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Research Article | Open Access
An Efficient Algorithm for the Reflexive Solution of the Quaternion Matrix Equation
We propose an iterative algorithm for solving the reflexive solution of the quaternion matrix equation . When the matrix equation is consistent over reflexive matrix , a reflexive solution can be obtained within finite iteration steps in the absence of roundoff errors. By the proposed iterative algorithm, the least Frobenius norm reflexive solution of the matrix equation can be derived when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate reflexive solution to a given reflexive matrix can be derived by finding the least Frobenius norm reflexive solution of a new corresponding quaternion matrix equation. Finally, two numerical examples are given to illustrate the efficiency of the proposed methods.
Throughout the paper, the notations and represent the set of all real matrices and the set of all matrices over the quaternion algebra . We denote the identity matrix with the appropriate size by . We denote the conjugate transpose, the transpose, the conjugate, the trace, the column space, the real part, the vector formed by the vertical concatenation of the respective columns of a matrix by , respectively. The Frobenius norm of is denoted by , that is, . Moreover, and stand for the Kronecker matrix product and Hadmard matrix product of the matrices and .
Let be a generalized reflection matrix, that is, and . A matrix is called reflexive with respect to the generalized reflection matrix , if . It is obvious that any matrix is reflexive with respect to . Let denote the set of order reflexive matrices with respect to . The reflexive matrices with respect to a generalized reflection matrix have been widely used in engineering and scientific computations [1, 2].
In the field of matrix algebra, quaternion matrix equations have received much attention. Wang et al.  gave necessary and sufficient conditions for the existence and the representations of P-symmetric and P-skew-symmetric solutions of quaternion matrix equations and . Yuan and Wang  derived the expressions of the least squares -Hermitian solution with the least norm and the expressions of the least squares anti--Hermitian solution with the least norm for the quaternion matrix equation . Jiang and Wei  derived the explicit solution of the quaternion matrix equation . Li and Wu  gave the expressions of symmetric and skew-antisymmetric solutions of the quaternion matrix equations and . Feng and Cheng  gave a clear description of the solution set of the quaternion matrix equation .
The iterative method is a very important method to solve matrix equations. Peng  constructed a finite iteration method to solve the least squares symmetric solutions of linear matrix equation . Also Peng [9–11] presented several efficient iteration methods to solve the constrained least squares solutions of linear matrix equations and , by using Paige’s algorithm  as the frame method. Duan et al. [13–17] proposed iterative algorithms for the (Hermitian) positive definite solutions of some nonlinear matrix equations. Ding et al. proposed the hierarchical gradient-based iterative algorithms  and hierarchical least squares iterative algorithms  for solving general (coupled) matrix equations, based on the hierarchical identification principle . Wang et al.  proposed an iterative method for the least squares minimum-norm symmetric solution of . Dehghan and Hajarian constructed finite iterative algorithms to solve several linear matrix equations over (anti)reflexive [22–24], generalized centrosymmetric [25, 26], and generalized bisymmetric [27, 28] matrices. Recently, Wu et al. [29–31] proposed iterative algorithms for solving various complex matrix equations.
However, to the best of our knowledge, there has been little information on iterative methods for finding a solution of a quaternion matrix equation. Due to the noncommutative multiplication of quaternions, the study of quaternion matrix equations is more complex than that of real and complex equations. Motivated by the work mentioned above and keeping the interests and wide applications of quaternion matrices in view (e.g., [32–45]), we, in this paper, consider an iterative algorithm for the following two problems.
Problem 1. For given matrices and the generalized reflection matrix , find , such that
Problem 2. When Problem 1 is consistent, let its solution set be denoted by . For a given reflexive matrix , find , such that
The remainder of this paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we introduce an iterative algorithm for solving Problem 1. Then we prove that the given algorithm can be used to obtain a reflexive solution for any initial matrix within finite steps in the absence of roundoff errors. Also we prove that the least Frobenius norm reflexive solution can be obtained by choosing a special kind of initial matrix. In addition, the optimal reflexive solution of Problem 2 by finding the least Frobenius norm reflexive solution of a new matrix equation is given. In Section 4, we give two numerical examples to illustrate our results. In Section 5, we give some conclusions to end this paper.
In this section, we provide some results which will play important roles in this paper. First, we give a real inner product for the space over the real field .
Theorem 3. In the space over the field , a real inner product can be defined as for . This real inner product space is denoted as .
Proof. (1) For , let , then
It is obvious that and .
(2) For , let and , then we have
(4) For and ,
All the above arguments reveal that the space over field with the inner product defined in (3) is an inner product space.
Let represent the matrix norm induced by the inner product . For an arbitrary quaternion matrix , it is obvious that the following equalities hold: which reveals that the induced matrix norm is exactly the Frobenius norm. For convenience, we still use to denote the induced matrix norm.
Let denote the quaternion matrix whose entry is 1, and the other elements are zeros. In inner product space , it is easy to verify that , , , is an orthonormal basis, which reveals that the dimension of the inner product space is .
Next, we introduce a real representation of a quaternion matrix.
