Research Article | Open Access
Least-Norm of the General Solution to Some System of Quaternion Matrix Equations and Its Determinantal Representations
We constitute some necessary and sufficient conditions for the system , , , , , to have a solution over the quaternion skew field in this paper. A novel expression of general solution to this system is also established when it has a solution. The least norm of the solution to this system is also researched in this article. Some former consequences can be regarded as particular cases of this article. Finally, we give determinantal representations (analogs of Cramer’s rule) of the least norm solution to the system using row-column noncommutative determinants. An algorithm and numerical examples are given to elaborate our results.
In the whole article, the notation is reserved for the real number field and stands for the set of all matrices over the quaternion skew field specifies its subset of matrices with a rank . For , let and designate the conjugate transpose, the column right space and the left row space of . dim illustrates the size of and dim =dim by , which is known as the rank of denoted by .
Definition 1. The Moore-Penrose inverse of , denoted by , is defined to be the unique solution to the following four matrix equationsMatrices satisfying (1) and (2) are known as reflexive inverses.
Note that the reflexive inverse is denoted most often by but sometimes by (see, e.g., ) that is different from the denotation of the Moore-Penrose by . We will use the denotation for the reflexive inverse.
Suppose refers an identity matrix with feasible size. In addition, , represent a pair of orthogonal projectors induced by , respectively, and , , , , and .
Quaternions were invented by Hamilton in 1843. Zhang presented a detail survey on quaternion matrices in . Quaternions provide a concise mathematical method for representing the automorphisms of three- and four-dimensional spaces. The representations by quaternions are more compact and quicker to compute than the representations by matrices . For this reason, an increasing number of applications based on quaternions are found in various fields, such as color imaging, geometry, mechanics, linear adaptive filter, altitude control and computer science, signal processing, in particular as quaternion-valued neural networks, etc. [5–10].
The research of matrix equations have both applied and theoretical importance. In particular, the Sylvester-type matrix equations have far reaching applications in singular system control , system design , robust control , feedback , perturbation theory , linear descriptor systems , neural networks , and theory of orbits .
Some recent work on generalized Sylvester matrix equations and their systems can be observed in [19–31]. In , Bao  examined the least-norm and extremal ranks of the least square solution to the quaternion matrix equationsWang et al.  examined the expression of the general solution to the systemAnd, as an application, the -symmetric and -skew-symmetric solution to has been established. Li et al.  established a novel expression to the general solution of system (4) and they computed the least-norm of general solution to (4). In , Wang et al.  constituted the expression of the general solution toand as an application they explored the -symmetric solution to the system
Some latest findings on the least-norm of matrix equations and -symmetric matrices can be consulted in [36–40]. Furthermore, our main system (6) is a special case of the following system:which has been investigated by Zhang in 2014. But the expressions provided for the , and in , we are in position to calculate the least-norm of the solutions with its determinantal representations. When some given matrices are zero in (8), then it becomes our system and we will give such kind of expressions in which the least-norm of the solutions can also be computed with its determinantal representations. It is worthy to note that Zhang examined (8) with complex settings and we will consider our system (6) with quaternion settings.
According to our best of knowledge, the least-norm of the general solution to system (6) is not investigated by any one. Motivated by the vast application of quaternion matrices and the latest interest of least-norm of matrix equations, we construct a novel expression of the general solution to system (6) and apply this to investigate the least-norm of the general solution to system (6) over in this paper. Observing that systems (3) and (4) are particular cases of our system (6), solving system (6) will encourage the least-norm to a wide class of problems in the collected work.
Since the general solutions of considered systems are expressed in term of generalized inverses, another goal of the paper is to give determinantal representations of the least-norm of the general solution to system (6) based on determinantal representations of generalized inverses.
