Systems of four coupled one sided Sylvester-type real quaternion matrix equations and their applications
Zhuo-Heng He, Qing-Wen Wang
TL;DR
This paper establishes solvability conditions and general solutions for coupled Sylvester-type quaternion matrix equations, extending existing results and illustrating findings with numerical examples.
Contribution
It introduces new solvability criteria and explicit solutions for coupled quaternion matrix equations, broadening the theoretical framework in this area.
Findings
Derived necessary and sufficient solvability conditions
Provided explicit general solutions when solvable
Extended known results in quaternion matrix equations
Abstract
In this paper, we derive some necessary and sufficient solvability conditions for some systems of one sided coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. We also give the expressions of the general solutions to these systems when they are solvable. Moreover, we provide some numerical examples to illustrate our results. The findings of this paper extend some known results in the literature.
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Taxonomy
TopicsMatrix Theory and Algorithms · Algebraic and Geometric Analysis · Mathematics and Applications
**Systems of four coupled one sided Sylvester-type real quaternion matrix equations and their applications111This research was supported by the grants from the National Natural Science Foundation of China (11571220).
- Corresponding author. Email address: [email protected] (Z.H. He); [email protected], [email protected] (Q.W. Wang)**
Zhuo-Heng Hea,b, Qing-Wen Wangb,∗
Department of Mathematics and Statistics, Auburn University, AL 36849-5310, USA
Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China
Abstract: In this paper, we derive some necessary and sufficient solvability conditions for some systems of one sided coupled Sylvester-type real quaternion matrix equations in terms of ranks and generalized inverses of matrices. We also give the expressions of the general solutions to these systems when they are solvable. Moreover, we provide some numerical examples to illustrate our results. The findings of this paper extend some known results in the literature.
Keywords: Quaternion; Sylvester-type equations; Moore-Penrose inverse; Rank; Solution; Solvability
**2010 AMS Subject Classifications: **15A03, 15A09, 15A24, 15B33
1. Introduction
Quaternions were introduced by Irish mathematician Sir William Rowan Hamilton in 1843. It is well known that quaternion algebra is an associative and noncommutative division algebra over the real number field. Quaternions and quaternion matrices have found a huge amount of applications in quantum physics, signal and color image processing, and so on (e.g. [3], [21], [27]-[30]). General properties of quaternions and quaternion matrices can be found in [48]. Quaternion matrix equations play an important role in mathematics and other disciplines, such as engineering, system and control theory. There have been many papers using various approaches to investigate many quaternion matrix equations (e.g. [9]-[11], [38]-[43], [45], [46], [49]).
The Sylvester-type matrix equations have wide applications in neural network [47], robust control ([4], [31]), output feedback control ([25], [26]), the almost noninteracting control by measurement feedback problem ([32], [44]), graph theory [7], and so on. Since Roth [23] first studied the one-sided generalized Sylvester matrix equation
[TABLE]
over the complex field in 1952, there have been many papers to discuss the generalized Sylvester matrix equations (e.g. [1], [2], [8], [13], [17], [18], [20], [24], [34]-[37], [44]). For instance, De Tern et al. ([5], [6]) considered the -Sylvester equation and . Quite recently, Dmytryshyn and Kågsträm [7] presented some solvability conditions of the following systems consisting of Sylvester and -Sylvester equations through the corresponding equivalence relations of the block matrices
[TABLE]
where each unknown is and all other matrices are of appropriate sizes. Jonsson and Kågsträm ([15], [16]) provided some effective approaches for solving one-sided and two-sided triangular Sylvester-type matrix equations.
The study on the coupled generalized Sylvester matrix equations is active in recent years. Lee and Vu [19] derived a consistency condition for the following system of mixed Sylvester matrix equations through the corresponding equivalence relations of the block matrices
[TABLE]
where and are given matrices over a field, and are unknowns. Wang and He [34] gave some computable necessary and sufficient solvability conditions for the system (1.1), and presented the general solution when (1.1) is solvable. Afterwards, He and Wang [14] provided some necessary and sufficient solvability conditions for the system of mixed Sylvester matrix equations
[TABLE]
where and are given complex matrices, and are unknowns. They also derived an expression of the general solution to the system (1.2). Recently, Wang and He [35] considered the following three systems of generalized coupled Sylvester matrix equations with four variables
[TABLE]
[TABLE]
[TABLE]
where and are given complex matrices, and are unknowns. He, Mauricio, Wang and De Moor [8] considered two sided coupled generalized Sylvester matrix equations with four variables
[TABLE]
where are given complex matrices, are unknowns. Very recently, He and Wang [13] derived the solvability conditions and the general solution to the system of the periodic discrete-time coupled Sylvester quaternion matrix equations
[TABLE]
where are given matrices, and are unknowns.