For an arbitrary quaternion matrix , a map , from to , can be defined as
By (9), it is easy to verify that
Finally, we introduce the commutation matrix.
A commutation matrix is a matrix which has the following explicit form: Moreover, is a permutation matrix and . We have the following lemmas on the commutation matrix.
Lemma 5 (see ). Let be a matrix. There is a commutation matrix such that
Lemma 6 (see ). Let be a matrix and a matrix. There exist two commutation matrices and such that
3. Main Results
3.1. The Solution of Problem 1
In this subsection, we will construct an algorithm for solving Problem 1. Then some lemmas will be given to analyse the properties of the proposed algorithm. Using these lemmas, we prove that the proposed algorithm is convergent.
Algorithm 7 (Iterative algorithm for Problem 1). (1)Choose an initial matrix .(2)Calculate (3)If , then stop and is the solution of Problem 1; else if and , then stop and Problem 1 is not consistent; else .(4)Calculate (5)Go to Step (3).
Lemma 8. Assume that the sequences and are generated by Algorithm 7, then and for .
Proof. Since and for , we only need to prove that and for .
Now we prove this conclusion by induction.
Step 1. We show that We also prove (17) by induction. When , we have Also we can write Now assume that conclusion (17) holds for , then And it can also be obtained that Therefore, the conclusion (17) holds for .
Step 2. Assume that and for and . We will show that We prove the conclusion (22) in two substeps.
Substep 2.1. In this substep, we show that It follows from Algorithm 7 that Also we can write Substep 2.2. In this substep, we prove the conclusion (22) in Step 2. It follows from Algorithm 7 that Repeating the above process (26), we can obtain Combining these two relations with (24) and (25), it implies that (22) holds. So, by the principle of induction, we know that Lemma 8 holds.
Proof. We also prove this conclusion by induction.
When , it follows from Algorithm 7 that This implies that (28) holds for .
Now it is assumed that (28) holds for , that is Then, when Therefore, Lemma 9 holds by the principle of induction.
From the above two lemmas, we have the following conclusions.
Remark 10. If there exists a positive number such that and , then we can get from Lemma 9 that Problem 1 is not consistent. Hence, the solvability of Problem 1 can be determined by Algorithm 7 automatically in the absence of roundoff errors.
Proof. In Section 2, it is known that the inner product space is -dimensional. According to Lemma 9, if , , then we have . Hence and can be computed. From Lemma 8, it is not difficult to get Then is an orthogonal basis of the inner product space . In addition, we can get from Lemma 8 that It follows that , which implies that is a solution of Problem 1.
3.2. The Solution of Problem 2
In this subsection, firstly we introduce some lemmas. Then, we will prove that the least Frobenius norm reflexive solution of (1) can be derived by choosing a suitable initial iterative matrix. Finally, we solve Problem 2 by finding the least Frobenius norm reflexive solution of a new-constructed quaternion matrix equation.
Lemma 12 (see ). Assume that the consistent system of linear equations has a solution then is the unique least Frobenius norm solution of the system of linear equations.
Lemma 13. Problem 1 is consistent if and only if the system of quaternion matrix equations is consistent. Furthermore, if the solution sets of Problem 1 and (34) are denoted by and , respectively, then, we have .
Proof. First, we assume that Problem 1 has a solution . By and , we can obtain and , which implies that is a solution of quaternion matrix equations (34), and .
Conversely, suppose (34) is consistent. Let be a solution of (34). Set . It is obvious that . Now we can write Hence is a solution of Problem 1. The proof is completed.
Lemma 14. The system of quaternion matrix equations (34) is consistent if and only if the system of real matrix equations is consistent, where , are submatrices of the unknown matrix. Furthermore, if the solution sets of (34) and (36) are denoted by and , respectively, then, we have .
Proof. Suppose that (34) has a solution Applying , , and in Lemma 4 to (34) yields which implies that is a solution of (36) and . Conversely, suppose that (36) has a solution By in Lemma 4, we have that Hence which implies that , and are also solutions of (36). Thus, is also a solution of (36), where Let Then it is not difficult to verify that We have that is a solution of (34) by in Lemma 4. The proof is completed.
Lemma 15. There exists a permutation matrix P(4n,4n) such that (36) is equivalent to
Proof. By , , and in Lemmas 4, 5, and 6, we have that
By Lemma 12, is the least Frobenius norm solution of matrix equations (46).
Noting (11), we derive from Lemmas 13, 14, and 15 that is the least Frobenius norm solution of Problem 1.
Using the above conclusion and considering Theorem 16, we propose the following theorem.
Theorem 17. Suppose that Problem 1 is consistent. Let the initial iteration matrix be where is an arbitrary quaternion matrix, or especially, , then the solution , generated by Algorithm 7, is the least Frobenius norm solution of Problem 1.
Now we study Problem 2. When Problem 1 is consistent, the solution set of Problem 1 denoted by is not empty. Then, For a given reflexive matrix , Let and , then Problem 2 is equivalent to finding the least Frobenius norm reflexive solution of the quaternion matrix equation By using Algorithm 7, let the initial iteration matrix , where is an arbitrary quaternion matrix in , or especially,