A Sylvester-type matrix equation over the Hamilton quaternions with an application

We derive the solvability conditions and a formula of a general solution to a Sylvester-type matrix equation over Hamilton quaternions. As an application, we investigate the necessary and sufficient conditions for the solvability of the quaternion matrix equation, which involves $\eta$-Hermicity. We also provide an algorithm with a numerical example to illustrate the main results of this paper.


Introduction
Let R stand for the real number field and H is called the Hamilton quaternion algebra, which is a non-commutative division ring.Hamilton quaternions and Hermitian quaternion matrices have been utilized in statistics of quaternion random signals [1], quaternion matrix optimization problems [2], signal and color image processing, face recognition [3,4], and so on.
Sylvester and Sylvester-type matrix equations have a large number of applications in different disciplines and fields.For example, the Sylvester matrix equation and the Sylvester-type matrix equation have been applied in singular system control [5], system design [6], perturbation theory [7], sensitivity analysis [8], H α -optimal control [9], linear descriptor systems [10], and control theory [11].
Roth [12] gave the Sylvester-type matrix Equation (1.2) for the first time over the polynomial integral domain.Baksalary and Kala [13] established the solvability conditions for Equation (1.2) and gave an expression of its general solution.In addition, Baksalary and Kala [14] derived the necessary and sufficient conditions for a two-sided Sylvester-type matrix equation to be consistent.Özgüler [15] studied (1.3) over a principal ideal domain.Wang [16] investigated (1.3) over an arbitrary regular ring with an identity element.Due to the wide applications of quaternions, the investigations on Sylvester-type matrix equations have been extended to H in the last decade (see, e.g., [21,22,20,17,18,19,24,23]).They are applied for signal processing, color-image processing, and maximal invariant semidefinite or neutral subspaces, etc. (see, e.g., [25,26,27,28]).For instance, the general solution to Sylvester-type matrix Equation (1.2) can be used in color-image processing.He [29] derived the matrix Equation (1.2) as an essential finding.Roman [25] established the necessary and sufficient conditions for Equation (1.1) to have a solution.Kychei [30] investigated Cramer's rules to drive the necessary and sufficient conditions for Equation (1.3) to be solvable.As an extension of Equations (1.2) and (1.3), Wang and He [31] gave the solvability conditions and the general solution to the Sylvester-type matrix equation over the complex number field C, which can be generalized to H and applicable in some Sylvestertype matrix equations over H (see, e.g., [32,29]).We know that in system and control theory, the more unknown matrices that a matrix equation has, the wider its application will be.Consequently, for the sake of developing theoretical studies and the applications mentioned above of Sylvester-type matrix equation and their generalizations, in this paper, we aim to establish some necessary and sufficient conditions for the Sylvester-type matrix equation to have a solution in terms of the rank equalities and Moore-Penrose inverses of some coefficient quaternion matrices in Equation (1.5) over H.We derive a formula of its general solution when it is solvable.It is clear that Equation (1.5) provides a proper generalization of Equation (1.4), and we carry out an algorithm with a numerical example to calculate the general solution of Equation (1.5).As a special case of Equation (1.5), we also obtain the solvability conditions and the general solution for the two-sided Sylvester-type matrix equation To the best of our knowledge, so far, there has been little information on the solvability conditions and an expression of the general solution to Equation (1.6) by using generalized inverses.
As usual, we use A * to denote the conjugate transpose of A. Recall that a quaternion matrix A, for η ∈ {i, j, k}, is said to be η-Hermitian if A = A η * , where A η * = −ηA * η [33].For more properties and information on η * -quaternion matrices, we refer to [33].We know that η-Hermitian matrices have some applications in linear modeling and statistics of quaternion random signals [1,33].As an application of Equation (1.5), we establish some necessary and sufficient conditions for the quaternion matrix equation to be consistent.Moreover, we derive a formula of the general solution to Equation (1.7) where ) over H.The rest of this paper is organized as follows.In Section 2, we review some definitions and lemmas.In Section 3, we establish some necessary and sufficient conditions for Equation (1.5) to have a solution.In addition, we give an expression of its general solution to Equation (1.5) when it is solvable.In Section 4, as an application of Equation (1.5), we consider some solvability conditions and the general solution to Equation (1.7), where Y i = Y η * i (i = 1, 3).Finally, we give a brief conclusion to the paper in Section 5.

