Matrix Equation AX Research Articles

Kaczmarz-type methods for solving matrix equations

In this paper, several Kaczmarz-type numerical methods for solving the matrix equations AX = B and XA = C are proposed, where the coefficient matrix A may be full rank or deficient rank. These methods are iterative methods without matrix multiplication. Theoretically, the convergence of these methods is proved. The numerical results show that these methods are more efficient than iterative methods involving matrix multiplication for high-dimensional matrices.

Matrix Inverse Problemunder the Central Master Submatrix Constraint AX = B of the Generalized Centersymmetric Solution

This paper begins by identifying a problem related to the solvability of the matrix equation AX=B, particularly within the realm of generalized centrosymmetric matrices. It exploits the distinctive properties inherent in generalized centrosymmetric matrices, primary focus involves investigating their solutions within the context of the matrix equation AX=B. This paper establishes the necessary and sufficient conditions for solvability while formulating a comprehensive expression for the generalized centrosymmetric solution. Furthermore, the paper delves into addressing the associated optimal approximation quandary concerning a specified matrix within the solution set.

RANKS OF SUBMATRICES IN THE REFLEXIVE SOLUTIONS OF SOME MATRIX EQUATIONS

Maximal and minimal ranks of the two submatrices X₁ and X₂ in the (skew-) Hermitian reflexive solution X=U[X₁ 00 X₂]U^{∗} of the matrix equation AXA^{∗}=C, in the reflexive solution of the matrix equation AXB=C are derived. Then necessary and sufficient conditions for these reflexive solutions to have special forms, and the general expressions of these reflexive solutions are achieved.

A General Approach to Sylvester-Polynomial-Conjugate Matrix Equations

Sylvester-polynomial-conjugate matrix equations unify many well-known versions and generalizations of the Sylvester matrix equation AX−XB=C which have a wide range of applications. In this paper, we present a general approach to Sylvester-polynomial-conjugate matrix equations via groupoids, vector spaces, and matrices over skew polynomial rings. The obtained results are applied to Sylvester-polynomial-conjugate matrix equations over complex numbers and quaternions. The main role in our approach is played by skew polynomial rings, which are well-known tools in algebra to provide examples of asymmetry between left-sided and right-sided versions of many ring objects.

General strong fuzzy solutions of fuzzy Sylvester matrix equations involving the BT inverse

This paper aims to solve the fuzzy Sylvester matrix equation (FSME) AX˜+X˜B=C˜, in which A, B are n×n and m×m crisp matrices, and C˜ is a n×m fuzzy matrix. Using the Kronecker product, we transform the FSME into an mn×mn fuzzy linear system. Firstly, we obtain the necessary and sufficient conditions for the existence of strong fuzzy solutions to the FSME by using the BT inverse of the coefficient matrix. Secondly, we investigate the existence of a nonnegative BT inverse by using its block structure. Next, we derive general strong fuzzy solutions to a class of FSME and establish an algorithm to obtain general strong fuzzy solutions to the FSME by the BT inverse. Finally, we give three numerical examples and an applied example in the economy to illustrate the proposed methods.

The solutions to some dual matrix equations

The solvability conditions for the dual matrix equation AXB = D, a pair of dual matrix equations AX = B,XC = D and the dual matrix equation AX + Y B = D are established by using the generalized inverses and the singular value decompositions of some real matrices, and the expressions of the general solutions to these dual matrix equations are presented. Finally, a numerical experiment is given to validate the accuracy of our result.

