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We provided a generalization of the existence and uniqueness of fixed points in partially ordered metric spaces for a monotone map. We applied the major results in the investigation of coupled fixed points for ordered pairs of two maps that met various monotone features, which included a mixed monotone property or a total monotone property. To ascertain necessary requirements for the existence and uniqueness of solutions to systems of matrix equations, the results regarding coupled fixed points for ordered pairs of maps were utilized. These results are illustrated with numerical examples. Some of the known results are a consequence of the results we obtained.

AbstractLet A be an m × n complex matrix and let B, C be n × m complex matrices. The main purpose of this article is to derive several new properties of the (B, C)-core inverse. Also, we introduce and investigate the dual (B, C)-core inverse in the context of rectangular matrices. The connection between (dual) (B, C)-core inverses and other generalized inverses is given. Finally, the minimum norm solution and least square solution to the equation are expressed by using the (B, C)-core inverse. [ABSTRACT FROM AUTHOR]

We provided a generalization of the existence and uniqueness of fixed points in partially ordered metric spaces for a monotone map. We applied the major results in the investigation of coupled fixed points for ordered pairs of two maps that met various monotone features, which included a mixed monotone property or a total monotone property. To ascertain necessary requirements for the existence and uniqueness of solutions to systems of matrix equations, the results regarding coupled fixed points for ordered pairs of maps were utilized. These results are illustrated with numerical examples. Some of the known results are a consequence of the results we obtained. [ABSTRACT FROM AUTHOR]

This paper mainly studies the least-squares solutions of matrix equation $ AXB = C $ under a semi-tensor product. According to the definition of the semi-tensor product, the equation is transformed into an ordinary matrix equation. Then, the least-squares solutions of matrix-vector and matrix equations respectively investigated by applying the derivation of matrix operations. Finally, the specific form of the least-squares solutions is given.

This paper mainly studies the least-squares solutions of matrix equation A X B = C under a semi-tensor product. According to the definition of the semi-tensor product, the equation is transformed into an ordinary matrix equation. Then, the least-squares solutions of matrix-vector and matrix equations respectively investigated by applying the derivation of matrix operations. Finally, the specific form of the least-squares solutions is given. [ABSTRACT FROM AUTHOR]

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 when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples. [ABSTRACT FROM AUTHOR]

It is given a characterization of being a matrix Q g (a 3 , b 3) (k) of linear combination of a matrix Q g (a 1 , b 1) (n) and a matrix Q g (a 2 , b 2) (m) , where a i , b i ∈ R ∗ , i = 1 , 2 , 3 , m , n , k ∈ Z , and Q g (a , b) (l) denotes an (a, b)-generalized Fibonacci Q-matrix with l ∈ Z . In addition, some examples are presented illustrating the main result. Finally, some applications of the main result obtained are given. [ABSTRACT FROM AUTHOR]

In this paper, we study a matrix equation involving the matrix geometric mean . We study several properties and inequalities for the unique solution of such an equation. We also use the weighted geometric mean to define a new quantum Hellinger divergence and show that the new quantum divergence satisfies the Data Processing Inequality. [ABSTRACT FROM AUTHOR]

The decomposition or approximation of a linear operator on a matrix space as a sum of Kronecker products plays an important role in matrix equations and low-rank modeling. The approximation problem in Frobenius norm admits a well-known solution via the singular value decomposition. However, the approximation problem in spectral norm, which is more natural for linear operators, is much more challenging. In particular, the Frobenius norm solution can be far from optimal in spectral norm. We describe an alternating optimization method based on semidefinite programming to obtain high-quality approximations in spectral norm, and we present computational experiments to illustrate the advantages of our approach. [ABSTRACT FROM AUTHOR]

Linear coupled matrix equations are implemented in various domains, including stability analysis of control systems and robust control. The solvability requirements and the solution of the linear coupled system were established in this study. Additionally, as applications of the system, we investigate the linear coupled quaternion system to be compatible and deduce the solvability criteria for matrix equations, including Hermicity. An algorithm and an example are given to illustrate the main results. [ABSTRACT FROM AUTHOR]

Matrix equations play an important place in some areas of sciences. Intensive studies aiming to solve some matrix equations and to explore some applications of these studies have been carried out recently. The contribution of the present paper falls within this framework. We investigate here an algorithm that converges to the positive solution of Σmj=1 Am-jXAj-1 = B, when A and B are two positive matrices. Application for some standard matrix equations, like Sylvester equation, is provided as well. Numerical examples showing the interest of this work and illustrating the theoretical study are also discussed. [ABSTRACT FROM AUTHOR]

Fuzzy numbers have many applications in various mathematical models in both linear and nonlinear forms. In the form of a nonlinear system, fuzzy nonlinear equations can be constructed in the form of matrix equations. Unfortunately, the matrix equations used to solve the problem of dual fully fuzzy nonlinear systems are still relatively few found in the publication of research results. This article attempts to solve a dual fully fuzzy nonlinear equation system involving triangular fuzzy numbers using Broyden’s method. This article provides the pseudocode algorithm and the implementation of the algorithm into the MATLAB program for the iteration process to be carried out quickly and easily. The performance of the given algorithm is the fastest in finding system solutions and provides a minimum error value.