Preliminaries
Throughout this paper, H m×n stands for the space of all m × n matrices over H.The symbol r(A) denotes the rank of A. I and 0 represent an identity matrix and a zero matrix of appropriate sizes, respectively.In general, A † stands for the Moore-Penrose inverse of A ∈ H l×k , which is defined as the solution of The following lemma is due to Marsaglia and Styan [34], which can be generalized to H.

Lemma 2.1 ([34]
).Let A ∈ H m×n , B ∈ H m×k , C ∈ H l×n , D ∈ H j×k and E ∈ H l×i be given.Then, we have the following rank equality:
Then, the following statements are equivalent: (1) The system has a solution. (2) In this case, the general solution to Equation (1.2) can be expressed as , where U 1 , U 2 , and U 3 are arbitrary matrices with appropriate sizes.
Then, the following statements are equivalent: (1) Equation (1.4) has a solution. (2) In this case, the general solution to Equation (1.4) can be expressed as where T 1 , ..., T 8 are arbitrary matrices with appropriate sizes over H.

Some Solvability Conditions and a Formula of the General Solution
In this section, we establish the solvability conditions and a formula of the general solution to Equation (1.5).We begin with the following lemma, which is used to reach the main results of this paper.
Then, the following statements are equivalent: (1) The system (2.1) is consistent. (2) (3) (4) In this case, the general solution to system (2.1) can be expressed as where V 1 , V 2 , and V 3 are arbitrary matrices with appropriate sizes over H.
Proof.(1) ⇔ (2) It follows from Lemma 2.3 that , where G and F are given in Lemma 2.3.
(1) ⇒ (3) If the system (2.1) has a solution, then there exists a solution X 0 such that It is easy to show that ( We now prove that X 1 in (3.2) is the general solution of the system (2.1).We prove it in two steps.We show that X 1 is a solution of system (2.1) in Step 1.In Step 2, if the system (2.1) is consistent, then the general solution to system (2.1) can be expressed as (3.2).
Step 1.In this step, we show that X 1 is a solution of system (2.1).Substituting X 1 in (3.2) into the system (2.1) yields where Step 2. In this step, we show that the general solution to the system (2.1) can be expressed as (3.2).It is sufficient to show that for an arbitrary solution, say, X 01 of (2.1), X 01 can be expressed in form (3.2).Put Hence, X 01 can be expressed as (3.2).To sum up, (3.2) is the general solution of the system (2.1).Now, we give the fundamental theorem of this paper.Theorem 3.2.Let A i , B i , and B (i = 1, 4) be given quaternion matrices with appropriate sizes over H. Set (3.7) Then, the following statements are equivalent: Proof.
(1) ⇔ (2) Equation (1.5) can be written as i.e., where A ii , B ii (i = 1, 3), and T 1 are defined by (3.4).In addition, when Equation (3.18) has a solution, we get the following: i.e., where C i , D i , E i (i = 1, 4) are defined by (3.5).When Equation (3.21) is solvable, we have that where Thus, according to Lemma 3.1, we have that the system (3.23) is consistent if and only if In this case, the general solution to system (3.23) can be expressed as where F 1 , F 2 are defined by (3.7) and V i , W i (i = 1, 3) are any matrices with the appropriate dimensions over H. Thus, system (3.23) has a solution if and only if (3.25) holds and there exist V i , W i (i = 1, 3) such that (3.26) equals to (3.27), namely where F , C ii and D ii (i = 1, 3) are defined by (3.6).It follows from Lemma 2.5 that Equation (3.28) has a solution if and only if In this case, the general solution to Equation (3.28) can be expressed as The specific proof is as follows.
Firstly, we prove that

and elementary transformations that
It follows from Lemma 3.1 and (3.24) that R F11 G 1 = 0 and R F22 G 2 = 0 are equivalent to According to Lemma 2.1, we have that According to Lemma 2.5 and (3.24), we have that (3.29) respectively.By Lemma 2.1, we have that   (2) ⇔ (3) We prove the equivalence in two parts.In the first part, we want to show that (3.30) to (3.37) are equivalent to (3.9) to (3.16), respectively.In the second part, we want to show that (3.47) is equivalent to (3.17).
Part 1.We want to show that (3.30) to (3.37) are equivalent to (3.9) to (3.16), respectively.It follows from Lemma 2.1 and elementary operations to (3.30) that Similarly, we can show that (3.31) to (3.33) are equivalent to (3.10) to (3.12), respectively.Now, we turn to prove that (3.34) is equivalent to (3.11).It follows from the Lemma 2.1 and elementary transformations that 21).
Next, we give the formula of general solution to matrix Equation (1.5) by using Moore-Penrose.According to Theorem 3.2, we get the following theorem: Theorem 3.3.Let matrix Equation (1.5) be solvable.Then, the general solution to matrix Equation (1.5) can be expressed as ) are arbitrary matrices with appropriate sizes over H,

