System Of Quaternion Matrix Equations Research Articles

A new system of sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra

In this paper, we derive some practical necessary and sufficient conditions for the existence of a solution to a new system of coupled two-sided Sylvester-like matrix equations with arbitrary number of equations and unknowns over the quaternion algebra. As applications, we give some practical necessary and sufficient conditions for the existence of an η-Hermitian solution to a system of quaternion matrix equations in terms of ranks. We also use graphs to represent linear mappings associated with some Sylvester-type systems. As a special case of the main theorem, we prove a conjecture is correct, which was proposed in [Linear Algebra Appl. 2016;496:549–593]. The main findings of this paper widely extend almost all the known results in the literature.

Some Properties of the Solution to a System of Quaternion Matrix 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.

Consistency and General Solutions to Some Sylvester-like Quaternion Matrix Equations

This article makes use of simultaneous decomposition of four quaternion matrixes to investigate some Sylvester-like quaternion matrix equation systems. We present some useful necessary and sufficient conditions for the consistency of the system of quaternion matrix equations in terms of the equivalence form and block matrixes. We also derive the general solution to the system according to the partition of the coefficient matrixes. As an application of the system, we present some practical necessary and sufficient conditions for the consistency of a ϕ-Hermitian solution to the system of quaternion matrix equations in terms of the equivalence form and block matrixes. We also provide the general ϕ-Hermitian solution to the system when the equation system is consistent. Moreover, we present some numerical examples to illustrate the availability of the results of this paper.

The Solvability of a System of Quaternion Matrix Equations Involving ϕ-Skew-Hermicity

Let H be the real quaternion algebra and Hm×n denote the set of all m×n matrices over H. For A∈Hm×n, we denote by Aϕ the n×m matrix obtained by applying ϕ entrywise to the transposed matrix AT, where ϕ is a non-standard involution of H. A∈Hn×n is said to be ϕ-skew-Hermicity if A=−Aϕ. 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 AiXi(Ai)ϕ+BiXi+1(Bi)ϕ=Ci,(i=1,2,3),A4X4(A4)ϕ=C4.

Solving a System of Sylvester-like Quaternion Matrix Equations

Using the ranks and Moore-Penrose inverses of involved matrices, in this paper we establish some necessary and sufficient solvability conditions for a system of Sylvester-type quaternion matrix equations, and give an expression of the general solution to the system when it is solvable. As an application of the system, we consider a special symmetry solution, named the η-Hermitian solution, for a system of quaternion matrix equations. Moreover, we present an algorithm and a numerical example to verify the main results of this paper.

An Iterative Algorithm for the Generalized Reflexive Solution Group of a System of Quaternion Matrix Equations

In the present paper, an iterative algorithm is proposed for solving the generalized (P,Q)-reflexive solution group of a system of quaternion matrix equations ∑l=1M(AlsXlBls+ClsXl˜Dls)=Fs,s=1,2,…,N. A generalized (P,Q)-reflexive solution group, as well as the least Frobenius norm generalized (P,Q)-reflexive solution group, can be derived by choosing appropriate initial matrices, respectively. Moreover, the optimal approximate generalized (P,Q)-reflexive solution group to a given matrix group can be derived by computing the least Frobenius norm generalized (P,Q)-reflexive solution group of a reestablished system of matrix equations. Finally, some numerical examples are given to illustrate the effectiveness of the algorithm.

An Exact Solution to a Quaternion Matrix Equation with an Application

In this paper, we establish the solvability conditions and the formula of the general solution to a Sylvester-like quaternion matrix equation. As an application, we give some necessary and sufficient conditions for a system of quaternion matrix equations to be consistent, and present an expression of the general solution of the system when it is solvable. We present an algorithm and an example to illustrate the main results of this paper. The findings of this paper generalize the known results in the literature.

