Consistency and General Solutions to Some Sylvester-like Quaternion Matrix Equations

: This article makes use of simultaneous decomposition of four quaternion matrixes to inves-tigate some Sylvester-like quaternion matrix equation systems. We present some useful necessary and sufﬁcient conditions for the consistency of the system of quaternion matrix equations in terms of the equivalence form and block matrixes. We also derive the general solution to the system according to the partition of the coefﬁcient matrixes. As an application of the system, we present some practical necessary and sufﬁcient conditions for the consistency of a φ -Hermitian solution to the system of quaternion matrix equations in terms of the equivalence form and block matrixes. We also provide the general φ -Hermitian solution to the system when the equation system is consistent. Moreover, we present some numerical examples to illustrate the availability of the results of this paper.


Introduction
It is well known that quaternion and quaternion matrixes have a great range of applications in color image processing ( [1,2]), quantum physics ( [3,4]) and signal processing ( [5]), etc. Quaternion matrix equations play an important role in mathematics and other domains, such as theoretical mechanics, optics, digital image processing, aerospace technology, etc. There are a great deal of papers from different aspects investigating quaternion matrix equation, such as solvability conditions, general solutions, extreme rank of solutions, minimum norm least squares solutions and their applications (e.g., [1,2,4,).
Kyrchei investigated two-sided generalized Sylvester matrix equation A 1 X 1 B 1 + A 2 X 2 B 2 = C over quaternion refer to Cramer's Rules and Moore-Penrose inverse [13]. Liu et al. [17] considered the existence and uniqueness of solutions of some quaternion matrix equations with three new defined real representions. Kyrchei deduced the minimum norm least squares solutions of some quaternion matrix equations [11]. Futorny et al. [32] utilized Roth's solvability criteria derived some solvability conditions for a few quaternion equations in terms of the equivalence of quaternion matrix. Xu et al. [28] provided some useful necessary and sufficient consitions and general solution to a constrained system of Sylvester-like matrix equations over the quaternion in terms of ranks and Moore-Penrose inverse of the coefficient matrixes. Mehany et al. [18] showed some solvaility conditions and general solution for three symmetrical coupled Sylvester-type matrix equations systems over quaternion from the aspects of ranks and generalized inverse. Liu et al. [15] gave solvability conditions and general solution to a Sylvester-like quaternion matrix equation A 1 X + YB 1 + A 2 Z 1 B 2 + A 3 Z 2 B 3 + A 4 Z 3 B 4 = C with five unknown matrixes. Wang et al. [27] formed some practical solvability conditions for a Sylvesterlike matrix equation system and gave an application involving an η-Hermitian solution. Dmytryshyn et al. [33] showed some solvability conditions for a class of quaternion matrix equations. He [10] gave some useful necessary and sufficient conditions for the existence of 1, 1, − 1 [34]. The rank of a quaternion matrix A is defined to be the right linear independent columns maximum number of A and is represented by symbol r(A). Note that A and PAQ have the same rank for any invertible matrixes P and Q with appropriate sizes. The identity matrix and zero matrix with appropriate size are labeled by I and 0, respectively.

Solvability Conditions to the System (1)
In this section, we deduce the consistency conditions of the system of quaternion matrix Equation (1), where A, B, C, D, E, F, G, H, Ω and Φ are given quaternion matrixes, through utilizing the equivalence canonical form of four matrixes. According to the matrix product order principle, we can observe that coefficient matrixes A, B have the same number of columns, A, C, D have the same number of rows, E, F, G have the same number of columns, and E, H have the same number of rows, so they can be formed in the following two quaternion matrix arrays where To solve the equation system, we first give the simultaneous decomposition and equivalence canonical form of four quaternion matrixes in the following lemma. Lemma 1. ( [9,26]) Given A ∈ H m×p , B ∈ H t×p , C ∈ H m×q , and D ∈ H m×s . Then, there exist nonsingular matrixes M, P, T, Q, and S such that The order of identity matrixes in (4) are shown in [9] directly.
We utilize Lemma 1 to transform the matrix arrays (3) into two simple forms such as (4) and and and M, P, T, Q, S, M 1 , P 1 , T 1 , Q 1 , S 1 are nonsingular. Hence, the system (1) is equivalent to where (Ω ij ) 18×18 and (Φ ij ) 8×8 have the same block rows as S a and S b , the same block columns as S e and S h , respectively. Then, substituting (9)-(11) into the system (8) becomes A result can be conclude that the system (1) is equivalent with the system (12). Substituting (9)-(11) into the system (12) yields and where The following theorem takes into account the solvability conditions to the system (1) from the level of the partion of the coefficient matrixes of equivalent matrix equation system (12) and deduces solvability condition in terms of ranks which is equivalent to block matrixes condition. Theorem 1. Consider the system of quaternion matrix Equation (1). Then the following statements are equivalent: (1) The system (1) is consistent.

Remark 1.
We use a new method which differs from the one presented in [10] to obtain our result.

A Numerical Example of the System (1)
In this section, we give an numerical example to demonstrate the availability of Theorem 1.

Example 1. Let
Consider the system (1). By a straightforward calculation, we have The system (1) is consistent since all the rank equalities in (15)-(21) hold. Moreover, it is easy to show that is a solution that satisfies the system (1) .

Solvability Conditions to the System (2) Involving φ-Hermicity
In this section, we provide some consistency conditions to the system of quaternion matrix Equation (2), to obtain a φ-Hermitian solution as an application of the system (1), where A, B, C, D, Ω = Ω φ and Θ = Θ φ are given quaternion matrixes, X and Y are φ-Hermitian unknowns.

Remark 2.
We utilize a new method which differs from the one in [10] to obtain our result.

A Numerical Example to System (2)
The goal of this section is to present a numerical example to the system (2) to illustrate the availability of the Theorem 2.

Example 2. Let
where φ(a) = a k * = −ka * k for a ∈ H. We consider the φ-Hermitian solution to the system (2). Check that The system (2) has a φ-Hermitian solution since all the rank equalities in (38)-(41) hold. Note that is a solution that satisfies the system (2) .

Conclusions
We have investigated a Sylvester-like quaternion matrix equation system (1) making use of simultaneous decomposition of four quaternion matrixes to deduce some useful equivalent conditions of equation system consistency in terms of the partion of the coefficient matrixes of equivalent matrix equation system, also derived ranks condition according to the partion condition. Therefore, we have provided the general solution to the system (1). Based on the result of the system (1), we have infered the consistency conditions and general φ-Hermitian solution to the system (2). We also give several numerical examples to illustrate the main outcome.