To our best knowledge, there has been little information on the solvability and the general solutions to the systems of four coupled one sided Sylvester-type real quaternion matrix equations with five unknowns. Motivated by the wide applications of generalized Sylvester matrix equations and real quaternion matrix equations and in order to improve the theoretical development of generalized Sylvester real quaternion matrix equations, we in this paper consider the solvability and the expressions of the general solutions to the following systems of four coupled one sided Sylvester-type real quaternion matrix equations
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where are given real quaternion matrices, and are unknowns. Note that the th equation and th equation in (1.10)-(1.30) have a common variable . The given real quaternion matrices located at the left of the variables and located at the right of the variables. Systems (1.1)-(1.5) are special cases of systems (1.10)-(1.30).
The remainder of the paper is organized as follows. In Section 2, we provide some known lemmas which are used in this paper. In Section 3,4,5,6,7, we present some solvability conditions to the systems of four coupled one sided Sylvester-type real quaternion matrix equations (1.10)-(1.30), respectively. We also derive the general solutions to the systems (1.10)-(1.30), respectively. Moreover, we give some numerical examples to illustrate our results.
Throughout this paper, let be the real number fields. Let be the set of all matrices over the real quaternion algebra
[TABLE]
For , the symbols and denote the conjugate transpose and the rank of , respectively. The identity matrix with appropriate size is denoted by . The Moore-Penrose inverse of , denoted by , is defined to be the unique solution to the following four matrix equations
[TABLE]
Furthermore, and stand for the two projectors and induced by , respectively. It is known that and .
2. Preliminaries
In this section, we review some lemmas which are used in the further development of this paper. The following lemma give the solvability conditions and general solution to the mixed Sylvester real quaternion matrix equations (1.1).
Lemma 2.1**.**
[34]** Let and be given. Set
[TABLE]
*Then the following statements are equivalent:
The mixed Sylvester real quaternion matrix equations (1.1) is consistent.
*
[TABLE]
[TABLE]
[TABLE]
In this case, the general solution to the mixed Sylvester real quaternion matrix equations (1.10) can be expressed as
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
and are arbitrary matrices over with appropriate sizes.
The solvability conditions and general solution to the mixed Sylvester real quaternion matrix equations (1.2) can be found in the following lemma.
Lemma 2.2**.**
[14]** Let and be given. Set
[TABLE]
*Then the following statements are equivalent:
The mixed generalized Sylvester real quaternion matrix equations (1.2) is consistent.
*
[TABLE]
**
[TABLE]
In this case, the general solution to the mixed generalized Sylvester real quaternion matrix equations (1.10) can be expressed as
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
and are arbitrary matrices over with appropriate sizes.
Based on Lemma 2.1, we can solve the following mixed Sylvester real quaternion matrix equations
[TABLE]
Lemma 2.3**.**
Let and be given. Set
[TABLE]
*Then the following statements are equivalent:
The mixed Sylvester real quaternion matrix equations (2.1) is consistent.
*
[TABLE]
[TABLE]
[TABLE]
In this case, the general solution to the mixed Sylvester real quaternion matrix equations (2.1) can be expressed as
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
and are arbitrary matrices over with appropriate sizes.
The following real quaternion matrix equation
[TABLE]
which play an important role in the construction of the solvability conditions and the general solution to the systems (1.10)-(1.30).
Lemma 2.4**.**
[12]**, [33] Let , and be given. Set
[TABLE]
Then the equation (2.2) is consistent if and only if
[TABLE]
*In this case, the general solution can be expressed as
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where are arbitrary matrices over with appropriate sizes.
The following lemma can be easily generalized to .
Lemma 2.5**.**
[22*]** Let , and be given. Then
*
3. **Some solvability conditions and the general solution to
system (1.10)**
In this section, we consider the solvability conditions and the general solution to the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.10). For simplicity, put
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Now we give the fundamental theorem of this section.
Theorem 3.1**.**
*Let and be given. Then the following statements are equivalent:
The system of one-sided coupled Sylvester-type real quaternion matrix equations (1.10) is consistent.
*
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
**
[TABLE]
[TABLE]
[TABLE]
In this case, the general solution to the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.10) can be expressed as
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
the remaining are arbitrary matrices over , and are the column numbers of and , respectively, and are the row numbers of and , respectively.
Proof.
