Iterative Algorithm for Solving a Class of Quaternion Matrix Equation over the Generalized -Reflexive Matrices
The matrix equation which includes some frequently investigated matrix equations as its special cases, plays important roles in the system theory. In this paper, we propose an iterative algorithm for solving the quaternion matrix equation over generalized -reflexive matrices. The proposed iterative algorithm automatically determines the solvability of the quaternion matrix equation over generalized -reflexive matrices. When the matrix equation is consistent over generalized -reflexive matrices, the sequence generated by the introduced algorithm converges to a generalized -reflexive solution of the quaternion matrix equation. And the sequence converges to the least Frobenius norm generalized -reflexive solution of the quaternion matrix equation when an appropriate initial iterative matrix is chosen. Furthermore, the optimal approximate generalized -reflexive solution for a given generalized -reflexive matrix can be derived. The numerical results indicate that the iterative algorithm is quite efficient.
1. Introduction and Notations
Quaternion matrix equation is one of the topics of very active research in matrix theory and applications, and some important results have been developed. For example, Kyrchei  derived the explicit representation formulas for the minimum norm least squares solutions of quaternion matrix equations , , . Yuan and Liao  derived the expressions of the least squares (-self-conjugate) solution with the least norm of the quaternion matrix equation . Song et al.  obtained the expressions of the explicit solutions of quaternion matrix equations and . He and Wang  studied the necessary and sufficient conditions for the existence of a solution to the quaternion matrix equation and derived a general solution when the equation is solvable.
The definition of generalized -reflexive matrix can be found in . A complex matrix is called -generalized reflexive (generalized anti-reflexive) if , where and are two generalized reflection matrices; that is, and . The generalized -reflexive matrices have been widely used in engineering and scientific computations [5–7]. Throughout, we use to denote the set of generalized -reflexive quaternion matrices.
The iterative method is a very important method to solve matrix equations. Peng et al. [8, 9] constructed iteration methods to solve the symmetric and reflexive solutions of matrix equations , . Ding and Chen [10, 11] developed iterative methods for solving a wide variety of matrix equations, based on the hierarchical identification principle . Peng [13–15] presented several efficient iterative methods to solve the constrained least squares solutions of linear matrix equations and , by using Paige's algorithm  as the frame method. Wang et al.  proposed iterative algorithms for solving the matrix equation . Xie et al.  presented a gradient based iterative algorithm for solving In , Li et al. proposed iterative algorithms to solve the minimal norm least squares solution to (1). Duan et al. [20–24] proposed iterative algorithms for the (Hermitian) positive definite solutions of some nonlinear matrix equations. Dehghan and Hajarian constructed iterative algorithms to solve several linear matrix equations over (anti-)reflexive [25, 26], generalized centrosymmetric [27, 28], and generalized bisymmetric [29, 30] matrices. Recently, Wu et al. [31–35] proposed iterative algorithms for solving various complex matrix equations. Wang et al.  derived an iterative method for finding the minimum-norm solution of the QLS problem in quaternionic quantum theory.
However, to our best knowledge, the generalized -reflexive solution of (1) over the quaternion algebra has not been considered so far. Due to the noncommutativity of , some well-known equalities for complex and real matrices no longer hold for quaternion matrices, which make the study of quaternion matrix equation more complex than that of real and complex equation. Motivated by the work mentioned previously and keeping the interests and wide applications of quaternion matrices in view (e.g., [37–47]), we, in this paper, consider an iterative method for the following two problems.
Problem 1. For given matrices , , , , , , , and the generalized reflection matrices , , find that , 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 matrix equation (2) plays important roles in the system theory [48–50]. Moreover, (2) obviously includes the matrix equations , , and as special cases, which have been investigated in [17, 28, 36, 51–53].
Throughout this paper, and represent the set of all real matrices and the set of all matrices over the quaternion algebra . We use , , , , , , and to denote the conjugate transpose, the transpose, the conjugate, the trace, the column space, the real part, and the vector formed by the vertical concatenation of the respective columns of a matrix ; respectively. We use to denote the Frobenius norm of , that is, . The Kronecker matrix product and Hadamard matrix product of the matrices and are denoted by and . We use to denote the identity matrix with the appropriate size.
In this section, we provide some results which will play important roles in this paper.
Firstly, we recall the definition of real inner product space. Then we define a real inner product in the space over the field .
Definition 3 (see ). A real inner product space is a vector space over the real field together with an inner product defined by a mapping satisfying the following three axioms for all vectors and all scalars .(1)Symmetry: ;(2)Linearity in the first argument: ;(3)Positive definiteness: for all .
