Three special kinds of least squares solutions for the quaternion generalized

In this paper, we propose an efficient method for some special solutions of the quaternion matrix equation AXB + CYD = E. By integrating real representation of a quaternion matrix with H-representation, we investigate the minimal norm least squares solution of the previous quaternion matrix equation over different constrained matrices and obtain their expressions. In this way, we first apply H-representation to solve quaternion matrix equation with special structure, which not only broadens the application scope of H-representation, but further expands the research idea of solving quaternion matrix equation. The algorithms only include real operations. Consequently, it is very simple and convenient, and it can be applied to all kinds of quaternion matrix equation with similar problems. The numerical example is provided to illustrate the feasibility of our algorithms.


Introduction
In this paper, we adopt the following notations. R represents the real number field; R n stands for the set of all real column vectors with order n; R m×n stands for the set of all m×n real matrices. C represents the complex number field; C n stands for the set of all complex column vectors with order n; C m×n stands for the set of all m × n complex matrices. The sets U n , U −n , V n , W n represent the set of all n × n tridiagonal symmetric matrices, tridiagonal skew-symmetric matrices, Brownian matrices, Generalized Rotation matrices, respectively. Q stands for the quaternion skew-field; Q n stands for the set of all quaternion column vectors with order n; Q m×n represents the set of all m × n quaternion matrices; HTQ n×n , AHTQ n×n , BQ n×n , MQ n×n represent the set of all n × n quaternion tridiagonal Hermitian matrices, quaternion tridiagonal anti-Hermitian matrices, quaternion Brownian matrices, quaternion Generalized Rotation matrices, respectively. I n represents the unit matrix with order n. For matrix A, A T , A H , A † stand for the transpose, the conjugate transpose, Moore-Penrose inverse of matrix A, respectively. ⊗ represents the Kronecker product of matrices. · represents the Frobenius norm of a matrix or Euclidean norm of a vector. For C = (c 1 , c 2 , . . . , c n ) ∈ R m×n , vec(C) means the vector operator, i.e., vec(C) = (c T 1 , c T 2 , . . . , c T k ) T . Matrix equations can be encountered in many areas, such as system theory, control theory, stability analysis, some fields of pure and applied mathematics and so on [1][2][3]. With the rapid development of these fields, more and more scholars are interested in matrix equations and have obtained many valuable results [4][5][6]. Now, we turn our attention to quaternion matrix equation. Quaternion matrix equations and their least squares solutions are widely applied in many fields, such as computer science, quantum mechanics, control theory, field theory and so on [7][8][9]. Therefore, many people are engaged in studying theoretical properties and numerical computations of quaternion matrix equations. By means of complex representation, Jiang et al. studied algebraic algorithm for quaternion least squares problem [10] and quaternion eigenvalue problem [11]; Yuan et al. studied the quaternion least squares problems for the quaternion matrix equations AXB + CXD = E [12], X − AXB = C [13]. By applying the real representation of quaternion matrices, Wang et al. proposed an iterative method for solving the quaternion least squares problem [14].
Consider the generalized Sylvester matrix equation If B and C are identity matrices, then the matrix Eq (1.1) reduces to the well-known Sylvester matrix Eq [15]. If C and D are identity matrices, then the matrix Eq (1.1) reduces to the wellknown Stein matrix equation. It has extensive application value in robust control, feedback control, pole assignment design, neural network and so on [16][17][18]. There are many important results about their solutions, for example, [19] and [20] considered the solvability condition for the complex and real matrix Eq (1.1), respectively. For the quaternion matrix Eq (1.1), [21] derived necessary and sufficient conditions for the existence of a solution or a unique solution using the method of complex representation of quaternion matrices; [12,22] studied η-Hermitian and η-anti-Hermitian solutions to the quaternion matrix equations AXB + CXD = E, AXB + CY D = E, respectively; [23] obtained the expression of solutions of a system of quaternion matrix equations including η-Hermicity. Also, it is worth noting that a number of important results on Sylvester operators have been obtained in recent years. For instance, [24] studied some features of slice semi-regular functions SξM(Ω) on a circular domain Ω and verified the equivalence of slice semi-regular functions via Sylvester operators; [25] applied the existing results to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. The name of Sylvester operator is due to the fact that, when dealing with matrices, equation S f,g (χ) = b is usually called Sylvester equation. In the most common use, Sylvester equations are special matrices equations, introduced by Sylvester himself [26], which are used in several subjects, including similarity, commutativity, control theory and differential equation [27]. In the quaternionic setting, such equations were studied with different purposes. In this paper, we consider the least squares problem with different constrains for quaternion matrix Eq (1.1) based on the real representation of quaternion matrices together with the H-representation method, which is able to transform a matrix-valued equation into a standard vector-valued equation with independent coordinates. The related problems are described as follows. Problem 1. Let A ∈ Q m×p , B ∈ Q p×n , C ∈ Q m×q , D ∈ Q q×n , E ∈ Q m×n , and The solution (X H , Y A ) in Problem 1 is called the minimal norm least squares tridiagonal mixed solution.
The solution (X B , Y B ) in Problem 2 is called the minimal norm least squares Brownian solution.
The solution (X M , Y M ) in Problem 3 is called the minimal norm least squares Rotation solution.
The remaining content of this paper is organized as follows. In Section 2, we study and recall some preliminary results with regard to the real representation of a quaternion matrix, and then introduce some matrix sets with special structures. In Section 3, we give the concept of H-representation and subsequently study its properties. In Section 4, on the basis of the real representation matrix of a quaternion matrix and H-representation of matrices with special structures, operational properties, the properties of Frobenius norm and Moore-Penrose generalized inverse, we can convert Problems 1-3 into the corresponding problems of the real matrix equation over free variables, and then the unique solution (X, Y) and expressions for special solution are established. In addition, the necessary and sufficient conditions for the quaternion matrix equation to have solution with special structure are included as corollaries. In Section 5, we provide numerical algorithms for solving Problems 1-3 by the results obtained in Section 4, and afterwards we present a numerical example to verify the feasibility of our proposed method. Finally, in Section 6, we put some conclusions.