In this article, we discuss the relation-theoretic aspects of weakly contractive mappings to prove fixed point results in the setting of metric spaces endowed with a certain binary relation. We also provide an example and an application to validate of our results. The results proved herewith unify, generalize, improve, extend, sharpen, subsume and enrich some well-known fixed point theorems of the existing literature. [ABSTRACT FROM AUTHOR]

Given the matrix equation A X + X B + f (X) C = D in the unknown n × m matrix X , we analyze existence and uniqueness conditions, together with computational solution strategies for f : R n × m → R being a linear or nonlinear function. We characterize different properties of the matrix equation and of its solution, depending on the considered classes of functions f. Our analysis mainly concerns small dimensional problems, though several considerations also apply to large scale matrix equations. [ABSTRACT FROM AUTHOR]

This paper solves two matrix equations problems completely which are originated from the problems of state feedback stabilization of first‐order and second‐order switched systems, respectively. Two parametric solutions are proposed for the both two problems. Comparison for the advantages and disadvantages between the two methods is presented from an unified point of view. It is disclosed that principle of each uncontrollable pole will lead to a linear relationship between the rows of solution matrix V. Examples show the proposed methods are convenient calculation. [ABSTRACT FROM AUTHOR]

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 when it is solvable. In addition, we also provide the least squares Hermitian solution in cases where the system of matrix equations is not consistent. To illustrate our main findings, in this paper we present two numerical algorithms and examples.

We find an algorithmic procedure that enables the computation and description of the structure of the isotropy subgroups of the group of complex orthogonal matrices with respect to the action of similarity on complex symmetric matrices. A key step in our proof is to solve a certain rectangular block upper triangular Toeplitz matrix equation. [ABSTRACT FROM AUTHOR]

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials. We also find a generating set for the vanishing ideal of these orbits. [ABSTRACT FROM AUTHOR]

We provide a new, concise derivation of necessary and sufficient conditions for a real matrix-normally distributed matrix X and we characterize the general matrix-normal covariance structure such that, given the symmetric positive-semidefinite matrices Ai, i = 1,2,...,m, the matrix quadratic forms X ′ A i X are distributed as independent noncentral Wishart matrices. [ABSTRACT FROM AUTHOR]

In this paper we present the closed-form expression for the solution of the linear matrix difference equation Xn+1 = AXn+XnB, X(0) = C, n=0,1,2,... which we obtained with the aid of the Kronecker sum, Kronecker product and Vec operator. [ABSTRACT FROM AUTHOR]

Abstract The goal of this manuscript is to establish quadruple fixed-point and coincidence-point consequences in the setting of generalized metric spaces equipped with vector-valued metrics and matrix equations. Moreover, some supportive examples and corollaries are presented here to support the theoretical results. Ultimately, the theoretical results are presented here to discuss some applications to support our study.

In this paper, the new type of Gradient Neural Network (GNN) model is proposed for the following linear system of matrix equations: A X = C , X B = D. The convergence analysis of given models is shown. The model is applied for the computation of the regular matrix inverse, as well as Moore-Penrose and Drazin generalized inverses. Some illustrative examples and simulations are given to verify theoretical results. [ABSTRACT FROM AUTHOR]

This paper investigates the properties of the ϕ -skew-Hermitian solution to the system of quaternion matrix equations involving ϕ -skew-Hermicity with four unknowns A i X i (A i) ϕ + B i X i + 1 (B i) ϕ = C i , (i = 1 , 2 , 3) , A 4 X 4 (A 4) ϕ = C 4 . We present the general ϕ -skew-Hermitian solution to this system. Moreover, we derive the β (ϕ) -signature bounds of the ϕ -skew-Hermitian solution X 1 in terms of the coefficient matrices. We also give some necessary and sufficient conditions for the system to have β (ϕ)-positive semidefinite, β (ϕ)-positive definite, β (ϕ)-negative semidefinite and β (ϕ)-negative definite solutions. [ABSTRACT FROM AUTHOR]