The Fučík spectrum for discrete systems and some nonlinear existence theorems

In this paper, we study the existence solutions to nonlinear Fučík problems of the form(1)Ax=αx+−βx−+g(x), where A is a symmetric n×n matrix, α,β are real numbers, and g:Rn→Rn is continuous. The nonlinear problem (1) is motivated by application to nonlinear oscillating systems such as the Tacoma Narrows Bridge.The paper begins by developing a qualitative picture of Fučík spectrum associated with the matrix equationAx=αx+−βx−. In this setting, we present two characterizations: first, we show that under appropriate assumptions the Fučík spectrum consists of curves bifurcating from points (λ,λ)∈R2, where λ is an eigenvalue of A; second, we give more global variational characterization of the Fučík curves. In both cases, we present various qualitative properties of the Fučík curves. The paper finishes by presenting two existence theorems for the nonlinear Fučík problem under mild assumptions on the nonlinear term g.

Constructing solutions of the Yang–Baxter-like matrix equation for singular matrices

Let A=UVH be an n×n complex singular matrix of rank r, where U, V are two n×r matrices of full column rank. Some sufficient and necessary conditions are derived for X being a nontrivial (commuting) solution of the quadratic matrix equation AXA=XAX such that we can construct infinitely many (commuting) solutions of the matrix equation. When VHU is nonsingular, all the solutions of the matrix equation are characterized and can be found and constructed by solving smaller matrix equations. In particular, all the commuting solutions can be obtained by finding the projections that commute with VHU. The method of constructing the solutions is free of computations of Jordan canonical forms and matrix inverses.

A new image restoration model associated with special elliptic quaternionic least-squares solutions based on LabVIEW

In this paper, we take advantage of the elliptic complex matrix representation of elliptic quaternion matrices. Then we obtain the methods of the elliptic quaternionic least-squares solution, the pure elliptic quaternionic least-squares solution, and the real least-squares solution with the least norm of the elliptic quaternion matrix equation AX=B. We also apply the newly obtained method of the pure elliptic quaternionic least-squares solution with the least norm to the color image restoration based on the LabVIEW program. In this context, we propose a new image restoration model called “ELSI image restoration model” associated with elliptic quaternionic least-squares solutions.

Solution Set of the Yang-Baxter-like Matrix Equation for an Idempotent Matrix

Given a complex idempotent matrix A, we derive simple, sufficient and necessary conditions for a matrix X being a nontrivial solution of the Yang-Baxter-like matrix equation AXA = XAX, discriminating commuting solutions from non-commuting ones. On this basis, we construct all the commuting solutions of the nonlinear matrix equation.

Numerical strategies for recursive least squares solutions to the matrix equation AX = B

The recursive solution to the Procrustes problem -with or without constraints- is thoroughly investigated. Given known matrices and , the proposed solution minimizes the square of the Frobenius norm of the difference when rows or columns are added to and . The proposed method is based on efficient strategies which reduce the computational cost by utilizing previous computations when new data are acquired. This is particularly useful in the iterative solution of an unbalanced orthogonal Procrustes problem. The results show that the computational efficiency of the proposed recursive algorithms is more significant when the dimensions of the matrices are large. This demonstrates the usefulness of the proposed algorithms in the presence of high-dimensional data sets. The practicality of the new method is demonstrated through an application in machine learning, namely feature extraction for image processing.

Developing Kaczmarz method for solving Sylvester matrix equations

The Kaczmarz method is one of the most simple and computationally efficient iterative methods for solving large-scale linear algebraic system of equations Ax=b, which at each iteration of it, only one row of coefficient matrix need to be known. The aim of this paper is to derive the matrix form of Kaczmarz method for finding the solution of Sylvester matrix equation. First, we present an iterative algorithm for finding the solution of the matrix equation AX=F by extending the Kaczmarz method and obtain the convergence of it for any initial matrix. Then, by applying the hierarchical approach, we obtain a new iterative algorithm for solving the Sylvester matrix equation AX+XB=C. Finally, three numerical examples are provided to illustrate the effectiveness of the proposed algorithm and compare with some similar gradient-based methods.

Finding Solutions to the Yang–Baxter-like Matrix Equation for Diagonalizable Coefficient Matrix

Let A be a diagonalizable complex matrix. In this paper, we discuss finding solutions to the Yang–Baxter-like matrix equation AXA=XAX. We then present a concrete example to illustrate the validity of the results obtained.

