On Hermitian Solutions of the Generalized Quaternion Matrix Equation AXB + CXD = E

The paper deals with the matrix equation AXB + CXD � E over the generalized quaternions. By the tools of the real representation of a generalized quaternion matrix, Kronecker product as well as vec-operator, the paper derives the necessary and suﬃcient conditions for the existence of a Hermitian solution and gives the explicit general expression of the solution when it is solvable and provides a numerical example to test our results. The paper proposes a uniﬁcated algebraic technique for ﬁnding Hermitian solutions to the mentioned matrix equation over the generalized quaternions, which includes many important quaternion algebras, such as the Hamilton quaternions and the split quaternions.


Introduction
In 1843, Irish Mathematician William Rowan Hamilton introduced the Hamilton quaternions. It is a great event in the history of mathematics. e set of Hamilton quaternions can form a skew field [1,2]. In 1849, James Cockle introduced the split quaternions. It can be used to express Lorentzian rotations, which is used in geometry and physics (see [3][4][5]). In this paper, we consider a more generalized case, that is, the generalized quaternions, which is in the form of [6] q � q 1 + q 2 i + q 3 j + q 4 k, q 1 , q 2 , q 3 , q 4 ∈ R. (1) where i 2 � u, j 2 � v, k 2 � ijk � −uv, ij � −ji � k, jk � −kj � −vi, ik � −ki � uj. In the paper, we only focus on the cases of 0 ≠ u, v ∈ R. We call the set Q G by the generalized quaternions. Obviously, Q G is a noncommutative 4-dimensional Clifford algebra. Specially, Q G is the Hamilton quaternion ring Q if u � v � −1, Q G is the split quaternion ring Q s if u � −1, v � 1, Q G is the nectarine ring Q n if u � 1, v � −1, Q G is the conectarine ring Q c if u � v � 1.
roughout this paper, let R m×n , C m×n , Q m×n G , SR n×n , and ASR n×n denote the set of all m × n real matrices, the set of all m × n complex matrices, the set of all m × n generalized quaternion matrices, the set of all n × n real symmetric matrices, and the set of all n × n real antisymmetric matrices, respectively. e identity matrix of order n is denoted by I n .
e zero matrix with suitable size is denoted by 0. We define the conjugate of q ∈ Q G as q � q 1 − q 2 i − q 3 j − q 4 k. For A � (a ij ) ∈ Q m×n G ; we use A � (a ij ), A T to denote the conjugate matrix, the transpose matrix of A, respectively.
We call a matrix A ∈ Q n×n G is Hermitian if A H � A, which we denote it by A ∈ HQ n×n G , where HQ n×n matrix equation over the generalized quaternions, which one may refer to [9,17,30,31]. Hermitian matrix has attracted lots of attentions because of its great importance. ere are some results about Hermitian solutions of matrix equations over several kinds of quaternion algebras (see [6,26,28,32]). For example, Yu et al. [6] studied Hermitian solutions to the generalized quaternion matrix equation AXB + CX * D � E by the real representation method; Yuan et al. [32] discussed Hermitian solutions to the split quaternion matrix equation AXB + CX D � E by using the complex representation method. Based on the work mentioned above, and inspired by the methods in ( [28,32]), we discuss the following problem: Problem I: given

Properties of the Generalized Quaternion Matrices
For Obviously, the map Φ A is an isomorphism of A, we denote by Φ A � A. Next, we propose a real matrix representation for the generalized quaternion matrix A ∈ Q m×n G : . For the generalized quaternion matrices A, B, C, D, E, F, G, and H with suitable sizes and the real number k, we have For the matrix A � (a ij ) ∈ Q m×n G , let a j � (a 1j , a 2j , . . . , a mj ) with j � 1, 2, . . . , n, we denote the vector vec(A) by vec(A) � a 1 , a 2 , . . . , a n T .
roughout the paper, we denote N n � I n 0 0 0 0 uI n 0 0 0 0 vI n 0 e following are some properties of generalized quaternion matrices.
Proof. When u � v � −1 and u � −1, v � 1, Q G is the quaternion ring and the split quaternion ring, we can refer to 2 Mathematical Problems in Engineering [26,32]), and the other cases can be easily obtained by direct calculation. Some important properties of A R and Φ A are as follows.
Proof. Since the proofs of (i), (ii), (iv), and (v) are easy, we only prove (iii). By direct calculation, we have us, where

The Structure of vec(AXB)
In the section, we investigate the structure of vec(AXB). For A ∈ C m×n , B ∈ C n×s , and C ∈ C s×t , it is well known that However, (11) cannot hold in the generalized quaternions for the noncommutative multiplication of the generalized quaternions. us, we need to study the structure of vec(Φ ABC ).

Let
Proof. By (iii) in Proposition 2,

Mathematical Problems in Engineering
where It follows from (11) that vec which completed our proof. Yuan et al. [26] studied the vec(Φ ABC ) over Q, while eorem 1 extends it to the result over Q G . As we can see that eorem 1 maps the product of generalized quaternion matrices into the product of real matrices by using the real representation method, by this way, we can convert a generalized quaternion matrix equation into a real one.
In the following, we introduce some definitions and useful lemmas.

The Hermitian Solutions
Based on our earlier discussion, we now pay our attention to Problem I. e following notation is necessary for deriving a solution to Problem I.
In the remaining of the paper, we set We also need the following lemma.
Lemma 2 (see [33]). e matrix equation Ax � b, with A ∈ R m×n and b ∈ R m , has a solution x ∈ R n if and only if where A + is the Moore-Penrose inverse of the matrix A. In this case, it has the general solution where y ∈ R n is an arbitrary vector, and it has the unique solution x � A + b for the case when rank(A) � n. e solution of the matrix equation Ax � b with the least norm is x � A + b.

en, Problem I has a solution X ∈ H E if and only if
If this condition satisfies, then where y ∈ R 2n 2 −n is an arbitrary vector. Furthermore, if (31) holds, then the generalized quaternion matrix equation (2) has a unique solution X ∈ H E if and only if rank(P) � 2n 2 − n. (33) In this case, Proof. By (ii) in Proposition 2 and eorem 3, we have By Lemma 2, Problem I has a solution X ∈ H E if and only if (31) holds. If this condition satisfies, then Also by (23), where y ∈ R 2n 2 −n is an arbitrary vector. We can draw the conclusion (32). Furthermore, if (31) holds, Problem I has a unique solution X ∈ H E if and only if at is, (33) holds. In the case, we obtain (34).
□ Mathematical Problems in Engineering

Example
In this section, we give two examples to illustrate our results.

Conclusion
In this paper, we provide a direct method to find Hermitian solutions of the generalized quaternion matrix equation AXB + CXD � E by using the real representation of generalized quaternion matrices, Kronecker product and vecoperator. We give the necessary and sufficient conditions for the existence of a Hermitian solution and also derive the general solution when the matrix equation is consistent. e paper proposes an algebraic technique for finding the Hermitian solutions to the above matrix equation over the generalized quaternions. e generalized quaternions include many important quaternion algebras, for instance, Q, Q s , Q n , and Q c , thus the paper actually proposes a unified technique to solve the Hermitian solution problems over the several quaternion algebras.
Data Availability e data that support the findings of this study are available from the corresponding author upon reasonable request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.