We separate this system of one-sided coupled Sylvester-type real quaternion matrix equations (1.10) into two parts
[TABLE]
and
[TABLE]
Observe that system (3.33) has the form of (1.1), and system (3.36) has the form of (1.2). We can solve the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.10) through the following three steps. In the first step, we consider the system (3.33). It follows from Lemma 2.1 that the system (3.33) is consistent if and only if
[TABLE]
[TABLE]
or
[TABLE]
In this case, the general solution to the system (3.33) can be expressed as
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
and are arbitrary matrices over with appropriate sizes.
In the second step, we consider the system (3.36). It follows from Lemma 2.2 that the system (3.36) is consistent if and only if
[TABLE]
[TABLE]
or
[TABLE]
In this case, the general solution to the system (3.36) can be expressed as
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
and are arbitrary matrices over with appropriate sizes.
In the third step, equating in (3.40) and in (3.48) gives
[TABLE]
i.e.,
[TABLE]
Now we want to solve the matrix equation (3.53). It follows from Lemma 2.4 that the matrix equation (3.53) is consistent if and only if
[TABLE]
Hence, the general solution to the matrix equation (3.53) can be expressed as
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where are arbitrary matrices over with appropriate sizes.
Now we want to prove that (3.54) (3.10)-(3.13). At first, we prove that is equivalent to (3.10). Applying Lemma 2.5 to gives
[TABLE]
Similarly, we can show that and are equivalent to (3.11), (3.12) and (3.13), respectively. ∎
Next we give an example to illustrate Theorem 3.1.
Example 1**.**
Given the quaternion matrices:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Now we consider the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.10). Check that
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
All the rank equalities in (3.7)-(3.13) hold. Hence, the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.10) is consistent. Note that
[TABLE]
[TABLE]
and
[TABLE]
satisfy the system (1.10).
Let and vanish in Theorem 3.1. Then we can obtain some necessary and sufficient conditions and general solution to the system of coupled generalized Sylvester real quaternion matrix equations (1.3).
Corollary 3.2**.**
[35]** Let and be given. Set
[TABLE]
*Then the following statements are equivalent:
The system of coupled generalized Sylvester real quaternion matrix equations (1.3) is consistent.
*
[TABLE]
[TABLE]
[TABLE]
[TABLE]
**
[TABLE]
In this case, the general solution to the coupled generalized Sylvester real quaternion matrix equations (1.3) can be expressed as
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
and are arbitrary matrices over with appropriate sizes.
4. **Some solvability conditions and the general solution to
system (1.15)**
In this section, we consider the solvability conditions and the general solution to the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.15). For simplicity, put
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Now we give the fundamental theorem of this section.
Theorem 4.1**.**
*Let and be given. Then the following statements are equivalent:
The system of one-sided coupled Sylvester-type real quaternion matrix equations (1.15) is consistent.
*
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
**
[TABLE]
[TABLE]
[TABLE]
In this case, the general solution to the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.15) can be expressed as
[TABLE]
[TABLE]
[TABLE]
[TABLE]
or
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
the remaining are arbitrary matrices over , and are the column numbers of and , respectively, and are the row numbers of and , respectively.
Proof.
We separate this system of one-sided coupled Sylvester-type real quaternion matrix equations (1.15) into two parts
[TABLE]
and
[TABLE]
Applying the main idea of Theorem 3.1, Lemma 2.1, Lemma 2.2, Lemma 2.4 and Lemma 2.5, we can prove Theorem 4.1. ∎
Now we give an example to illustrate Theorem 4.1.
Example 2**.**
Given the quaternion matrices:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Now we consider the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.15). Check that
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
All the rank equalities in (4.1)-(4.7) hold. Hence, the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.15) is consistent. Note that
[TABLE]
[TABLE]
and
[TABLE]
satisfy the system (1.15).
Let and vanish in Theorem 4.1. Then we can obtain some necessary and sufficient conditions and general solution to the system of coupled generalized Sylvester real quaternion matrix equations (1.4).
Corollary 4.2**.**
[35]** Let and be given. Set
[TABLE]
*Then the following statements are equivalent:
The system of coupled generalized Sylvester real quaternion matrix equations (1.4) is consistent.
*
[TABLE]
**
[TABLE]
In this case, the general solution to the coupled generalized Sylvester real quaternion matrix equations (1.4) can be expressed as
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
and are arbitrary matrices over with appropriate sizes.
5. **Some solvability conditions and the general solution to
system (1.20)**
In this section, we consider the solvability conditions and the general solution to the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.20). For simplicity, put
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Now we give the fundamental theorem of this section.
Theorem 5.1**.**
*Let and be given. Then the following statements are equivalent:
The system of one-sided coupled Sylvester-type real quaternion matrix equations (1.20) is consistent.