In uncertainty quantification, it is commonly required to solve a forward model consisting of a partial differential equation (PDE) with a spatially varying uncertain coefficient that is represented as an affine function of a set of random variables, or parameters. Discretizing such models using stochastic Galerkin finite element methods (SGFEMs) leads to very high-dimensional discrete problems that can be cast as linear multi-term matrix equations (LMTMEs). We develop efficient computational methods for approximating solutions of such matrix equations in low rank. To do this, we follow an alternating energy minimization (AEM) framework, wherein the solution is represented as a product of two matrices, and approximations to each component are sought by solving certain minimization problems repeatedly. Inspired by proper generalized decomposition methods, the iterative solution algorithms we present are based on a rank-adaptive variant of AEM methods that successively computes a rank-one solution component at each step. We introduce and evaluate new enhancement procedures to improve the accuracy of the approximations these algorithms deliver. The efficiency and accuracy of the enhanced AEM methods is demonstrated through numerical experiments with LMTMEs associated with SGFEM discretizations of parameterized linear elliptic PDEs. [ABSTRACT FROM AUTHOR]

The paper investigates the class of square matrices which have idempotent Moore–Penrose inverse. A number of original characteristics of the class are derived and new properties identified. Additionally, some isolated results scattered in the literature are recaptured and rederived in an extended or generalized form. The research was inspired by some matrix equations (occurring, e.g. in physics) which either directly involve matrices with an idempotent Moore–Penrose inverse or whose solvability under the assumption that the matrices involved have an idempotent Moore–Penrose inverse provides an emerging insight into the corresponding theory. [ABSTRACT FROM AUTHOR]

In this paper, we construct a parametric family of iterative methods to compute the inverse of a nonsingular matrix. This class is free of inverse operators. We prove the third-order of convergence under some conditions involving the parameter of the family. Moreover, a dynamical analysis is made for the first time to a matrix iterative method, finding intervals of stability, that include but are wider than those found in the convergence analysis. Numerical tests on large random matrices confirm the results found. [ABSTRACT FROM AUTHOR]

In this article, we discuss the relation-theoretic aspects of weakly contractive mappings to prove fixed point results in the setting of metric spaces endowed with a certain binary relation. We also provide an example and an application to validate of our results. The results proved herewith unify, generalize, improve, extend, sharpen, subsume and enrich some well-known fixed point theorems of the existing literature.

An efficient strategy for the numerical solution of time-fractional diffusion-reaction problems is devised. A standard finite difference discretization of the space derivative is initially applied which results in a linear stiff term. Then a rectangular product-integration (PI) rule is implemented in an implicit-explicit (IMEX) framework in order to implicitly treat this linear stiff term and handle in an explicit way the non-linear, and usually non-stiff, term. The kernel compression scheme (KCS) is successively adopted to reduce the overload of computation and storage need for the persistent memory term. To reduce the computational effort the semidiscretized problem is described in a matrix-form, so as to require the solution of Sylvester equations only with small matrices. Theoretical results on the accuracy, together with strategies for the optimal selection of the main parameters of the whole method, are derived and verified by means of numerical experiments carried out in two-dimensional domains. The computational advantages with respect to other approaches are also shown and some applications to the detection of pattern formation are illustrated at the end of the paper. [ABSTRACT FROM AUTHOR]

Let H be the real quaternion algebra and H m × n denote the set of all m × n matrices over H. For A ∈ H m × n , we denote by A ϕ the n × m matrix obtained by applying ϕ entrywise to the transposed matrix A T , where ϕ is a non-standard involution of H. A ∈ H n × n is said to be ϕ -skew-Hermicity if A = − A ϕ . In this paper, we provide some necessary and sufficient conditions for the existence of a ϕ -skew-Hermitian solution to the system of quaternion matrix equations with four unknowns A i X i (A i) ϕ + B i X i + 1 (B i) ϕ = C i , (i = 1 , 2 , 3) , A 4 X 4 (A 4) ϕ = C 4 . [ABSTRACT FROM AUTHOR]

In this paper, we present a new algorithm Global-DGMRES to find the Drazin inverse solution of the linear matrix equation AXB = C, where at least one of the matrices A or B is rank deficient. This method is based on oblique projection process, onto matrix Krylov subspaces. Also, we study convergence properties of this algorithm. The matrix equation AXB = C is a mathematical model for deblurring problems and the Global-DGMRES method helps us to reconstruct blurred images corresponding to this matrix equation. Moreover, by numerical results, we compare the proposed method with Global-LSMR and Global-LSQR. [ABSTRACT FROM AUTHOR]

