Related papers: Toward Solution of Matrix Equation X=Af(X)B+C
The treated matrix equation $(1+ae^{-\frac{\|X\|}{b}})X=Y$ in this short note has its origin in a modelling approach to describe the nonlinear time-dependent mechanical behaviour of rubber. We classify the solvability of…
This paper presents an efficient method for obtaining the least squares Hermitian solutions of the reduced biquaternion matrix equation $(AXB, CXD) = (E, F )$. The method leverages the real representation of reduced biquaternion matrices.…
Let $B$ and $C$ be square complex matrices. The differential equation \begin{equation*} x''(t)+Bx'(t)+Cx(t)=f(t) \end{equation*} is considered. A solvent is a matrix solution $X$ of the equation $X^2+BX+C=\mathbf0$. A pair of solvents $X$…
One well-known necessary and sufficient condition for equality in the Frobenius rank inequality due to Tian and Styan is that the matrix equation B = BCX + YAB be solvable for X and Y. We develop an algorithm to construct the matrices X and…
In this paper, the consimilarity of complex matrices is generalized for the split quaternions. In this regard, coneigenvalue and coneigenvector are defined for split quaternion matrices. Also, the existence of solution to the split…
The Sylvester equation $AX-XB=C$ is considered in the setting of quaternion matrices. Conditions that are necessary and sufficient for the existence of a unique solution are well-known. We study the complementary case where the equation…
This paper provides an overview of the necessary and sufficient conditions for guaranteeing the unique solvability of absolute value equations. In addition to discussing the basic form of these equations, we also address several…
We study solution X of the stochastic equation X = AX +B, where A is a random matrix and B,X are random vectors, the law of (A,B) is given and X is independent of (A,B). The equation is meant in law, the matrix A is 2x2 upper triangular,…
In this article we apply proper splittings of matrices to develop an iterative process to approximate solutions of matrix equations of the form TX = W. Moreover, by using the partial order induced by positive semidefinite matrices, we…
Generalized equations are problems emerging in contexts of modern variational analysis as an adequate formalism to treat such issues as constraint systems, optimality and equilibrium conditions, variational inequalities, differential…
This paper investigates the conditions that guarantee unique solvability and unsolvability for the generalized absolute value equations (GAVE) given by $Ax - B \vert x \vert = b$. Further, these conditions are also valid to determine the…
Consider the nonlinear matrix equation X-sum_{i=1}^{m}A_{i}^{*}X^{-1}A_{i}=Q. This paper shows that there exists a unique positive definite solution to the equation without any restriction on A_{i}. Three perturbation bounds for the unique…
In this paper, which is a follow-up to [A. Borobia, R. Canogar, F. De Ter\'an, Mediterr. J. Math. 18, 40 (2021)], we provide a necessary and sufficient condition for the matrix equation $X^\top AX=B$ to be consistent when $B$ is symmetric.…
We present a circulant and skew-circulant splitting (CSCS) iterative method for solving large sparse continuous Sylvester equations $AX + XB = C$, where the coefficient matrices $A$ and $B$ are Toeplitz matrices. A theoretical study shows…
The T-congruence Sylvester equation is the matrix equation $AX+X^{\mathrm{T}}B=C$, where $A\in\mathbb{R}^{m\times n}$, $B\in\mathbb{R}^{n\times m}$, and $C\in\mathbb{R}^{m\times m}$ are given, and $X\in\mathbb{R}^{n\times m}$ is to be…
This paper studies algebraic properties of Hermitian solutions and Hermitian definite solutions of the two types of matrix equation $AX = B$ and $AXA^* = B$. We first establish a variety of rank and inertia formulas for calculating the…
We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the…
The nonlinear eigen-problem $ Ax+F(x)=\lambda x$ is studied where $A$ is an $n\times n$ irreducible Stieltjes matrix. Under certain conditions, this problem has a unique positive solution. We show that, starting from a multiple of the…
In this article, a system of Yang-Baxter-type matrix equations is studied, $XAX=BXB$, $XBX=AXA$, which "generalizes" the matrix Yang-Baxter equation and exhibits a broken symmetry. We investigate the solutions of this system from various…
We construct the general solution of the equation $w_t+\sum\limits_{k=1}^nw_{x_k}\rho^{(k)}(w)=\rho(w)+[w,T\tilde\rho(w)]$, for the $N\times N$ matrix $w$, where $T$ is any constant diagonal matrix, $n, N \in \NN_+$ and $\rho^{(k)}, \rho,…