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 o...
Given the linear matrix equation AXB=C, we partition it into the form A1X11B1+A1X12B2+A2X21B1+A2X22B2=C, and then pre- and post-multiply both sides of the equation by the four orthogonal projectors generated from the coefficient matrices A1, A1, B1,...
Nonsymmetric differential matrix Riccati equations arise in many problems related to science and engineering. This work is focusing on the sensitivity of the solution to perturbations in the matrix coefficients and the initial condition. Two approach...
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.
This paper investigates the Sylvester-transpose matrix equation A⋉X+XT⋉B=C, where all mentioned matrices are over an arbitrary field. Here, ⋉ is the semi-tensor product, which is a generalization of the usual matrix product defined...
We derive the solvability conditions and a formula of a general solution to a Sylvester-type matrix equation over Hamilton quaternions. As an application, we investigate the necessary and sufficient conditions for the solvability of the quaternion ma...
We derive a double-optimal iterative algorithm (DOIA) in an m-degree matrix pencil Krylov subspace to solve a rectangular linear matrix equation. Expressing the iterative solution in a matrix pencil and using two optimization techniques, we determine...
In this paper, we establish the solvability conditions and the formula of the general solution to a Sylvester-like quaternion matrix equation. As an application, we give some necessary and sufficient conditions for a system of quaternion matrix equat...
This survey provides a comprehensive overview of the solutions to the matrix equation AXB=C over real numbers, complex numbers, quaternions, dual quaternions, dual split quaternions, and dual generalized commutative quaternions, including various spe...
A randomized block Kaczmarz method and a randomized extended block Kaczmarz method are proposed for solving the matrix equation AXB=C, where the matrices A and B may be full-rank or rank-deficient. These methods are iterative methods without matrix m...
Dual generalized commutative quaternions have broad application prospects in many fields. Additionally, the matrix equation AXB=C has important applications in mathematics and engineering, especially in control systems, economics, computer science, a...
We introduce a gradient iterative scheme with an optimal convergent factor for solving a generalized Sylvester matrix equation ∑i=1pAiXBi=F, where Ai,Bi and F are conformable rectangular matrices. The iterative scheme is derived from the gradient...
In this paper, we explore the least-norm solution to the classical matrix equation AXB=C over the dual quaternion algebra, where A, B, and C are given matrices, while X remains the unknown matrix. We begin by transforming the definition of the Froben...
In this paper, we study the bisymmetric and skew bisymmetric solutions of quaternion generalized Lyapunov equation. With the help of semi-tensor product of matrices, some new conclusions on the expansion rules of row and column of matrix product on q...
This paper investigates new solution sets for the Yang–Baxter-like (YB-like) matrix equation involving constant entries or rational functional entries over complex numbers. Towards this aim, first, we introduce and characterize an essential cla...
In this paper, we propose a new iterative algorithm to solve the matrix equation AXB + CXT D = E. The algorithm can obtain the minimal Frobenius norm solution or the least-squares solution with minimal Frobenius norm. Our algorithm is better than Alg...
This paper proposes a class of randomized Kaczmarz and Gauss–Seidel-type methods for solving the matrix equation AXB=C, where the matrices A and B may be either full-rank or rank deficient and the system may be consistent or inconsistent. These...
We propose a new iterative method for solving a generalized Sylvester matrix equation A1XA2+A3XA4=E with given square matrices A1,A2,A3,A4 and an unknown rectangular matrix X. The method aims to construct a sequence of approximated solutions convergi...
In this paper, we transform the problem of solving the Sylvester matrix equation into an optimization problem through the Kronecker product primarily. We utilize the adaptive accelerated proximal gradient and Newton accelerated proximal gradient meth...
The Poisson equation frequently emerges in many fields of science and engineering. As exact solutions are rarely possible, numerical approaches are of great interest. Despite this, a succinct discussion of a systematic approach to constructing a flex...
In this short communication, an algorithm for efficiently solving a sparse matrix equation, which arises frequently in the field of distributed control and estimation theory, is proposed. The efficient algorithm stems from the fact that the sparse eq...
(R,S)-(skew) symmetric matrices have numerous applications in civil engineering, information theory, numerical analysis, etc. In this paper, we deal with the (R,S)-(skew) symmetric solutions to the quaternion matrix equation AXB=C. We use a real repr...
In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation AXB = C, and present the general solution expression when the solvability conditions are met. As an application, we de...
We derive a conjugate-gradient type algorithm to produce approximate least-squares (LS) solutions for an inconsistent generalized Sylvester-transpose matrix equation. The algorithm is always applicable for any given initial matrix and will arrive at...
Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative...
In this paper, we introduce the matrix Mittag–Leffler function, which is a generalization of the multivariate Mittag–Leffler function, in order to investigate the uniqueness of the solutions to a fractional nonlinear partial integro-diffe...