In [33–35], the following real inner product was presented to solve some complex matrix equations for . It is easy to verify that if we let , (5) also defines a real inner product in over . We denote this real inner product space by . From the equalities we know that the induced matrix norm by the introduced inner product is exactly the Frobenius norm. For convenience, we still use to denote the induced matrix norm. Let denote the 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 the well-known real representation of a quaternion matrix. For an arbitrary quaternion matrix , a map , from to , can be defined as Let and be two arbitrary quaternion matrices with appropriate size. We know that satisfies the following properties: It follows a simple verification that also satisfies the following two properties which is useful for some deduction in this paper: where and is a matrix whose elements are all equal to one with the same size as .
Finally, we recall some results about 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 4 (see ). Let be an matrix. There is a commutation matrix such that
Lemma 5 (see ). Let be an matrix and a matrix. There exist two commutation matrices and such that
3. An Iterative Algorithm for Solving Problem 1
In this section, we firstly construct an algorithm for solving Problem 1. Then we show that if the problem is consistent, the sequence generated by the introduced algorithm converges to a solution of Problem 1 within at most iteration steps.
Algorithm 6. Input matrices ,, , , , , , generalized reflection matrices , , .
If , or and , stop; otherwise .
Go to Step .
The following Lemmas show some properties of Algorithm 6.
Lemma 7. The sequences , , and generated by Algorithm 6 satisfy
Lemma 8. Assume that the sequences and are generated by Algorithm 6, and then and for .
Proof. Since and for , we only need to prove that and for .
Now we prove this conclusion by induction.
First, we prove When , from Lemma 7 we have From Lemma 7 and conclusion (24), we have Now assume that conclusion (23) holds for ; from Lemma 7, we have From Lemma 7 and conclusion (26), it can also be obtained that Therefore, by the principle of induction, conclusion (23) holds.
Next, assume that and for and . We will prove that In order to prove conclusion (28), we first prove that From Lemma 7 we have From Lemma 7 and conclusion (30), we have Also we have Repeating the previous process (32), we have Combining these two relations with (30) and (31), it implies that (28) holds. So, by the principle of induction, we know that Lemma 8 holds.
Remark 10. Lemma 9 implies that if there exists a positive integer such that and , then Problem 1 is not consistent. Therefore, the solvability of Problem 1 can be determined by Algorithm 6 in the absence of round-off errors.
Theorem 11. Assume that Problem 1 is consistent, and then using Algorithm 6 for any generalized ()-reflexive initial matrix , a solution of Problem 1 can be obtained within finite iteration steps in the absence of round-off errors.
Proof. If , from Lemma 9 we can obtain . Thus and can be computed by Algorithm 6. According to Lemma 8, we have , . Considering that the inner product space is -dimensional, we know that the set of is an orthogonal basis of the inner product space . Also from Lemma 8, we have for , which implies that ; that is, is a solution of Problem 1.
4. The Solution of Problem 2
In this section, we will prove that the sequence generated by Algorithm 6 converges to the least Frobenius norm generalized ()-reflexive solution of the quaternion matrix equation when an appropriate initial iterative matrix is chosen. Then, we solve Problem 2 by finding the least Frobenius norm generalized ()-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. If Problem 1 is consistent, then the following system of real linear equations is consistent:
Furthermore, if the solution sets of Problem 1 and (37) are denoted by and , respectively, then,
Proof. If Problem 1 is consistent, let be a solution of Problem 1, then we have and , which implies that is a solution of quaternion matrix equations
By (8), (9), and (12), we can derive that the quaternion matrix equations (39) is equivalent to the following real matrix equations:
By the operation properties of , , , and Lemma 4, it is not difficult to verify that matrix equations (40) is equivalent towhich implies that is a solution of (37). So system of real linear equations (37) is consistent.
From the previous procedure, the proof of (38) is trivial.
Proof. By (8), (9), (10), and (12) and Lemmas 4 and 5, we haveBy Lemma 12, is the least Frobenius norm solution of (37).
Noting (11), we derive from Lemma 13 that is the least Frobenius norm solution of Problem 1.
Theorem 16. When Problem 1 is consistent, if we choose the following matrix as the initial iteration matrix where is arbitrary, or especially, , then the solution , generated by Algorithm 6, is exactly the least Frobenius norm solution of Problem 1.
Now we consider Problem 2. When Problem 1 is consistent, the solution set of Problem 1 is not empty. For a given generalized ()-reflexive matrix , let , ; then the least Frobenius norm generalized -reflexive solution of the following quaternion matrix equation is exactly the solution of According to Theorem 16, if we choose the initial iteration matrix , where is arbitrary, or especially, choosing, the least Frobenius norm generalized -reflexive solution of (46) can be obtained. Then the solution of Problem 2 is .
In this section, we give two examples to illustrate the performance of the proposed algorithm.
Example 17. Consider the quaternion matrix equation with the matrices