Solvability conditions and general solutions to some quaternion matrix equations

This paper studies some quaternion matrix equations. We use the equivalence canonical form to present some practical necessary and sufficient conditions for the existence of a solution to the system of quaternion matrix equations in terms of ranks We derive the general solution to the above system in terms of block matrices. As an application of the system, we give some practical necessary and sufficient conditions for the existence of a ϕ‐Hermitian solution to the system of quaternion matrix equations where X, Y, and Z are ϕ‐Hermitian unknowns. We also provide the general ϕ‐Hermitian solution to the system when the solvability conditions are satisfied. Moreover, we present some numerical examples to illustrate the results of this paper.

A Quaternion Matrix Equation with Two Different Restrictions

In this paper, we use the equivalence canonical form of four quaternion matrices to consider two systems of quaternion matrix equations. We derive some practical necessary and sufficient conditions for the existence of a solution to these two systems in terms of ranks of the quaternion matrices involved. We also give the general solutions to the systems when the solvability conditions are satisfied. Moreover, we present some numerical examples to illustrate the results of this paper.

Constraint Solution of a Classical System of Quaternion Matrix Equations and Its Cramer’s Rule

Constraint Solution of a Classical System of Quaternion Matrix Equations and Its Cramer’s Rule

On the general solutions to some systems of quaternion matrix equations

In this paper, we consider a system of three coupled quaternion matrix equations $$A_{i}X_{i}+Y_{i}B_{i}+C_{i}ZD_{i}=E_{i}$$, where $$A_{i},B_{i},C_{i},D_{i},$$ and $$E_{i}$$ are given matrices, $$X_{i},Y_{i},$$ and Z are unknowns $$(i=1,2,3)$$. We establish some practical necessary and sufficient conditions for the existence of a solution to this system, and present the general solution to the system. As applications of this system, we give some solvability conditions and general solutions to some systems of quaternion matrix equations involving $$\eta $$-Hermicity. Some examples are given to illustrate the main results. Some of the findings of this paper extend some known results in the literature.

$$\eta $$-Hermitian Solution to a System of Quaternion Matrix Equations

For $$\eta \in \{{\mathbf {i}},{\mathbf {j}},{\mathbf {k}}\}$$ , a real quaternion matrix A is said to be $$\eta $$ -Hermitian if $$A=A^{\eta *},$$ where $$A^{\eta *}=-\eta A^{*}\eta $$ , and $$A^{*}$$ stands for the conjugate transpose of A. In this paper, we present some practical necessary and sufficient conditions for the existence of an $$\eta $$ -Hermitian solution to a system of constrained two-sided coupled real quaternion matrix equations and provide the general $$\eta $$ -Hermitian solution to the system when it is solvable. Moreover, we present an algorithm and a numerical example to illustrate our results.

Explicit formulas and determinantal representation for η-skew-Hermitian solution to a system of quaternion matrix equations

Some necessary and sufficient conditions for the existence of the ?-skew-Hermitian solution quaternion matrix equations the system of matrix equations with ?-skew-Hermicity, A1X = C1, XB1 = C2, A2Y = C3, YB2 = C4, X = -X?*; Y=-Y?*, A3XA?*3 + B3YB?*3=C5, are established in this paper by using rank equalities of the coefficient matrices. The general solutions to the system and its special cases are provided when they are consistent. Within the framework of the theory of noncommutative row-column determinants, we also give determinantal representation formulas of finding their exact solutions that are analogs of Cramer?s rule. A numerical example is also given to demonstrate the main results.

Some new results on a system of Sylvester-type quaternion matrix equations

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

Cramer’s Rule for the General Solution to a Restricted System of Quaternion Matrix Equations

In this paper, we mainly study the Cramer’s rule for the general solution to the restricted system of quaternion matrix equations and the Cramer’s rule for the general solution to the restricted generalized Sylvester quaternion matrix equation $$\begin{aligned} AXB+CYD=E,\text { }\mathcal {R}_{r}\left( X\right) \subset T_{1} ,\mathcal {N}_{r}\left( X\right) \supset S_{1},\mathcal {R}_{r}\left( Y\right) \subset T_{2},\mathcal {N}_{r}\left( Y\right) \supset S_{2} \end{aligned}$$respectively. The findings of this paper extend some known results in the literature.