Manifold expressions of all solutions of the Yang–Baxter-like matrix equation for rank-one matrices

Let A be a complex rank-one matrix, we derive simple sufficient and necessary conditions for a complex matrix X being a (commuting, non-commuting) solution of the quadratic matrix equation AXA=XAX. On the basis, we construct the manifolds, each of which is a product of one free parameter matrix and another one to two matrices fixed by A, to express concisely all the (commuting) solutions of the matrix equation.

Solutions of the Matrix Equation AX + YB = C with Triangular Coefficients

Solutions of the Matrix Equation AX + YB = C with Triangular Coefficients

The solution of the matrix equation [formula omitted] and the system of matrix equations [formula omitted] with [formula omitted

In this paper, the solvability conditions for the matrix equation AXB=D and a pair of matrix equations AX=C,XB=D with the constraint X*X=Ip are deduced by applying the spectral and singular value decompositions of matrices, and the expressions of the general solutions to these matrix equations are also provided. Furthermore, the associated optimal approximate problems to the given matrices are discussed and the optimal approximate solutions are derived. Finally, two numerical experiments are given to validate the accuracy of the results.

Solving Complex Fuzzy Linear Matrix Equations

In this paper, a kind of complex fuzzy linear matrix equation A X ˜ B = C ˜ , in which C ˜ is a complex fuzzy matrix and A and B are crisp matrices, is investigated by using a matrix method. The complex fuzzy matrix equation is extended into a crisp system of matrix equations by means of arithmetic operations of fuzzy numbers. Two brand new and simplified procedures for solving the original fuzzy equation are proposed and the correspondingly sufficient condition for strong fuzzy solution are analysed. Some examples are calculated in detail to illustrate our proposed method.

A global variant of the COCR method for the complex symmetric Sylvester matrix equation AX + XB = C

Complex symmetric Sylvester matrix equations appear in many applications, such as the numerical solution of the complex Helmholtz equations. In this paper, by designing a global complex symmetric M-Lanczos process we develop a global variant of the conjugate A-orthogonal conjugate residual method (Gl-COCR) for solving the Sylvester matrix equation AX+XB=C with complex symmetric coefficient matrices. To obtain the smooth and monotone convergence behavior, we also propose a smoothed Gl-COCR method, denoted by SGl-COCR. Finally, numerical examples are given to illustrate the performances of our methods.

The Sherman–Morrison–Woodbury formula for generalized linear matrix equations and applications

AbstractWe discuss the use of a matrix‐oriented approach for numerically solving the dense matrix equation AX + XAT + M1XN1 + … + MℓXNℓ = F, with ℓ ≥ 1, and Mi, Ni, i = 1, … , ℓ of low rank. The approach relies on the Sherman–Morrison–Woodbury formula formally defined in the vectorized form of the problem, but applied in the matrix setting. This allows one to solve medium size dense problems with computational costs and memory requirements dramatically lower than with a Kronecker formulation. Application problems leading to medium size equations of this form are illustrated and the performance of the matrix‐oriented method is reported. The application of the procedure as the core step in the solution of the large‐scale problem is also shown. In addition, a new explicit method for linear tensor equations is proposed, that uses the discussed matrix equation procedure as a key building block.

Standard form of Matrices Over Quadratic Rings with Respect to the (z,k)-Equivalence and the Structure of Solutions of Bilateral Matrix Linear Equations

We introduce the notion of (z,k)-equivalence of matrices over quadratic rings. The established standard form of matrices with respect to the (z,k)-equivalence is used to describe the structure of solutions of the matrix equation AX + YB = C over Euclidean quadratic rings. We prove the existence of solutions with minimum Euclidean norm and show that the analyzed equation has finitely many solutions of this kind over Euclidean imaginary quadratic rings.