The General Solution of Quaternion Matrix Equation Having -Skew-Hermicity and Its Cramer’s Rule
We determine some necessary and sufficient conditions for the existence of the -skew-Hermitian solution to the following system over the quaternion skew field and provide an explicit expression of its general solution. Within the framework of the theory of quaternion row-column noncommutative determinants, we derive its explicit determinantal representation formulas that are an analog of Cramer’s rule. A numerical example is also provided to establish the main result.
Throughout this paper, the real number field and the complex field are denoted by and , respectively, and for the quaternion algebraThe set of all matrices of dimension over is represented by . An identity matrix with conformable size is denoted by . For any matrix over , and stand for the column right space and the row left space of , respectively. denotes the dimension of . By , we have , which is known as rank of and denoted by . For any quaternion , its conjugate is . So, represents the conjugate transpose of . means the Moore-Penrose inverse of , i.e., the exclusive matrix satisfying More results on generalized inverses can be seen in [2, 3]. Furthermore, and are couple of the projectors induced by , respectively. It is evident that and .
The notion of quaternions was first explored by an Irish mathematician Sir William Rowan Hamilton in . Quaternions have ample use in diverse areas of mathematics like computation, geometry and algebra; see, e.g. [5–8]. The researcher in  used quaternion in the field of computer graphics. Very recently, quaternion matrices have secured a vital role in control theory, mechanics, altitude control, quantum physics and signal processing; see, e.g., [10, 11]. In skeletal animation systems, quaternions are mostly practiced to interpolate between joint orientations specified with key frames or animation curves . The comprehensive study on quaternions can be found in .
The Hermitian solutions to some classical matrix equations were examined by numerous researchers in various papers by adopting different approaches. For instance, Khatri and Mitra  and Groß in  computed some necessary and sufficient condition for the Hermitian solution to some significant matrix equation with their general solution when they are consistent.
A matrix is said to be -Hermitian and skew--Hermitian [16, 17] if and , where , respectively. Applications of -Hermitian matrices in the fields of convergence analysis in statistical signal processing and linear modelling can be viewed in [16, 18, 19]. The singular value decomposition of the -Hermitian matrix was examined in . Very recently, a researcher in  determined the skew--Hermitian solution to some significant matrix equations including and gave general solution to these equation when they are consistent.
He and Wang  gave the general solution of bearing -Hermicity over . An iterative algorithm for determining the -Hermitian and -skew-Hermitian solutions to the quaternion generalized Sylvester equation were established in . For more related papers on -Hermicity, one may refer to [23–28]. Notice that Sylvester-type matrix equations have a huge amount of practical applications in feedback control [29, 30], robust control , pole/eigenstructure assignment design , neural network , and so on.
Keeping in view the latest development and the application of skew--Hermitian matrices in the fields like convergence analysis in statistical signal processing and linear modelling and quaternion matrices in mind, in this paper, we explore the -skew-Hermitian solution of the equation when it is consistent. We not only obtain the explicit form of the -skew-Hermitian solution in terms of generalized inverses but also give its explicit determinantal representation formulas that are an analog of Cramer’s rule. Our proposed Cramer’s rule is based on the theory of row-column noncommutative determinants introduced in , by using determinantal representations of the Moore-Penrose inverse matrix . Within the framework of the theory of noncommutative row-column determinants, determinantal representations of various generalized quaternion inverses and generalized inverse solutions to quaternion matrix equations have been derived by one of the authors (see, e.g., [36–46]) and by other researchers (see, e.g., [47–51]).
Observe that the system (3) is a particular case of our system (5). It is noteworthy to say that the researcher in  made nice effort to calculate the -skew-Hermitian solution of (3) with its general solution but he did not represent the necessary and sufficient conditions in terms of rank equalities and we will overcome this issue in this paper as well. Moreover, we give explicit determinantal representation formulas for the solution to (5).
The remaining part of this paper is composed as follows. In Section 2, we start with some provisions from generalized inverses and its determinantal representations which have vital role during the construction of the main results of this paper. Necessary and sufficient conditions for the general solution to (5), where and are -skew-Hermitian, are presented in Section 3. A particular case of (5) is also examined in this section. In Section 4, we give Cramer’s rule to Eq.(5). Based on obtained explicit determinantal representation formulas, in Section 5, we present a numerical example to glorify the notion established in this paper. Finally, in Section 6, we provide a conclusion to this article.
2. Preliminaries. Some Provisions from Generalized Inverses and Its Determinantal Representations
We begin with some famous results which will be used in the remaining part of this paper.
Lemma 1 (see ). Let and be known. Then (1).(2).(3).
