Determinantal Representations of General and (Skew-)Hermitian Solutions to the Generalized Sylvester-Type Quaternion Matrix Equation
In this paper, we derive explicit determinantal representation formulas of general, Hermitian, and skew-Hermitian solutions to the generalized Sylvester matrix equation involving -Hermicity over the quaternion skew field within the framework of the theory of noncommutative column-row determinants.
Let and stand for the set of all matrices and for its subset of matrices with rank , respectively, over the quaternion skew fieldwhere is the real number field. For , the symbol stands for conjugate transpose (Hermitian adjoint) of . A matrix is Hermitian if .
The Moore-Penrose inverse of is called the unique matrix satisfying the following four equationsIt is denoted by .
The two-sided generalized Sylvester matrix equation has been well studied in matrix theory. For instance, Huang  obtained necessary and sufficient conditions for the existence of solutions to (3) with over the quaternion skew field. Baksalary and Kala  derived the general solution to (3) expressed in terms of generalized inverses which has been extended to an arbitrary division ring and on any regular ring with identity in [3, 4]. Ranks and independence of solutions to (3) were explored in . In  expressions, as well as necessary and sufficient conditions, were given for the existence of the real and pure imaginary solutions to the consistent quaternion matrix equation (3).
The high research activities on Sylvester-type matrix equations can be observed lately. In particular, we note the following papers concerning methods of their computing solutions. Liao et al.  established a direct method for computing its approximate solution using the generalized singular value decomposition and the canonical correlation decomposition. Efficient iterative algorithms were presented to solve a system of two generalized Sylvester matrix equations in  and to solve the minimum Frobenius norm residual problem for a system of Sylvester-type matrix equations over generalized reflexive matrix in .
Systems of periodic discrete-time coupled Sylvester quaternion matrix equations , systems of quaternary coupled Sylvester-type real quaternion matrix equations , and optimal pole assignment of linear systems by the Sylvester matrix equations  have been explored. Some constraint generalized Sylvester matrix equations [13, 14] were studied recently.
Special solutions to Sylvester-type quaternion matrix equations have been actively studied. Roth’s solvability criteria for some Sylvester-type matrix equations were extended over the quaternion skew field with a fixed involutive automorphism in . Şimşek et al.  established the precise solutions on the minimum residual and matrix nearness problems of the quaternion matrix equation for centrohermitian and skew-centrohermitian matrices. Explicit solutions to some Sylvester-type quaternion matrix equations (with -conjugation) were established by means of Kronecker map and complex representation of a quaternion matrix in [17, 18]. The expressions of the least squares solutions to some Sylvester-type matrix equations over nonsplit quaternion algebra  and Hermitian solutions over a split quaternion algebra  were derived. Solvability conditions and general solution for some generalized Sylvester real quaternion matrix equations involving -Hermicity were given in [21, 22].
Many authors have paid attention also to the Sylvester-type matrix equation involving -Hermicity Chang and Wang  derived expressions for the general symmetric solution and the general minimum-2-norm symmetric solution to the matrix equation (4) within the real settings. Xu et al.  have given a representation of the least-squares Hermitian (skew-Hermitian) solution to the matrix equation (4). Zhang  obtained a representation of the general Hermitian nonnegative-definite (respectively positive-definite) solution to (4) within the complex settings. Yuan et al.  derived the expression of Hermitian solution for the matrix nearness problem associated with the quaternion matrix equation (4). Wang et al.  gave a necessary and sufficient condition for the existence and an expression for the re-nonnegative definite solution to (4) over by using the decomposition of pairwise matrices. Wang et al.  established the extreme ranks for the general (skew-)Hermitian solution to (4) over .
Motivated by the vast application of quaternion matrices and the latest interest of Sylvester-type quaternion matrix equations, the main goal of the paper is to derive explicit determinantal representation formulas of the general, Hermitian, and skew-Hermitian solutions to (4) based on determinantal representations of the Moore-Penrose inverse.