*
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
**
[TABLE]
[TABLE]
[TABLE]
In this case, the general solution to the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.20) can be expressed as
[TABLE]
[TABLE]
[TABLE]
or
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
the remaining are arbitrary matrices over , and are the column numbers of and , respectively, and are the row numbers of and , respectively.
Proof.
We separate this system of one-sided coupled Sylvester-type real quaternion matrix equations (1.20) into two parts
[TABLE]
and
[TABLE]
Applying the main idea of Theorem 3.1, Lemma 2.1, Lemma 2.2, Lemma 2.4 and Lemma 2.5, we can prove Theorem 5.1. ∎
Now we give an example to illustrate Theorem 5.1.
Example 3**.**
Given the quaternion matrices:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Now we consider the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.20). Check that
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
All the rank equalities in (5.1)-(5.7) hold. Hence, the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.20) is consistent. Note that
[TABLE]
[TABLE]
and
[TABLE]
satisfy the system (1.20).
Let and vanish in Theorem 5.1. Then we can obtain some necessary and sufficient conditions and general solution to the system of coupled generalized Sylvester real quaternion matrix equations (1.5).
Corollary 5.2**.**
[35]** Let and be given. Set
[TABLE]
*Then the following statements are equivalent:
The system of coupled generalized Sylvester real quaternion matrix equations (1.5) is consistent.
*
[TABLE]
[TABLE]
[TABLE]
[TABLE]
**
[TABLE]
In this case, the general solution to the coupled generalized Sylvester real quaternion matrix equations (1.5) can be expressed as
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
and are arbitrary matrices over with appropriate sizes.
6. **Some solvability conditions and the general solution to
system (1.25)**
Our goal of this section is to give some necessary and sufficient conditions and the general solution to the system (1.25). Set
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Now we give the fundamental theorem of this section.
Theorem 6.1**.**
*Let and be given. Then the following statements are equivalent:
The system of one-sided coupled Sylvester-type real quaternion matrix equations (1.25) is consistent.
*
[TABLE]
[TABLE]
[TABLE]
[TABLE]
**
[TABLE]
[TABLE]
In this case, the general solution to the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.25) can be expressed as
[TABLE]
[TABLE]
[TABLE]
or
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
the remaining are arbitrary matrices over , and are the column numbers of and , respectively, and are the row numbers of and , respectively.
Proof.
We separate this system of one-sided coupled Sylvester-type real quaternion matrix equations (1.25) into two parts
[TABLE]
and
[TABLE]
Applying the main idea of Theorem 3.1, Lemma 2.2, Lemma 2.4 and Lemma 2.5, we can prove Theorem 6.1. ∎
Now we give an example to illustrate Theorem 6.1.
Example 4**.**
Given the quaternion matrices:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Now we consider the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.25). Check that
[TABLE]
[TABLE]
[TABLE]
[TABLE]
All the rank equalities in (6.1)-(6.1) hold. Hence, the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.25) is consistent. Note that
[TABLE]
[TABLE]
and
[TABLE]
satisfy the system (1.25).
7. **Some solvability conditions and the general solution to
system (1.30)**
In this section, we consider the solvability conditions and the general solution to the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.30). For simplicity, put
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Theorem 7.1**.**
*Let and be given. Then the following statements are equivalent:
The system of one-sided coupled Sylvester-type real quaternion matrix equations (1.30) is consistent.
*
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
**
[TABLE]
[TABLE]
[TABLE]
In this case, the general solution to the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.30) can be expressed as
[TABLE]
[TABLE]
[TABLE]
[TABLE]
where
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
the remaining are arbitrary matrices over , and are the column numbers of and , respectively, and are the row numbers of and , respectively.
Proof.
We separate this system of one-sided coupled Sylvester-type real quaternion matrix equations (1.30) into two parts
[TABLE]
and
[TABLE]
Applying the main idea of Theorem 3.1, Lemma 2.1, Lemma 2.3, Lemma 2.4 and Lemma 2.5, we can prove Theorem 7.1. ∎
Now we give an example to illustrate Theorem 7.1.
Example 5**.**
Given the quaternion matrices:
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
Now we consider the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.30). Check that
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
[TABLE]
All the rank equalities in (7.1)-(7.7) hold. Hence, the system of one-sided coupled Sylvester-type real quaternion matrix equations (1.30) is consistent. Note that
[TABLE]
[TABLE]
and
[TABLE]
satisfy the system (1.30).
8. Conclusion
We have provided some necessary and sufficient conditions for the existence and the general solutions to the systems of four coupled one sided Sylvester-type real quaternion matrix equations (1.10)-(1.30), respectively. Moreover, we have presented some numerical examples. It is worthy to say that the main results of this paper can be generalized to an arbitrary division ring with an involutive antiautomorphism.
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