Lemma 2 (see ). Let be given. Then (1)(2)(3)(4)(5)(6)
Lemma 3 (see ). Let , and be given matrices with right sizes over . Then (1)(2)(3)
Remark 4. Since for any for all , and , then elements of the main diagonal of an η1-Hermitian matrix must be as and a pair of elements which are symmetric with respect to the main diagonal can be represented as Similarly, elements of the main diagonal of an -skew-Hermitian matrix must be as and a pair of elements which are symmetric with respect to the main diagonal can be represented as where for all .
For , we define row determinants and column determinants. Suppose is the symmetric group on the set .
Definition 5 (see ). The th row determinant of is defined for all by putting where is the left-ordered permutation. It means that its first cycle from the left starts with , while other cycles start from the left with the minimal of all the integers which are contained in it, and the order of disjoint cycles (except for the first one) is strictly conditioned by increase from left to right of their first elements, .
Definition 6 (see ). The th column determinant of is defined for all by putting where is the right-ordered permutation. It means that its first cycle from the right starts with , while other cycles start from the right with the minimal of all the integers which are contained in it, and the order of disjoint cycles (except for the first one) is strictly conditioned by increase from right to left of their first elements, .
Let be the th column and be the th row of . Suppose denotes the matrix obtained from by replacing its th row with the row , and denotes the matrix obtained from by replacing its th column with the column .
We state some properties of row-column determinants needed below.
Lemma 7 (see ). If the th row of is a left linear combination of other row vectors, i.e., , where and for all and , then
Lemma 8 (see ). If the th column of is a right linear combination of other column vectors, i.e., , where and for all and , then
Lemma 9 (see ). Let . Then , for all .
Lemma 10. Let . Then for all
So, Lemma 10 gives the following features of -Hermitian and -skew-Hermitian matrices.
Lemma 11. If is -Hermitian, then If is -skew-Hermitian, then for all .
Remark 12. Since  for Hermitian we have the determinant of a Hermitian matrix is defined by putting for all .
Properties of the determinant of a Hermitian matrix are similar to the properties of an usual (commutative) determinant and they have been completely explored by row-column determinants in .
The following notation will be used for determinantal representations of the Moore-Penrose inverse. Let and be subsets with . Let be a submatrix of whose rows and columns are indexed by and , respectively. Similarly, let be a principal submatrix of whose rows and columns are indexed by . If is Hermitian, then denotes the corresponding principal minor of . The collection of strictly increasing sequences of integers chosen from is denoted by for . For fixed and , let , .
Denote the th column and the th row of by and , respectively.
Theorem 13 (see ). If , then the Moore-Penrose inverse have the following determinantal representations or
Remark 14. Note that for an arbitrary full-rank matrix , a row-vector , and a column-vector we put, respectively, (i)if , then for all ;(ii)if , then for all .
Corollary 15. If , then the Moore-Penrose inverse , , and have the following determinantal representations, respectively,
Corollary 16 (see ). If , then the projection matrix has the determinantal representation where and are the th column and the th row of .
Corollary 17 (see ). If , then the projection matrix has the determinantal representation where and are the th row and the th column of .
Lemma 18 (see ). Let , , be known and be unknown. Then the matrix equation is consistent if and only if . In this case, its general solution can be expressed as where , are arbitrary matrices over with appropriate dimensions.
Lemma 19 (see ). Let , . Then the partial solution to (35) have determinantal representations, where are the column vector and the row vector, respectively. and are the th row and the th column of .
Corollary 20 (see ). Let , be known and be unknown. Then is consistent if and only if . In this case, its general solution can be expressed as , where is an arbitrary matrix over with appropriate dimensions. The partial solution has the following determinantal representation, where is the th column of .
Corollary 21 (see ). Let , be given, and be unknown. Then is solvable if and only if and its general solution is , where is a any matrix with conformable dimension. Moreover, its partial solution has the determinantal representation, where is the th row of .
3. Some Solvability Conditions and the General Solution to (5)
In this section, we provide some necessary and sufficient conditions for the system (5) to have a solution , where and Furthermore, its general solution is also constructed when the solvability conditions are accomplished.
Theorem 22. Let and be given coefficient matrices in (5) over with conformable sizes. Denote Then (1)The system (5) has a solution , where and .(2)The coefficient matrices in (5) satisfy (3)(4)are equivalent statements. Under these conditions, the general solution to the system (5) can be demonstrated as where and are arbitrary matrices over .
Proof. Obviously, (2)(3).
Now we show (2) (4). By means of Lemmas 1 and 2, we have Similarly, Now we show (1): If the system (5) has a solution , where and . Then (2)(1): Now we prove that mentioned in (44)–(46), respectively, is a solution of (5) under (42). Obviously, and represented in (44) and (45) are -skew-Hermitian. Now putting and in the following expression and simplifying, we have Now we show that in (50). By Lemma 3, equation (42) and , we have By using (51), can be written as follows: i.e., By using (53) and , we can write By using , (42) and (51), the term in (54) can be expressed as