On Hermitian Solutions of the Generalized Quaternion Matrix Equation
The paper deals with the matrix equation 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.
In 1843, Irish Mathematician William Rowan Hamilton introduced the Hamilton quaternions. It is a great event in the history of mathematics. The 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 [3–5]). In this paper, we consider a more generalized case, that is, the generalized quaternions, which is in the form of where . In the paper, we only focus on the cases of . We call the set by the generalized quaternions. Obviously, is a noncommutative 4-dimensional Clifford algebra. Specially, is the Hamilton quaternion ring if , is the split quaternion ring if , is the nectarine ring if , is the conectarine ring if .
Throughout this paper, let , , , , and denote the set of all real matrices, the set of all complex matrices, the set of all generalized quaternion matrices, the set of all real symmetric matrices, and the set of all real antisymmetric matrices, respectively. The identity matrix of order is denoted by . The zero matrix with suitable size is denoted by 0. We define the conjugate of as . For ; we use , to denote the conjugate matrix, the transpose matrix of , respectively. is the conjugate transpose of . We call a matrix is Hermitian if , which we denote it by , where is the set of all Hermitian generalized quaternion matrices with the size of .
In recent decades, different kinds of matrix equations over some quaternion algebras had been studied, such as the , , , , and , , and over the real/complex fields or some quaternion algebras (see [7–29] and references cited therein). For now, only few papers explored some fundamental properties and 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. There 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.  studied Hermitian solutions to the generalized quaternion matrix equation by the real representation method; Yuan et al.  discussed Hermitian solutions to the split quaternion matrix equation 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 , , find the solution set
2. Properties of the Generalized Quaternion Matrices
For any with , we define
Obviously, the map is an isomorphism of , we denote by . Next, we propose a real matrix representation for the generalized quaternion matrix :
For , the Kronecker product of and is defined as . For the generalized quaternion matrices with suitable sizes and the real number , we have
For the matrix , let with , we denote the vector by
Throughout the paper, we denote
The following are some properties of generalized quaternion matrices.
Proposition 1. Let,,, and. Then,(i) does not hold in general(ii) in general(iii) in general(iv) in general(v) in general(vi) in general
Proof. When and , is the quaternion ring and the split quaternion ring, we can refer to [26, 32]), and the other cases can be easily obtained by direct calculation.
Some important properties of and are as follows.
Proposition 2. Let , , , and . Then,(i) if and only if , if and only if (ii), , , (iii), (iv)If is invertible, then (v)
Proof. Since the proofs of (i), (ii), (iv), and (v) are easy, we only prove (iii). By direct calculation, we haveThus,whereNow, it is easy to verify .
3. The Structure of
In the section, we investigate the structure of . For , , and , it is well known that
However, (11) cannot hold in the generalized quaternions for the noncommutative multiplication of the generalized quaternions. Thus, we need to study the structure of .
Theorem 1. Let , , and , where , and . Then,
Proof. By (iii) in Proposition 2,whereIt follows from (11) thatThus,which completed our proof.
Yuan et al.  studied the over , while Theorem 1 extends it to the result over . As we can see that Theorem 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.
Definition 1. For the matrix , let , , , , we denote
Definition 2. For the matrix , let , , , , we denote
Lemma 1 (see ). Suppose , thenwhere is represented as (4), and the matrix is of the following form:where is the -th column of the identity matrix of order .where is represented as (5), and the matrix is of the following form:where is the -th column of . Obviously, , .
By Lemma 1, we have the following.
Theorem 2. For , thenin which
Theorem 3. Let , , and , where , and . Then,
4. The Hermitian Solutions
Based on our earlier discussion, we now pay our attention to Problem I. The following notation is necessary for deriving a solution to Problem I. Let , , , , and . In the remaining of the paper, we set
We also need the following lemma.
Lemma 2 (see ). The matrix equation , with and , has a solution if and only ifwhere is the Moore–Penrose inverse of the matrix . In this case, it has the general solutionwhere is an arbitrary vector, and it has the unique solution for the case when . The solution of the matrix equation with the least norm is .
Theorem 4. Let , , and . Then, Problem I has a solution if and only if
If this condition satisfies, thenwhere is an arbitrary vector.
In this case,
Proof. By (ii) in Proposition 2 and Theorem 3, we haveBy Lemma 2, Problem I has a solution if and only if (31) holds. If this condition satisfies, thenAlso by (23),where is an arbitrary vector. We can draw the conclusion (32). Furthermore, if (31) holds, Problem I has a unique solution if and only ifThat is, (33) holds. In the case, we obtain (34).
In this section, we give two examples to illustrate our results.
Example 1. Consider the Hamilton quaternion matrix equation , whereObviously, the Hamilton quaternions mean . By (4) and (28), we easily getBy Theorem 4 and MATLAB, calculating the formula gives a solution
In this paper, we provide a direct method to find Hermitian solutions of the generalized quaternion matrix equation by using the real representation of generalized quaternion matrices, Kronecker product and vec-operator. 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. The paper proposes an algebraic technique for finding the Hermitian solutions to the above matrix equation over the generalized quaternions. The generalized quaternions include many important quaternion algebras, for instance, , , , and , thus the paper actually proposes a unified technique to solve the Hermitian solution problems over the several quaternion algebras.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This research was supported by Macao Science and Technology Development Fund (No. 185/2 017/A3) and The Joint Research and Development Fund of Wuyi University, Hong Kong and Macao (2019WGALH20).
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