Basic definitions
We start by recalling the usual Kronecker product. Definition 2.1. For any two matrices A = (a i j ) ∈ R m×n , B ∈ R p×q , the Kronecker product of A and B is defined as We now turn to recall the standard representation of a quaternion.
Definition 2.2. A quaternion q ∈ Q is represented as where q 1 , q 2 , q 3 , q 4 ∈ R, and three imaginary units i, j, k satisfy We recall a standard norm in this setting.
Definition 2.4. [28] The Frobenius norm of A = A 1 + A 2 i + A 3 j + A 4 k is defined as Definition 2.5. [28] For A = A 1 + A 2 i + A 3 j + A 4 k ∈ Q m×p , its real representation matrix − → A is defined as follows: According to the matrix blocks of the real representation matrix For the sake of convenience, we use − → A c to represent the first column block of − → A, i.e., Next, we investigate some properties of − → A c , which will be used in the sequel.
We will now present four special matrix sets. We refer to [29] for all the concepts involved in this paper.
Definition 2.6. A tridiagonal symmetric matrix P ∈ U n is an n × n matrix with the following form Definition 2.7. A tridiagonal skew-symmetric matrix P ∈ U −n is an n × n matrix with the following form Specifically, the form is as follows: Definition 2.9. A Generalized Rotation matrix P ∈ W n is an n × n matrix with the following form