Determinantal representation of a solution gives a direct method of its finding analogous to classical Cramer’s rule that has important theoretical and practical significance. However, determinantal representations are not so unambiguous even for generalized inverses within the complex or real settings. Through looking for their more applicable explicit expressions, there are various determinantal representations of generalized inverses (see, e.g., [29–31]). By virtue of noncommutativity of quaternions, the problem for determinantal representation of generalized quaternion inverses is even more complicated, and only now it can be solved due to the theory of column-row determinants introduced in [32, 33]. Within the framework of the theory of row-column determinants, determinantal representations of various generalized quaternion inverses, namely, the Moore-Penrose inverse , the Drazin inverse , the W-weighted Drazin inverse , and the weighted Moore-Penrose inverse , have been derived by the author. These determinantal representations were used to obtain explicit representation formulas for the minimum norm least squares solutions  and weighted Moore-Penrose inverse solutions  to some quaternion matrix equations and explicit determinantal representation formulas of both Drazin and W-weighted Drazin inverse solutions to some restricted quaternion matrix equations and quaternion differential matrix equations [40–42]. Recently, determinantal representations of solutions to some systems of quaternion matrix equations [43, 44] and, in , two-sided generalized Sylvester matrix equation (3) have been derived by the author as well.
Other researchers also used the row-column determinants in their developments. In particular, Song derived determinantal representations of the generalized inverse  and the Bott-Duffin inverse . Song et al. obtained the Cramer rules for the solutions of restricted matrix equations  and for the generalized Stein quaternion matrix equation , and so forth. Moreover, Song et al.  have just recently considered determinantal representations of the general solution to the generalized Sylvester matrix equation (3) over using row-column determinants as well. But their approach differs from ours because for determinantal representations of solutions we use only coefficient matrices of the equation, while in  supplementary matrices have been constructed and used.
The paper is organized as follows. In Section 2, we start with some remarkable results which have significant role during the construction of the main results of this paper. Elements of the theory of row-column determinants are given in Section 2.1, determinantal representations of the Moore-Penrose inverse and of the general solution to the quaternion matrix equation and its special cases are considered in Section 2.2, and the explicit determinantal representation of the general solution to (3) previously obtained within the framework of the theory of row-column determinants is in Section 2.3. The main results of the paper, namely, explicit determinantal representation formulas of the general, Hermitian, skew-Hermitian solutions to (4), are derived in Section 3. In Section 4, a numerical example to illustrate the main results is considered. Finally, in Section 5, the conclusions are drawn.
We commence with the following preliminaries which have crucial function in the construction of the chief outcomes of the following sections.
2.1. Elements of the Theory of Row-Column Determinants
Due to noncommutativity of quaternions, a problem of defining a determinant of matrices with noncommutative entries (which is also defined as noncommutative determinants) has been unsolved for a long time. There are several versions of the definition of noncommutative determinant (see, e.g., [51–56]). But any of the previous noncommutative determinants has not fully retained those properties which it has owned for matrices with commutative entries. Moreover, if functional properties of noncommutative determinant over a ring are satisfied, then it takes on a value in its commutative subset. This dilemma can be avoided thanks to the theory of row-column determinants.
Suppose is the symmetric group on the set . Let . Row determinants of along each row can be defined as follows.
Definition 1 (see ). The th row determinant of is defined for all by puttingwhere and for all and .
Similarly, for a column determinant along an arbitrary column, we have the following definition.
Definition 2 (see ). The th column determinant of is defined for all by puttingwhere and for and .
So an arbitrary quaternion matrix inducts a set from row determinants and column determinants that are different in general. Only for Hermitian , we have ,which enables defining the determinant of a Hermitian matrix by puttingfor all .
Its 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. We note the following that will be required below.
Lemma 3. Let . Then , .