Determinantal representation of a solution gives a direct method of its finding analogous to the classical Cramer’s rule that has important theoretical and practical significance. Through looking for their more applicable explicit expressions, there are various determinantal representations of generalized inverses even with the complex or real entries, in particular for the Moore-Penrose inverse (see, e.g., [42–44]). By virtue of noncommutativity of quaternions, the problem for determinantal representation of generalized quaternion inverses is more complicated, and only now it can be solved due to the theory of column-row determinants introduced in [45, 46]. 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 our authors (see, e.g.[47–54]) and by other researchers (see, e.g. [55–57]). Moreover, Song et al.  have just recently considered determinantal representations of general solution to the two-sided coupled generalized Sylvester matrix equation over obtained using the theory of row-column determinants as well. But their proposed approach differs from our proposed. In , for determinantal representations of the general solution to the equation supplementary matrices have been used that not always easy to get. While, by proposed method only coefficient matrices of the equations are used. More detailed Cramer’s rule to solutions and (skew-)Hermitian solutions of some systems of matrix equations and generalized Sylvester matrix equation over are recently explored in [59, 60] and [61, 62], respectively.
The remainder of our article is directed as follows. In Section 2, we commence with some needed known results about systems of matrix equations and determinantal representations of the Moore-Penrose inverse and of solutions to the quaternion matrix equations. In Section 3, we provide a new expression of the general solution to our system (6) and present an algorithm with an example. We discuss the least-norm of the general solution to (6) over in Section 4. In Section 5, determinantal representations of the general solution to (6) are derived and other example to elaborate obtained Cramer’s Rule to system (6) with data from the example in Section 3 is given. As expected, we get the same solution. Finally, in Section 6, the conclusions are drawn.
We commence with the following lemmas which have crucial function in the construction of the chief outcomes of the following sections.
2.1. The General Solution to System (6)
Lemma 2 (see ). Let be given. Then
Lemma 3 (see ). Let , , and be known matrices over with right sizes. Then
Lemma 4 (see ). Let be matrices over and Thenwhere , are any fixed reflexive inverses, and stand for the projectors , induced by , , respectively.
Remark 5. Since Moore-Penrose inverses are reflexive inverses, this lemma can be used for Moore-Penrose inverses without any changes. It has taken place in (, Lemma 2.4). But for more credibility, we prove this lemma below for the Moore-Penroses inverse as well.
Lemma 6 (see ). Suppose thatis consistent linear matrix equation, where , , , and , respectively. Then (1)The general solution of the homogeneous equation, can be expressed by where and are general solution of the following four homogeneous matrix expressions By computing the value of unknowns in the above equations and using them in and , we have where , , , , , and ; the matrices are free to vary over (2) Assume that the matrix expression (11) is solvable, then its general solution can be expressed as where and are any pair of particular solutions to (11).It can also be written as
Lemma 7 (see ). Let , , , and be given and to be determined. Then the systemis consistent if and only ifUnder these conditions, the general solution to (18) can be established as where is a free matrix over with accordant dimension.
Lemma 8 (see ). Let , , , , , , , ,, , , , , be given and , to be determined. Denote Then the following conditions are tantamount: (1)System (6) is resolvable.(2)The conditions in (19) are met and(3)The equalities in (19) and (22) are satisfied and In these conditions, the general solution to system (6) can be written aswhere are free matrices over with agreeable dimensions.
2.2. Determinantal Representations of Solutions to the Quaternion Matrix Equations
Due to noncommutativity of quaternions there is a problem of a determinant of matrices with noncommutative entries (which are also defined as noncommutative determinants). There are several versions of defining of noncommutative determinants (e.g., see [68–70]). But any of the previous noncommutative determinants has not fully retained those properties which it owned for matrices with real settings. Moreover, if functional properties of a noncommutative determinant over a ring are satisfied, then it takes on a value in its commutative subset. This dilemma can be avoided due to the theory of row-column determinants.
For , we define row determinants and column determinants. Suppose is the symmetric group on the set .
Definition 9 (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 , 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 10 (see ). The th column determinant of is defined for all by putting noindent where is the right-ordered permutation. It means that its first cycle from the right starts with , 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, .
Since  for Hermitian we havethe determinant of a Hermitian matrix is defined by puttingIts properties are similar to the properties of an usual (commutative) determinant and they have been completely explored in  by using row and column determinants that are so defined only by construction.
For determinantal representations of the Moore-Penrose inverse, we shall use the following notations. Let and be subsets of the order . Let be a submatrix of whose rows are indexed by and the columns indexed by . Similarly, let be a principal submatrix of whose rows and columns indexed by . If is Hermitian, then is the corresponding principal minor of . For , the collection of strictly increasing sequences of integers chosen from is denoted by . For fixed and , let denotes the collection of sequences of row indexes that contain the index , and denotes the collection of sequences of column indexes that contain .
