Let i, j, k be the quaternion units and let A a square real matrix. is said to η-Hermitian if −η A*η = A, where η ∈ {i, k} A* conjugate transpose of A. Denote η* − A*η. Following Horn Zhang's recent research on matrices (A generalization complex AutonneTakagi factorization matrices, Linear Multilinear Algebra, DOI:10.1080/03081087.2011.618838), we consider matrix equation involving η-Hermicity, i.e. Y Z are required η-Hermitian. We provide some necessary sufficient conditions for existence...

In this paper, we consider an expression of the general solution to classical system matrix equationsWe present a necessary and sufficient condition for existence by using generalized inverses. We give when it is solvable. As applications, derive some conditions consistence systemand systemwhere means conjugate transpose. also expressions solutions systems.

In this paper, we establish a different approach for solving the system of three coupled two-sided Sylvester-type quaternion matrix equations AiXiBi+CiXi+1Di=Ei, i=1,3¯. We give some new necessary and sufficient conditions existence solution to in terms Moore-Penrose inverses matrices involved. show that these solvability are equivalent with which were presented recent paper [Linear Algebra Appl. 2016;496:549–593]. The general is given when satisfied. Applications discussed include...

Abstract We establish necessary and sufficient conditions for the solvability to matrix equation present an expression of general solution (Equation1) when it is solvable. As applications, we discuss consistence where * means conjugate transpose, provide explicit (Equation2). also study extremal ranks X 3 4 inertias in (Equation1). In addition, obtain classical have Re-nonnegative definite, Re-nonpositive Re-positive definite Re-negative solutions. The findings this article extend related...

We in this paper derive necessary and sufficient conditions for the system of periodic discrete-time coupled Sylvester matrix equations [Formula: see text] over quaternion algebra to be consistent terms ranks generalized inverses coefficient matrices. also give an expression general solution when it is solvable. The findings generalize some known results literature.

Let $\mathbb{H}^{m\times n}$ be the space of $m\times n$ matrices over $\mathbb{H}$, where $\mathbb{H}$ is real quaternion algebra. $A_{\phi}$ $n\times m$ matrix obtained by applying $\phi$ entrywise to transposed $A^{T}$, $A\in\mathbb{H}^{m\times and a nonstandard involution $\mathbb{H}$. In this paper, some properties Moore-Penrose inverse are given. Two systems mixed pairs Sylvester equations $A_{1}X-YB_{1}=C_{1},~A_{2}Z-YB_{2}=C_{2}$ $A_{1}X-YB_{1}=C_{1},~A_{2}Y-ZB_{2}=C_{2}$ considered,...

Let be the set of all matrices over real quaternion algebra. , where is conjugate . We call that -Hermitian, if ; -bihermitian, in this paper, present solvability conditions and general -Hermitian solution to a system linear matrix equations. As an application, we give necessary sufficient for systemto have -bihermitian solution. establish expression when it solvable. also obtain criterion -bihermitian. Moreover, provide algorithm numerical example illustrate theory developed paper.

In this paper, the pure product singular value decomposition (PSVD) for four quaternion matrices is given. The system of coupled Sylvester-type matrix equations with five unknowns $X_{i}A_{i}-B_{i}X_{i+1}=C_{i}$ considered by using PSVD approach, where $A_{i},B_{i},$ and $C_{i}$ are given compatible sizes $(i=1,2,3,4)$. Some necessary sufficient conditions existence a solution to derived. Moreover, general presented when it solvable.