Algorithm with a Numerical Example
In this section, we give Algorithm 3.4 with a numerical example to illustrate the main results.(1) Input the quaternion matrices A i , B i (i = 1, 4) and B with conformable shapes.

Computation directly yields
All rank equalities in (3.9) to (3.17) hold.Hence, according to Theorem 3.2, Equation (1.5) has a solution.Moreover, by Theorem 3.3, we have that Remark 3.6.Chu et al. gave potential applications of the maximal and minimal ranks in the discipline of control theory (e.g., [36,37,38]).We may consider the rank bounds of the general solution of Equation (1.5).

The General Solution to equation with η-Hermicity
In this section, as an application of (1.5), we establish some necessary and sufficient conditions for quaternion matrix Equation (1.7) to have a solution and derive a formula of its general solution involving η-Hermicity.Theorem 4.1.Let A i (i = 1, 4) and B be given matrices with suitable sizes over H, B = B η * .Set Then, the following statements are equivalent: In this case, the general solution to Equation (1.7) can be expressed as is a solution of (4.1).Conversely, if (4.1) has a solution, say It is easy to show that (1.7) has a solution Letting A 1 and B 1 vanish in Theorem 3.2, it yields to the following result.
Corollary 4.2.Let A ii , B ii (i = 1, 3), and T 1 be given matrices with appropriate sizes over H. Set Then, the following statements are equivalent: (1) Equation (1.6) is consistent.
(2) R C i E i = 0, E i L D i = 0 (i = 1, 4), R E 22 EL E 33 = 0.In this case, the general solution to Equation (1.6) can be expressed as

Conclusions
We have established the solvability conditions and an exact formula of a general solution to quaternion matrix Equation (1.5).As an application of Equation (1.5), we also have established some necessary and sufficient conditions for Equation (1.7) to have a solution and derived a formula of its general solution involving η-Hermicity.The quaternion matrix Equation (1.5) plays a key role in studying the solvability conditions and general solutions of other types of matrix equations.For example, we can use the results on Equation (1.5) to investigate the solvability conditions and the general solution of the following system of quaternion matrix equations where Y 1 , Y 2 and Y 3 are unknown quaternion matrices and the others are given.
It is worth mentioning that the main results of (1.5) are available over not only R and C but also any division ring.Moreover, inspired by [39], we can investigate Equation (1.5) in tensor form.

Lemma 3 . 1 .
Let A 11 , B 11 , C 11 , and D 11 be given matrices with suitable sizes over H, A 11 L A 22 = 0 and R B 11 B 22 = 0. Set

Algorithm 3 . 4 .
Algorithm for computing the general solution of Equation(1.5)
By Lemma 2.4, Equation (3.18) is consistent if and only if there exist Y i .18) Clearly, Equation (1.5) is solvable if and only if Equation (3.18) has a solution.
3) are any matrices with appropriate dimensions over H. Hence, Equation (3.18) has a solution if and only if there exist Y i (i = 1, 3) in Equation (3.18) such that Equation (3.20) is solvable.According to Equation (3.20), we have that A 11 Y 1 B 11 + A 22 Y 2 B 22 = T 1 − A 33 Y 3 B 33 .
(3.21)Hence, Equation(3.20)is consistent if and only if Equation (3.21) is solvable.It follows from Lemma 2.5 that Equation (3.21) has a solution if and only if there exists Y 3 in Equation (3.21) such that 1 , T 1 are defined by (3.4), T = T 1 − A 33 Y 3 B 33 and U j (j = 4, 8) are any matrices with the appropriate dimensions over H.It is easy to infer that 12 , U 21 , U 31 , U 32 , U 33 , U 41 , and U 42 are arbitrary matrices with appropriate sizes over H, m is the column number of A 4 and n is the row number of B 4 .