are considered. The coefficients of the equations are third-order polynomial matrices of simple structure. For these equations, the conditions of existence and the method of finding a solution, which is a pair of a non-singular numerical matrix and a reversible polynomial matrix, are specified. Cite as: B. Z. Shavarovskii, "Non-singular solutions of one class of matrix equations over a polynomial ring," Prykl. Probl. Mekh. Mat. , Issue 1 7 , 1 7 – 26 (201 9 ) (in Ukrainian), https://doi.org/10.15407/apmm2019.17.17-26

Sums and products of units and idempotents in duo rings

An element in a ring R is said to be clean (respectively unit-regular) if it is a sum (respectively product) of an invertible element and an idempotent. It is known that all elements in R are clean, if all of them are unit-regular. We study some classes of unit-regular and clean matrices over a duo ring. It is shown that a right adequate duo domain is almost 2-good ring. Cite as: A. I. Gatalevych, M. I. Kuchma, Non-singular solutions of one class of matrix equations over a polynomial ring, Prykl. Probl. Mekh. Mat. , Issue 17, 42–46 (2019) (in Ukrainian), https://doi.org/10.15407/apmm2019.17.42-46

An accelerated technique for solving the positive definite solutions of a class of nonlinear matrix equations

Nonlinear matrix equations (NMEs) are encountered in many applications of control and engineering problems. In this work, we establish a complete study for a class of nonlinear matrix equations. With the aid of Sherman Morrison Woodbury formula, we have shown that any equation in this class has the maximal positive definite solution under certain conditions. Furthermore, a thorough study of properties about this class of matrix equations is provided. An acceleration of iterative method with R-superlinear convergence is then designed to solve the maximal positive definite solution. Two numerical experiments demonstrate that our methods perform efficiently and reliably.

Dykstra’s Algorithm for the Optimal Approximate Symmetric Positive Semidefinite Solution of a Class of Matrix Equations

Dykstra’s alternating projection algorithm was proposed to treat the problem of finding the projection of a given point onto the intersection of some closed convex sets. In this paper, we first apply Dykstra’s alternating projection algorithm to compute the optimal approximate symmetric positive semidefinite solution of the matrix equations AXB = E, CXD = F. If we choose the initial iterative matrix X0 = 0, the least Frobenius norm symmetric positive semidefinite solution of these matrix equations is obtained. A numerical example shows that the new algorithm is feasible and effective.

The Generalized Bisymmetric (Bi-Skew-Symmetric) Solutions of a Class of Matrix Equations and Its Least Squares Problem

The solvability conditions and the general expression of the generalized bisymmetric and bi-skew-symmetric solutions of a class of matrix equations(AX=B,XC=D)are established, respectively. If the solvability conditions are not satisfied, the generalized bisymmetric and bi-skew-symmetric least squares solutions of the matrix equations are considered. In addition, two algorithms are provided to compute the generalized bisymmetric and bi-skew-symmetric least squares solutions. Numerical experiments illustrate that the results are reasonable.

Least Squares Symmetrizable Solutions for a Class of Matrix Equations

In this paper, we discuss least squares symmetrizable solutions of matrix equations (AX = B, XC = D) and its optimal approximation solution. With the matrix row stacking, Kronecker product and special relations between two linear subspaces are topological isomorphism, and we derive the general solutions of least squares problem. With the invariance of the Frobenius norm under orthogonal transformations, we obtain the unique solution of optimal approximation problem. In addition, we present an algorithm and numerical experiment to obtain the optimal approximation solution.

Optimal Approximate Solution of a Class of Matrix Equation

The optimal approximate solution of a class of matrix equation is considered. Based on the generalized singular value decomposition, the expression of the optimal approximate solution to a given matrix pair is derived in the symmetric solution set.

Solution to a Class of Matrix Equations with k-Involutary Symmetrices

In this paper, we investigate the solvability of matrix equations with -involutary symmetric matrix , the general solution of which is obtained when it is solvable. Meantime, the associated optimal approximation problem for some given matrix is also considered under some particular hypothesis.

Solution to a Class of Matrix Equations with k-Involutary Symmetrices

Solution to a Class of Matrix Equations with k-Involutary Symmetrices

Centro-Symmetric Minimal Rank Solutions and Its Optimal Approximation to a Class of Matrix Equation

Centro-Symmetric Minimal Rank Solutions and Its Optimal Approximation to a Class of Matrix Equation

Unified parametrization for the solutions to the polynomial diophantine matrix equation and the generalized Sylvester matrix equation

The polynomial Diophantine matrix equation and the generalized Sylvester matrix equation are important for controller design in frequency domain linear system theory and time domain linear system theory, respectively. By using the so-called generalized Sylvester mapping, right coprime factorization and Bezout identity associated with certain polynomial matrices, we present in this note a unified parametrization for the solutions to both of these two classes of matrix equations. Moreover, it is shown that solutions to the generalized Sylvester matrix equation can be obtained if solutions to the Diophantine matrix equation are available. The results disclose a relationship between the polynomial Diophantine matrix equation and generalized Sylvester matrix equation that are respectively studied and used in frequency domain linear system theory and time domain linear system theory.

An iterative algorithm for solving a class of matrix equations

In this paper, an iterative algorithm is presented to solve the Sylvester and Lyapunov matrix equations. By this iterative algorithm, for any initial matrix X1, a solution X* can be obtained within finite iteration steps in the absence of roundoff errors. Some examples illustrate that this algorithm is very efficient and better than that of [1] and [2].

Positive definite solutions of some matrix equations

In this paper we investigate some existence questions of positive semi-definite solutions for certain classes of matrix equations known as the generalized Lyapunov equations. We present sufficient and necessary conditions for certain equations and only sufficient for others.

Distribution of the spectrum and representation of solutions of degenerate dynamical systems

We propose algebraic methods for the investigation of the spectrum and structure of solutions of degenerate dynamical systems. These methods are based on the construction and solution of new classes of matrix equations. We prove theorems on the inertia of solutions of the matrix equations, which generalize the well-known properties of the Lyapunov equation.

Extension of the spectral-transform method for solving nonlinear evolution equations and related conservation laws

The spectral-transform method for solving nonlinear evolution equations is extended to certain classes of matrix equations with linearlyx-dependent coefficients and the existence of the conservation laws for the new class of nonlinear evolution equations is investigated. A remarkably explicit representation is, moreover, obtained for the conserved quantities of the «old» classes of nonlinear evolution equations (withx-independent coefficients).

On a class of matrix equations

On a class of matrix equations