Let be the th column and be the th row of , respectively. Suppose denotes the matrix obtained from by replacing its th column with the column-vector , and denotes the matrix obtained from by replacing its th row with the row-vector . We denote the th row and the th column of by and , respectively.
Lemma 11 (see ). If , then the Moore-Penrose inverse have the following determinantal representations,and
Remark 12. For an arbitrary full-rank matrix , a row-vector , and a column-vector , we put
Corollary 13. If , then the projection matrix has the determinantal representation where is the th column of .
Corollary 14. If , then the projection matrix has the determinantal representationwhere is the th row of .
Lemma 15 (see ). Let , , be known and be unknown. Then the matrix equationis consistent if and only if . In this case, its general solution can be expressed aswhere are arbitrary matrices over with appropriate dimensions.
Lemma 16 (see ). Let , . Then the minimum norm least squares solution to (38) have determinantal representations, or where are the column vector and the row vector, respectively. and are the th row and the th column of .
Corollary 17. 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 is an arbitrary matrix over with appropriate dimensions. Its minimum norm least squares solution has the following determinantal representation, where is the th column of .
Corollary 18. Let , be given, and be unknown. Then the equation is solvable if and only if and its general solution is , where is a any matrix with conformable dimension. Moreover, its minimum norm least squares solution has the determinantal representation, where is the th row of .
3. A New Expression of the General Solution to System (6)
First, we show that Lemma 4 is true for the Moore-Penrose inverses.
Lemma 19. Let be matrices over and Thenwhere , are the Moore-Penrose inverses, and , , , , , and are projectors with respect to the corresponding Moore-Penrose inverses.
Proof. In (, Lemma 2.4), it is proved that for fixed reflexive inverses and , the reflexive inverse can be expressed as follows, We choose , as the Moore-Penrose inverses, and as the projector with respect to the Moore-Penrose inverse and show that the obtained matrixis the Moore-Penrose inverse of . For this, it is enough to proof that satisfies the conditions (3) and (4) in Definition 1.
Since by Lemma 3, , then can be expressed as So, Since condition (3) is satisfied by components, namely, it follows that satisfies condition (3) as well; i.e., .
Similar, it can be shown that satisfies condition (4). Hence, the Moore-Penrose inverse of can be expressed by (49). From this (46) immediately follow.
The equations (47) can be proved similarly.
Now we demonstrate the principal theorem of this section.
Theorem 20. Assume that , , , , , , , , , , , and system (6) is solvable, then the general solution to our system can be formed aswhere are free matrices over with allowable dimensions.
Proof. Our proof contains three parts. At the first step, we show that the matrices and have the forms ofwhere and are any pair of particular solution to system (6), , , , and are free matrices of able shapes over , are solutions to system (6). At the second step, we display that any couple of solutions and to system (6) can be established as (56) and (57), respectively. At the end, we confirm that and are a couple of particular solutions to system (6).
Now we prove that a couple of matrices and having the shape of (56) and (57), respectively, are solutions to system (6). Observe that It is evident that having the form (56) is a solution of , and and having the form (57) is a solution to . Now we are left to show that is satisfied by and given in (56) and (57). By Lemma 4, we haveandObserve that and by using (61) and (62), we arrive that Conversely, assume that and are any couple of solutions to our system (6). By Lemma 7, we have Observe that producesOn the same lines, we can getIt is manifest that and defined in (66)-(67) are also solution pair ofSince Hence by Lemma 6, and can be written aswhere and are a couple of special solutions to (68) and and are free matrices with agreeable dimensions. Using (70) and (71) in (66) and (67), respectively, we get where and It is evident that and are a couple of solutions to system (6). It is clear that and can be represented by (56) and (57), respectively. Lastly, by putting , and equal to zero in (24) and (25), we conclude that and are special solutions to system (6). Hence the expressions (54) and (55) represent the general solution to system (6) and the theorem is completed.
Comment 1. We have established a novel expression of the general solution to system (6) in Theorem 20 which is different from one created in . With the help of this novel expression, we can explore the least-norm of the general solution which can not be studied with the help of the expression given in , which is one of the advantage of our new expression.
Now we discuss some special cases of our system.
If and disappear in Theorem 20, then we gain the following conclusion.
Corollary 22. Denote , , , , ,