Solving a System of Sylvester-like Quaternion Matrix Equations

: Using the ranks and Moore-Penrose inverses of involved matrices, in this paper we establish some necessary and sufﬁcient solvability conditions for a system of Sylvester-type quaternion matrix equations, and give an expression of the general solution to the system when it is solvable. As an application of the system, we consider a special symmetry solution, named the η -Hermitian solution, for a system of quaternion matrix equations. Moreover, we present an algorithm and a numerical example to verify the main results of this paper.


Introduction
In 1952, Roth [1] studied the following one-sided generalized Sylvester matrix equation for the first time which is widely used in system and control theory. Since then, many researches have paid attention to Sylvester-type matrix equations (e.g., [2][3][4][5]) because of their wide range of applications, such as in descriptor system control theory [6], neural networks [7], robust, feedback [8], graph theory [9] and other areas. For instance, Baksalary and Kala [10] established a necessary and sufficient condition for Equation (1) to have a solution and gave an expression of its general solution. In [11], Baksalary and Kala give a solvability condition for the equation Wang investigated Equation (2) over arbitrary regular rings with identity [12]. In 1843, the very famous mathematician Hamilton discovered the quaternion. It is well known that quaternion algebra, denoted by H, is an associative and non-commutative division algebra over the real number field R, where H = {a 0 + a 1 i + a 2 j + a 3 k|i 2 = j 2 = k 2 = ijk = −1, a 0 , a 1 , a 2 , a 3 ∈ R}.
In the last decade, the study of Sylvester-type matrix equations was extended to H (e.g., [21][22][23][24][25][26][27][28]). In 2012, Wang and He [29] presented the necessary and sufficient conditions for the Sylvester-type matrix equation to be consistent and derived the expression of its general solution, which can be easily generalized to H. For the Sylvester-type matrix equations with multiple variables and multiple equations, Wang [4] gave a solvability condition and the general solution to the system of Sylvester-type matrix equations Zhang [30] investigated the necessary and sufficient conditions for the solvability of the following system of Sylvester-like matrix equations and presented a formula of its general solution. We note that Equations (1)- (5) are the special cases of the following Sylvester-type quaternion matrix equations where A i , B i , C j , D k (i = 1, 8, j = 1, 3, k = 1, 5) are given matrices over H; X, Y, Z, V, W are unknown.
Motivated by the work mentioned above, in this paper we aim to investigate the solvability conditions and the general solutions to a more general system of a Sylvestertype quaternion matrix equation, Equation (6). In 2011, Took et al. [31] defined a special class of symmetric matrices, named η-Hermitian. For η ∈ {i, j, k}, a quaternion matrix A is called η-Hermitian if A = A η * , where A η * = −η A * η , A * is the conjugate and transpose matrix of A. It is well known that η-Hermitian matrices have some applications in linear modeling (e.g., [32][33][34]) and so on.
As an application of (6), we derive the solvability conditions and an expression of the η-Hermitian solution to the system of matrix equations where A i (i = 1, 2, 4, 6, 8), B 1 , B 2 , B 4 , D 5 are given matrices over H; X and Y are η-Hermitian matrices over H. We organize the rest of this article as follows: In Section 2, we introduce the basic knowledge of quaternions and Moore-Penrose inverse of a quaternion matrix, and review some matrix equations. In Section 3, we establish the solvability conditions for the system of (6) in terms of the Moore-Penrose inverses and the ranks of the coefficients' quaternion matrices in (6). In Section 4, we give an expression of the general solution to the system of (6), and illustrate the main results using a numerical example. In Section 5, we give some solvability conditions and an expression of the η-Hermitian solution to the system (7). Finally, we present a brief conclusion in Section 6 to end this paper.

Preliminaries
Let R and H m×n stand for the real number field and the set of all m × n matrix spaces over the quaternion algebra, respectively. The symbols r(A), A * , I and 0 are denoted by the rank of a given quaternion matrix A, the conjugate transpose of A, an identity matrix, and a zero matrix with appropriate sizes, respectively. The Moore-Penrose inverse of A ∈ H l×k is defined to be the unique matrix, denoted by A † , satisfying The following lemma was given by Marsaglia and Stynan [35], which is also available over H. Lemma 1 ([35]). Let A ∈ H m×n , B ∈ H m×k , C ∈ H l×n , D ∈ H j×k and E ∈ H l×i . Then,

Lemma 2 ([36]
). Let A 1 and C 1 be known matrices with feasible dimensions over H. Then, the matrix equation A 1 X = C 1 has a solution if and only if R A 1 C 1 = 0. In this case, its general solution is expressed as where T 1 is an arbitrary matrix of an appropriate size.

Lemma 3 ([36]
). Let B 1 and D 1 be known matrices with allowable dimensions over H. Then, the matrix equation YB 1 = D 1 has a solution if and only if D 1 L B 1 = 0. In this case, its general solution is where T 2 is an arbitrary matrix of an appropriate size.