Least-Norm of the General Solution to Some System of Quaternion Matrix Equations and Its Determinantal Representations

We constitute some necessary and sufficient conditions for the system A1X1=C1, X1B1=C2, A2X2=C3, X2B2=C4, A3X1B3+A4X2B4=Cc, to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.

Cramer’s rules for the system of quaternion matrix equations with η-Hermicity

The system of two-sided quaternion matrix equations with η-Hermicity, A1XA1η* = C1, A2XA2η* = C2 is considered in the paper. Using noncommutative row-column determinants previously introduced by the author, determinantal representations (analogs of Cramer’s rule) of a general solution to the system are obtained. As special cases, Cramer’s rules for an η-Hermitian solution when C1 = Cη*1 and C2 = Cη*2 and for an η-skew-Hermitian solution when C1 = −Cη*1 and C2 = −Cη*2 are also explored.

Structure, Properties and Applications of Some Simultaneous Decompositions for Quaternion Matrices Involving $$\phi $$ ϕ -Skew-Hermicity

Let $${\mathbb {H}}$$ be the real quaternion algebra and $${\mathbb {H}}^{m\times n}$$ denote the set of all $$m\times n$$ matrices over $${\mathbb {H}}$$ . For $$A\in {\mathbb {H}}^{m\times n},$$ we denote by $$A_{\phi }$$ the $$n\times m$$ matrix obtained by applying $$\phi $$ entrywise to the transposed matrix $$A^{t},$$ where $$\phi $$ is a nonstandard involution of $${\mathbb {H}}$$ . $$A\in {\mathbb {H}}^{n\times n}$$ is said to be $$\phi $$ -skew-Hermitian if $$A=-A_{\phi }$$ . In this paper, we investigate and analyze in detail the structure and properties of a simultaneous decomposition for six quaternion matrices involving $$\phi $$ -skew-Hermicity: where A and F are $$\phi $$ -skew-Hermitian. Using this simultaneous decomposition, we give some practical necessary and sufficient conditions for the existence of a $$\phi $$ -skew-Hermitian solution (X, Y, Z) to the system of quaternion matrix equations Apart from proving an expression for the general $$\phi $$ -skew-Hermitian solution to this system, we derive the $$\beta (\phi )$$ -signature bounds of the $$\phi $$ -skew-Hermitian solution in terms of the coefficient matrices. Moreover, we obtain necessary and sufficient conditions for the system to have $$\beta (\phi )$$ -positive definite, $$\beta (\phi )$$ -positive semidefinite, $$\beta (\phi )$$ -negative definite and $$\beta (\phi )$$ -negative semidefinite solutions. We also give some numerical examples to illustrate our results.

Cramer’s rule for a system of quaternion matrix equations with applications

In this paper, we investigate Cramer’s rule for the general solution to the system of quaternion matrix equationsA1XB1=C1,A2XB2=C2,and Cramer’s rule for the general solution to the generalized Sylvester quaternion matrix equationAXB+CYD=E,respectively. As applications, we derive the determinantal expressions for the Hermitian solutions to some quaternion matrix equations. The findings of this paper extend some known results in the literature.

The Least Square Solution with the Least Norm to a System of Quaternion Matrix Equations

We in this paper derive the expression of the least square solution with the least norm to the system of generalized Sylvester quaternion matrix equations $$\begin{aligned} A_1X = E_1 ,\quad XB_1 = E_2, \quad C_1Y = E_3 ,\quad YD_1 = E_4,\quad A_2XB_2+C_2YD_2=E_5, \end{aligned}$$ where X, Y are unknown quaternion matrices and the others are given quaternion matrices.