In this work, we introduce a mixed type of quadratic-additive (QA) functional equation and obtain its general solution. The objective of this work is to investigate the Ulam–Hyers stability of this quadratic-additive (QA) functional equation in...
Dual algebra plays an important role in kinematic synthesis and dynamic analysis, but there are still few studies on dual quaternion matrix theory. This paper provides an efficient method for solving the QLY least squares problem of the dual quaterni...
It is known that the excitations in graphene-like materials in external electromagnetic field are described by solutions of a massless two-dimensional Dirac equation which includes both Hermitian off-diagonal matrix and scalar potentials. Up to now,...
In this paper, we study the nonlinear matrix equation (NME) X+∑i=1mAi*X−1Ai=Q. We transform this equation into an equivalent zero-point equation, then we use Newton’s iteration method to solve the equivalent equation. Under some mild...
This research introduces three novel zeroing neural network (ZNN) models for addressing the time-varying Yang–Baxter-like matrix equation (TV-YBLME) with arbitrary (regular or singular) real time-varying (TV) input matrices in continuous time....
The goal of this article is to investigate a new solver in the form of an iterative method to solve X+A∗X−1A=I as an important nonlinear matrix equation (NME), where A,X,I are appropriate matrices. The minimal and maximal solutions of th...
This paper introduces a digital adaptive control framework for large-scale multivariable systems, integrating matrix linear Diophantine equations with block pole placement. The main innovation lies in adaptively relocating the full eigenstructure usi...
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-polyn...
In this paper, we present a stable and accurate high-order methodology for the symmetric matrix form (SMF) of the elastic wave equation. We use an accurate high-order upwind finite difference method to define spatial discretization. Then, an efficien...
Several types of linear and nonlinear singularly perturbed time-delay differential systems are considered. Asymptotic stability of the linear systems and asymptotic stability of the trivial solution of the nonlinear systems, valid for any sufficientl...
A fuzzified matrix space consists of a collection of matrices with a fuzzy structure, modeling the cases of uncertainty on the part of values of different matrices, including the uncertainty of the very existence of matrices with the given values. Th...
Thin-film organic solar cell (OSC) performances have been investigated in detail by improved analytical computation in this work. The generation of excitons inside OSC has been estimated by using the optical transfer matrix method (OTMM) to include t...
In this paper, we investigate the iterative methods for the solution of different types of nonlinear matrix equations. More specifically, we consider iterative methods for the minimal nonnegative solution of a set of Riccati equations, a nonnegative...
We consider when the quaternion matrix equation AXB+CXD=E has a reflexive (or anti-reflexive) solution with respect to a given generalized reflection matrix. We adopt a real representation method to derive the solutions when it is solvable. Moreover,...
This article makes use of simultaneous decomposition of four quaternion matrixes to investigate some Sylvester-like quaternion matrix equation systems. We present some useful necessary and sufficient conditions for the consistency of the system of qu...
Using the ranks and Moore-Penrose inverses of involved matrices, in this paper we establish some necessary and sufficient solvability conditions for a system of Sylvester-type quaternion matrix equations, and give an expression of the general solutio...
We propose a heuristic method to solve polynomial matrix equations of the type ∑k=1makXk=B, where ak are scalar coefficients and X and B are square matrices of order n. The method is based on the decomposition of the B matrix as a linear combinat...
This paper considers the Hermitian solutions of a new system of commutative quaternion matrix equations, where we establish both necessary and sufficient conditions for the existence of solutions. Furthermore, we derive an explicit general expression...
In this work, we focus on analyzing the location and separation of the solutions of the simplest quadratic matrix equation. For this, we use the qualitative properties that we can deduce of the study of the convergence of iterative processes. This st...
Sylvester Matrix Equations (SME) play a central role in applied mathematics, particularly in systems and control theory. A fuzzy theory is normally applied to represent the uncertainty of real problems where the classical SME is extended to Fully Fuz...
This paper investigates the existence and convergence of solutions for linear and nonlinear matrix equations. This study explores the potential of convex (α,β)-generalized contraction mappings in geodesic spaces, ensuring the existence of...
In the era of big data and multimedia communication, securing color images against unauthorized access and attacks is a pressing challenge. While quaternion-based models provide a unified representation for color images, most existing encryption sche...
In this paper, we study the necessary and sufficient conditions for a system of matrix equations to have a solution and a Hermitian solution. As an application, we establish the necessary and sufficient conditions for a classical matrix system to hav...
This work introduces a unified framework for analyzing linear delay differential Sylvester matrix equations with noncommuting coefficients. The methodology employs a Kronecker product-based vectorization to transform the system, yielding explicit clo...
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