Related papers: Toward Solution of Matrix Equation X=Af(X)B+C

We consider smoothing equations of the form $$X ~\stackrel{\mathrm{law}}{=}~ \sum_{j \geq 1} T_j X_j + C$$ where $(C,T_1,T_2,\ldots)$ is a given sequence of random variables and $X_1,X_2,\ldots$ are independent copies of $X$ and independent…

A novel approach for solving the general absolute value equation $Ax+B|x| = c$ where $A,B\in \mathbb{R}^{m\times n}$ and $c\in \mathbb{R}^m$ is presented. We reformulate the equation as a feasibility problem which we solve via the method of…

Nonlinear matrix equations arise in many practical contexts related to control theory, dynamical programming and finite element methods for solving some partial differential equations. In most of these applications, it is needed to compute…

In this note, we establish a new closed formula for the solution of homogeneous second-order linear difference equations with constant coefficients by using matrix theory. This, in turn, gives new closed formulas concerning all sequences of…

We show the closed-form solution to the maximization of trace(A'R), where A is given and R is unknown rotation matrix. This problem occurs in many computer vision tasks involving optimal rotation matrix estimation. The solution has been…

The matrix equation $XA + AX^T = 0$, which has relevance to the study of Lie algebras, was recently studied by De Teran and Dopico. They reduced the study of this equation to several special cases and produced explicit solutions in most…

We show that the global minimum solution of $\lVert A - BXC \rVert$ can be found in closed-form with singular value decompositions and generalized singular value decompositions for a variety of constraints on $X$ involving rank, norm,…

We derive several explicit formulae for finding infinitely many solutions of the equation $AXA=XAX$, when $A$ is singular. We start by splitting the equation into a couple of linear matrix equations and then show how the projectors…

We derive necessary and sufficient conditions for the existence of the exact solution to the Sylvester-type quaternion tensor system $ \mathcal{A}_i\ast_{N}\mathcal{X}_i+ \mathcal{Y}_i\ast_{M}\mathcal{B}_i+\mathcal{C}_i\ast_{N}…

In this article, we establish a class of new accelerated modulus-based iteration methods for solving the linear complementarity problem. When the system matrix is an $H_+$-matrix, we present appropriate criteria for the convergence…

Nonlinear matrix equations play a crucial role in science and engineering problems. However, solutions of nonlinear matrix equations cannot, in general, be given analytically. One standard way of solving nonlinear matrix equations is to…

In this paper, we consider the solvability problems for the fully coupled forward-backward stochastic difference equations (FBS{\Delta}Es) on spaces related to discrete time, finite state processes. On one hand, we provide the necessary and…

We present a symbolic-numeric method to refine an approximate isolated singular solution $\hat{\mathbf{x}}=(\hat{x}_{1}, ..., \hat{x}_{n})$ of a polynomial system $F=\{f_1, ..., f_n\}$ when the Jacobian matrix of $F$ evaluated at…

Let $n,\alpha\geq 2$. Let $K$ be an algebraically closed field with characteristic $0$ or greater than $n$. We show that the dimension of the variety of pairs $(A,B)\in {M_n(K)}^2$, with $B$ nilpotent, that satisfy $AB-BA=A^{\alpha}$ or…

Given two $k\times n$ matrices $A$ and $B$, we describe a couple of methods to solve the matrix equation $XA=BY$, where $X$ is an invertible $k\times k$ matrix, and $Y$ is an $n\times n$ permutation matrix, both of which we want to…

It is known that $C(X)$ is algebraically closed if $X$ is a locally connected, hereditarily unicoherent compact Hausdorff space. For such spaces, we prove that if $F:C(X) \to C(X)$ is given by an everywhere convergent power series with…

We study the continuous solutions of several classical functional equations by using the properties of the spaces of continuous functions which are invariant under some elementary linear trans-formations. Concretely, we use that the sets of…

Theoretical and computational properties of a vector equation $Ax-\|x\|_1x=b$ are investigated, where $A$ is an invertible $M$-matrix and $b$ is a nonnegative vector. Existence and uniqueness of a nonnegative solution is proved. Fixed-point…

We are interested in the numerical solution of nonsymmetric linear systems arising from the discretization of convection-diffusion partial differential equations with separable coefficients and dominant convection. Preconditioners based on…

This paper investigates functional equations arising from perturbations of Cauchy differences. We study equations of the form \[ f(x+y)-f(x)-f(y)=B(x,y) \quad \text{or} \quad f(xy)-f(x)f(y) = B(x,y) \] where $B$ is a biadditive mapping, and…