The goal of this manuscript is to establish quadruple fixed-point and coincidence-point consequences in the setting of generalized metric spaces equipped with vector-valued metrics and matrix equations. Moreover, some supportive examples and corollaries are presented here to support the theoretical results. Ultimately, the theoretical results are presented here to discuss some applications to support our study. [ABSTRACT FROM AUTHOR]

The current study investigates the solvability conditions and the general solution of three symmetrical systems of coupled Sylvester-like quaternion matrix equations. Accordingly, the necessary and sufficient conditions for the consistency of these systems are determined, and the general solutions of the systems are thereby deduced. An algorithm and a numerical example are constructed over the quaternions to validate the results of this paper. [ABSTRACT FROM AUTHOR]

We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields. We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials. We also find a generating set for the vanishing ideal of these orbits. [ABSTRACT FROM AUTHOR]

Consider an ordered Banach space and f,gf,g two self-operators defined on the interior of its positive cone. In this article, we prove that the equation f(X)=g(X)f(X)=g(X) has a positive solution, whenever f is strictly α\alpha -concave g-monotone or strictly (−α)(-\alpha )-convex g-antitone with g super-homogeneous and surjective. As applications, we show the existence of positive definite solutions to new classes of nonlinear matrix equations.

In this paper, we introduce the concept of Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction by unifying the definitions of Suzuki type $\mathcal{Z}$-contraction and $\mathcal{Z_\mathcal{R}}$-contraction and also provide examples to highlight the genuineness of our newly introduced contraction over earlier mentioned ones. Chiefly, we prove an existence and corresponding uniqueness fixed point results for Suzuki type $\mathcal{Z_\mathcal{R}}$-contraction employing an amorphous binary relation on metric spaces without completeness and also furnish an illustrative example to demonstrate the utility of our main results. Finally, we utilize our main results to discuss the existence and uniqueness of solutions of a family of nonlinear matrix equations.

The standard power-mismatch Newton method is still frequently used for computing load flow due to its simplicity and generality. In this paper, a matrix-based generalization for the usual power flow equations to an arbitrary number of phases is derived. The proposed equations enable computing power injections and the Jacobian matrix in terms of submatrices that compose the network admittance matrix. Besides the more compact representation, another advantage of the proposed generalization is execution time reduction compared to the standard scalar formulation. Simulations are carried out to demonstrate the time reduction achieved via the proposed equations.

This paper investigates the properties of the ϕ-skew-Hermitian solution to the system of quaternion matrix equations involving ϕ-skew-Hermicity with four unknowns AiXi(Ai)ϕ+BiXi+1(Bi)ϕ=Ci,(i=1,2,3),A4X4(A4)ϕ=C4. We present the general ϕ-skew-Hermitian solution to this system. Moreover, we derive the β(ϕ)-signature bounds of the ϕ-skew-Hermitian solution X1 in terms of the coefficient matrices. We also give some necessary and sufficient conditions for the system to have β(ϕ)-positive semidefinite, β(ϕ)-positive definite, β(ϕ)-negative semidefinite and β(ϕ)-negative definite solutions.

We explore algebraic strategies for numerically solving linear elliptic partial differential equations in polygonal domains. To discretize the polygon by means of structured meshes, we employ Schwarz–Christoffel conformal mappings, leading to a multiterm linear equation possibly including Hadamard products of some of the terms. This new algebraic formulation allows us to clearly distinguish between the role of the discretized operators and that of the domain meshing. Various algebraic strategies are discussed for the solution of the resulting matrix equation. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating some system of mixed coupled generalized Sylvester operator equations. The block operator matrix decomposition is used to find the necessary and sufficient conditions for the solvability to these systems. The solutions of the system are expressed in terms of the Moore{Penrose inverses of the coefficient operators. [ABSTRACT FROM AUTHOR]

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We show that the discrete operator stemming from time-space discretization of evolutionary partial differential equations can be represented in terms of a single Sylvester matrix equation. A novel solution strategy that combines projection techniques with the full exploitation of the entry-wise structure of the involved coefficient matrices is proposed. The resulting scheme is able to efficiently solve problems with a tremendous number of degrees of freedom while maintaining a low storage demand as illustrated in several numerical examples. [ABSTRACT FROM AUTHOR]

We analyze the solutions of one-sided matrix equations over algebraically closed fields of characteristic zero in a certain transcendental extension of the main field that are generated by a set of invertible matrices. We establish conditions under which the roots corresponding to different invertible matrices from this set coincide. [ABSTRACT FROM AUTHOR]