Lemma 4 ([37]
). Let A 1 , B 1 , C 1 and C 2 be the given matrices. Then, the system of matrix equations is consistent if and only if In this case, its general solution is where T 3 is an arbitrary matrix of an appropriate size.
where U 1 , U 2 and U 3 are arbitrary matrices with appropriate sizes over H.

Lemma 6 ([38]). Consider the following matrix equation over H
where A i , B i (i = 1, 4), B are given and the others are unknown. Let Then, the following statements are equivalent: In this case, the general solution to Equation (8) can be expressed as represents any matrix with appropriate dimensions over H, where U 11 , U 12 , U 21 , U 31 , U 32 , U 33 , U 41 and U 42 are any matrix with appropriate dimensions over H.

Solvability Conditions to the System (6)
The goal of this section is to give the necessary and sufficient conditions for the existence of a solution to system (6).
Then, the following statements are equivalent: (1) System (6) has a solution. (2) and (3) (22) holds and Proof The proof is divided into three parts: • Firstly, we divide the system (6) into the following: and consider the solvability conditions and the general solution to the system of matrices of Equation (36). For more information, see Step 1. • Secondly, substituting the W and Z obtained in the first step into Equation (37) yields where A 11 , B 11 and C 11 are defined by (9); T 3 and T 4 are unknowns. For more information, see Step 2. • Finally, by substituting the X, Y, Z, and V obtained from the above two steps into Equation (38), we obtain a matrix equation with the following form where A ii , B ii (i = 2, 5) and C 22 are given by (9)-(12); T 1 , T 2 , T 5 , U 1 and U 3 are unknowns. For more information, see Step 3.
We can obtain the results from the following steps: First, we consider the solvability conditions and the expression of the general solutions to the system of the matrix Equation (36).
Step 1. It follows from Lemmas 2-4 that system (36) has a solution if and only if (22) holds and In this case, the general solution to system (36) can be written as where T i (i = 1, 5) are arbitrary matrices over H with appropriate sizes.

has a solution if and only if
In this case, the general solution to Equation (39) can be expressed as where U 1 , U 2 and U 3 are any matrix with appropriate sizes over H.
Step 3. By substituting X, Y, V in (42) and Z in (46) into (38), we obtain Equation (40). By using Lemma 6, Equation (40) is consistent if and only if namely, In this case, the general solution to Equation (40) can be expressed as 1,8) are any matrix with suitable dimensions over H, where U 11 , U 12 , U 21 , U 31 , U 32 , U 33 , U 41 and U 42 are any matrix with suitable dimensions over H.
To sum up, the system of matrices of Equation (6)  (2) ⇔ (3) We divide it into three parts to prove its equivalence. (24) and (25) hold. According to Lemma 1, it is easy to show that (41) holds if and only if (24) and (25) hold.
Note that are the special solution to the equations respectively. Then, we have that It follows from Lemma 1 and elementary operations to (47) that Similarly, we can prove that R P 2 E 2 = 0 ⇔ (28), R P 3 E 3 = 0 ⇔ (29), R P 4 E 4 = 0 ⇔ (30) and E i L Q i = 0 (i = 1, 4) hold if and only if (31) to (34) hold, respectively. Next, we show that R M 22 EL M 33 = 0 ⇔ (35). According to Lemma 1 and elementary operations, we have that

The General Solution to the System (6)
In this section, we give an expression for the general solution of Equation (6) by using the Moore-Penrose inverse. According to the proof of Theorem 1, we obtain the following theorem: The general solution to system (6) can be expressed as follows when the solvability conditions are met: 1,8) are arbitrary matrices with appropriate sizes.
where U 11 , U 12 , U 21 , U 31 , U 32 , U 33 , U 41 and U 42 are arbitrary matrices over H of appropriate sizes.
Next, we discuss the special cases of the system of matrices of Equation (6). Letting A 3 , B 3 , A 5 , B 5 and D 4 vanish yields the following: Corollary 1. Suppose that A i , B i , C j , D j (i = 1, 4, j = 1, 5) and E 1 are given, denote Then, the following statements are equivalent: In this case, the general solution to system (5) can be expressed as where V i , W j (i = 1, 5, j = 1, 3) are arbitrary matrices over H with appropriate sizes. Remark 1. The above corollary is from the important findings of [30].
In this case, the general solution to system (4) can be expressed as where W 1 , W 2 and W 3 are arbitrary matrices over H of appropriate sizes.

Remark 2.
The above corollary is from the vital investigation of [4].
Finally, we give Algorithm 1 and an example to illustrate the main results of this paper.

Example 1. Consider the matrix of Equation
Computing directly yields r(A i B i ) = r(A i ) = 2, r C j D j = r C j = 2 (i = 1, 4, j = 1, 3), All rank equations hold. Thus, according to Theorem 1, the system of matrix equations has a solution, and the general solution to the system can be expressed as

The General Solution to the System (7) with η-Hermicity
As an application of the results of system (6), we study the necessary and sufficient conditions for system (7) to have a solution involving η-Hermicity and derive a formula of its general solution, where X, Y are η-Hermitian matrices.