H-Representation and its properties
In this section, we will briefly introduce the notion of H-representation and the related properties. Besides, we will also analyze the structure of four special matrix sets mentioned above by means of H-representation and present their H-representation matrices.   1) The H-representation of ψ(X) for X ∈ X is not unique because of the fact that the matrix H may be different owing to the basis choices of X. Apparently, when the basis of X is fixed, the H-representation matrix H, as well as X, is uniquely determined; 2) ψ is used here only as a function name for the convenience of defining its inverse in the sequel.
In what follows, based on the special matrix sets defined in Section 2, we present some simple examples to elucidate Definition 3.1. 3×3 , then dim(X) = 5. If we select the following basis of X It is then easy to compute Example 3.2. Let X = V 3 , then dim(X) = 7. If we select the following basis of X Then it is easy to compute Example 3.3. Let X = W 3 , then dim(X) = 3. If we select the following basis of X Then we can obtian In this paper, we are interested in the H-representation for X = U n /U −n /V n /W n . For X = U n , we select a standard basis throughout this paper as where E i j = (e lk ) n×n with e i j = e ji = 1 and the other entries are zeros. Clearly, for the above given bases, if X = U n , then for any X n = (x i j ) n×n ∈ X, we havẽ X n = (x 11 , x 21 , x 22 , x 32 , . . . , x n−1,n−1 , x n,n−1 , x nn ) T . (3.1) For X = U −n , we select a standard basis throughout this paper as where E i j = (e lk ) n×n with e i j = −1, e ji = 1 and the other entries are all zeros. For the above given bases, if X = U −n , then for any X −n = (x i j ) n×n ∈ X, we havẽ For the convenience of description, the following Z and W both represent X. Similarly, for Z = V n , we select a standard basis as where F ii = ( f lk ) n×n with f ii = 1, and F i j , F ji are n × n matrices with f i j = 1, f ji = 1 for ∀ j > i, respectively, and the other entries are zeros. Based on above bases, for any Z n = (z i j ) n×n ∈ Z, we havẽ Z n = (z 11 , z 21 , z 12 , z 22 , z 32 , z 23 , . . . , z n−1,n−1 , z n,n−1 , z n−1,n , z n,n ) T . (3.3) Likewise, for W = W n , we select a standard basis as with D 11 = I n and D i1 = Based on above bases, if W = W n , then for any W n = (w i j ) n×n ∈ W, we havẽ As soon as a standard basis is given,X n ,X −n ,Z n , andW n are uniquely determined by X n , X −n , Z n and W n , respectively. Thus, we can state the following definition: Definition 3.2. We define σ 1 : X n = (x i j ) n×n ∈ U n →X n , whereX n is defined in (3.1), σ 2 : X −n = (x i j ) n×n ∈ U −n →X −n , whereX −n is defined in (3.2), τ : Z n = (z i j ) n×n ∈ V n →Z n , whereZ n is defined in (3.3), and φ : W n = (w i j ) n×n ∈ W n →W n , whereW n is defined in (3.4).
Remark 3.2. ψ, σ 1 , σ 2 , τ and φ are obviously invertible in the sense that for any (ν, It should be noted that ψ, σ 1 , σ 2 , τ and φ are defined on different domains. Note that ψ(X n ) is a column vector formed by all elements of X n , while σ 1 (X n ), σ 2 (X −n ), τ(Z n ), φ(W n ) are column vectors formed by different nonzero elements of X n , X −n , Z n , W n , respectively. For clarity, we denote the H-matrix in H-representation corresponding to X = U n by H 1 n , the H-matrix in H-representation corresponding to X = U −n by H 1 −n , the H-matrix in H-representation corresponding to X = V n by H 2 n , the H-matrix in H-representation corresponding to X = W n by H 3 n . The following corollary is obvious from Definitions 3.1 and 3.2.

The solutions for Problems 1-3
In this section, we solve Problems 1-3 via the real representation of quaternion matrices and Hrepresentation. We first convert above least squares problems into corresponding problems of real matrix equation by using the real representation, then in order to reduce the size of original problems, we remove the redundancy and extract effective elements through H-representation. Finally, we obtain the solutions of Problems 1-3. [31] The least squares solutions of the linear system of equations Ax = b, with A ∈ R m×n and b ∈ R m can be represented as where, y ∈ R n is an arbitrary vector. The minimal norm least squares solution of the linear system of equations Ax = b is A † b. When Ax = b is compatible, the general solution can be represented as where, y ∈ R n is an arbitrary vector. Ax = b has a unique solution if and only if rank(A) = n.
In this case, the unique solution is x = A † b.
Theorem 4.3. Suppose A ∈ Q m×p , B ∈ Q p×n , C ∈ Q m×q , D ∈ Q q×n , E ∈ Q m×n be given. Hence the set T L of Problem 1 can be expressed as

