Related papers: Solvability and uniqueness criteria for generalize…
Many applications in applied mathematics and control theory give rise to the unique solution of a Sylvester-like matrix equation associated with an underlying structured matrix operator $f$. In this paper, we will discuss the solvability of…
We provide a characterization for a periodic system of generalized Sylvester and conjugate-Sylvester equations, with at most one generalized conjugate-Sylvester equation, to have a unique solution when all coefficient matrices are square…
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…
In this paper, we derive some necessary and sufficient solvability conditions for some systems of one sided coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. We also give the…
In this paper, a class of new Sylvester-like absolute value equation (AVE) $AXB-|CXD|=E$ with $A,C\in \mathbb{R}^{m\times n}$, $B,D\in \mathbb{R}^{p\times q}$ and $E\in \mathbb{R}^{m\times q}$ is considered, which is quite distinct from the…
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…
We give a new elementary proof of existence and uniqueness of a solution to the Sylvester equation $AX-XB=Y$
Sylvester-type matrix equations have applications in areas including control theory, neural networks, and image processing. In this paper, we establish the necessary and sufficient conditions for the system of Sylvester-type quaternion…
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…
In this paper, we prove a conjecture which was presented in a recent paper [Linear Algebra Appl. 2016; 496: 549--593]. We derive some practical necessary and sufficient conditions for the existence of a solution to a system of coupled…
We consider the convolution equation $F*X=B$, where $F\in\mathbb{R}^{3\times 3}$ and $B\in\mathbb{R}^{m\times n}$ are given, and $X\in\mathbb{R}^{m\times n}$ is to be determined. The convolution equation can be regarded as a linear system…
This work is to provide a comprehensive treatment of the relationship between the theory of the generalized (palindromic) eigenvalue problem and the theory of the Sylvester-type equations. Under a regularity assumption for a specific matrix…
In this paper we represent a new form of condition for the consistency of the matrix equation AXB=C. If the matrix equation AXB=C is consistent, we determine a form of general solution which contains both reproductive and non-reproductive…
The quaternionic equations ax-xb=0 and ax-xb=c are investigated, which are called homogeneous and inhomogeneous Sylvester equations, respectively. Conditions for the existence of solutions are provided. In addition, the general and nonzero…
In this paper, we provide some solvability conditions in terms of ranks for the existence of a general solution to a system of $k$ Sylvester-type quaternion matrix equations with $3k+1$ variables…
In this paper, we discussed the unique solvability of the two absolute value matrix equations. The unique solvability condition $\rho (\vert A^{-1} B \vert)<1$ is provided for the generalized absolute value matrix equation (GAVME) $AX + B…
W.E. Roth (1952) proved that the matrix equation $AX-XB=C$ has a solution if and only if the matrices $\left[\begin{matrix}A&C\\0&B\end{matrix}\right]$ and $\left[\begin{matrix}A&0\\0&B\end{matrix}\right]$ are similar. A. Dmytryshyn and B.…
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 consider the uniqueness of solution (i.e., nonsingularity) of systems of $r$ generalized Sylvester and $\star$-Sylvester equations with $n\times n$ coefficients. After several reductions, we show that it is sufficient to analyze periodic…
Sylvester equations $AX-XB=C$ have unique solutions for all $C$ when the spectra of $A$ and $B$ are disjoint. Here $A$ and $B$ are bounded operators in Banach spaces. We discuss the existence of polynomials $p$ such that the spectra of…