1)
And then, the minimal norm least squares solution (X H , Y A ) of Problem 1 satisfies where J , K have the same structure with the J, K, respectively. Since 2, 3, 4), in light of Corollary 3.1, we can derive For the convenience of what follows, let us denote Then we can obtain Thus AXB + CY D − E assume its minimum value For the real matrix equation by Lemma 4.1, its least squares solution can be represented as Corollary 4.4. Let A ∈ Q m×p , B ∈ Q p×n , C ∈ Q m×q , D ∈ Q q×n , E ∈ Q m×n be given, and G 1 be defined as in (4.3). Then (1.1) has a solution X ∈ HTQ p×p , Y ∈ AHTQ q×q , if and only if If (4.5) holds, the solutions set of (1.1) can be represented as Proof. According to the proof of Theorem 4.3, Lemma 4.2 and the definition of Moore-Penrose generalized inverse, we have thus (1.1) has tridiagonal mixed solution (X, Y) if and only if So we get the formula in (4.5). Under the condition that (4.5) is established, the solution (X, Y) of (1.1) satisfies Moreover, the solution (X, Y) of (1.1) satisfies Similarly, we can deduce (4.6) by multiplying both sides of the above equation by the matrix H 1 . At the same time, the unique tridiagonal mixed solution (4.7) can also be obtained.
In what follows, we concentrate on Problems 2 and 3. By Theorem 3.1, for (X, Y) with special structure, we can give its H-representation matrix, which will help us extract effective elements and reduce the complexity of operations. Based on the above ideas, the following conclusions can be easily obtained.
Theorem 4.5. Suppose A ∈ Q m×p , B ∈ Q p×n , C ∈ Q m×q , D ∈ Q q×n , E ∈ Q m×n . Hence the set B L of Problem 2 can be represented as

8)
and then, the minimal norm least squares solution (X B , Y B ) of Problem 2 satisfies where Corollary 4.6. Let A ∈ Q m×p , B ∈ Q p×n , C ∈ Q m×q , D ∈ Q q×n , E ∈ Q m×n be given. G 2 is defined in Theorem 4.5. Then (1.1) has a solution X ∈ BQ p×p , Y ∈ BQ q×q , if and only if If (4.10) holds, the Brownian solutions set of (1.1) can be represented as Similar to Theorem 4.5, when X ∈ MQ p×p , Y ∈ MQ q×q , we can give H 3 , G 3 for studying Problem 3.
Theorem 4.7. Suppose A ∈ Q m×p , B ∈ Q p×n , C ∈ Q m×q , D ∈ Q q×n , E ∈ Q m×n . Hence the set M L of Problem 3 can be expressed as 12) and then, the minimal norm least squares solution (X M , Y M ) of Problem 3 satisfies where Corollary 4.8. Let A ∈ Q m×p , B ∈ Q p×n , C ∈ Q m×q , D ∈ Q q×n , E ∈ Q m×n be given. G 3 is defined in Theorem 4.7. Then (1.1) has a solution X ∈ MQ p×p , Y ∈ MQ q×q , if and only if (4.14) If (4.14) holds, the Rotation solutions set of (1.1) can be expressed as

Algorithms and numerical experiments
In this section, on the basis of the discussions in Section 4, we propose the algorithms of solving Problems 1-3, and then give a numerical example to prove the feasibility of the proposed algorithms.
solution of the quaternion matrix Eq (1.1) over X ∈ HTQ p×p /BQ p×p /MQ p×p and Y ∈ AHTQ q×q / BQ q×q /MQ q×q . By Algorithms 5.1-5.3, for s ∈ {1, 2, 3}, we compute (X s , Y s ). Let m = p = n = q = 2K and the error ε = log 10 ( (X s , Y s ) − (X s , Y s ) ). The relation between K and the error ε is shown in Figure 1. According to Figure 1, we obtain that the errors ε are all no more than -9 for s ∈ {1, 2, 3}, which confirms the difference between the numerical solution and the exact solution is tiny. In other words, these three figures of Figure 1 are very similar, which is consistent with the actual situation. Therefore, our proposed algorithms are very feasible.

Conclusions
In this paper, by combining the real representation of quaternion matrices with H-representation, we convert the least squares problem of the quaternion matrix Eq (1.1) into a corresponding problem of the real matrix equation over free variables. Then we derive the expression of the minimal norm least squares solution for the quaternion matrix Eq (1.1) over different constrained matrices as in Problems 1-3. Our resulting expressions are expressed only by real matrices, and the algorithms only involve real operations. The final example shows that our proposed method is feasible and convenient to analyze such a matrix problem with special structures.