2.2. Determinantal Representations of the Moore-Penrose Inverse with Applications to Some Quaternion Matrix Equations
For introducing determinantal representations of the Moore-Penrose inverse, the following notations will be used.
Let and be subsets of the order . denotes a submatrix of whose rows are indexed by and the columns indexed by . So denotes a principal submatrix of with rows and columns indexed by . If is Hermitian, then denotes the corresponding principal minor of .
Let denote the collection of strictly increasing sequences of integers chosen from for all . Then, for fixed and , the collection of sequences of row indexes that contain the index is denoted by ; similarly, the collection of sequences of column indexes that contain the index is denote by .
Let be the th column and be the th row of . Suppose denote the matrix obtained from by replacing its th column with the column and denote the matrix obtained from by replacing its th row with the row . Denote by and the th column and the th row of , respectively.
Theorem 4 (see ). If , then the Moore-Penrose inverse have the following determinantal representations,
Remark 5. For an arbitrary full-rank matrix , a column vector and a row vector we put
Remark 6. First note that . Because of symbol equivalence, we shall use the denotation as well. So by Lemma 3, for the Hermitian adjoint matrix determinantal representations of its Moore-Penrose inverse are
Corollary 7. If , then the projection matrix has the determinantal representations where and are the th column and th row of , respectively.
Corollary 8. If , then the projection matrix has the determinantal representation where and are the th row and the th column of .
Determinantal representations of orthogonal projectors and induced from can be derived similarly.
Theorem 9 (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 allowable dimensions.
Theorem 10 (see ). Let , . Then the partial solution to (16) has determinantal representations,orwhereare the column vector and the row vector, respectively. and are the th row and the th column of .
Corollary 11. 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 an allowable dimension. The partial solution has the following determinantal representation,where is the th column of .
Corollary 12. Let , be given and be unknown. Then the equation is solvable if and only if and its general solution is , where is any matrix with an allowable dimension. Moreover, its partial solution has the determinantal representation, where is the th row of .
2.3. Determinantal Representations of the General Solution to the Sylvester Matrix Equation (3)
Lemma 13 (see ). Let , , , , . Put , , . Then the following results are equivalent. (i)Eq. (3) has a pair solution , where , .(ii), , , .(iii), .(iv), , , . In that case, the general solution to (3) can be expressed aswhere , , , and are arbitrary matrices over obeying agreeable dimensions.
Lemma 14 (see ). If is Hermitian and idempotent, then for any matrix the following equations hold
Since , , and are projectors, then by Lemma 14 the simplifications of (23) and (24) are as follows: By putting , and as zero-matrices, we obtain the partial solution to (3),The following theorem gives determinantal representations of (27)-(28).
Theorem 15 (see ). Let , , , , , , . Then the pair solution (27)-(28), , , to (3) by the components has the determinantal representation, as follows.
(i)or where are the column vector and the row vector, respectively. and are the th row and the th column of .
(ii) where is th row of . are the column vector and the row vector, respectively. and are the th row and the th column of .
(iii)where is the th column of , is the th column of , and are the row vector and the column vector, respectively. and are the th column and the th row of .
(iv) orwhere are the column vector and the row vector, respectively. and are the th row and the th column of .
(v) where are the row vector and the column vector, respectively. and are the th column and the th row of .
3. Determinantal Representations of the General and (Skew-)Hermitian Solutions to (4)
Lemma 16. Let , , . Put , . Then the following results are equivalent.(i)Equation (4) has a pair solution , where , .(ii), , , .(iii), .(iv), , . In that case, the general solution to (4) can be expressed as follows: where , , , and are arbitrary matrices over with allowable dimensions.
Theorem 17. Let , , , . Then the partial pair solution (43)-(44) to (4), , , by the componentspossesses the following determinantal representations:
(i)orwhere are the row vector and the column vector, respectively; and are the th column and the th row of .
(ii)where is the th row of and is such that are the column vector and the row vector, respectively. and are the th row and the th column of and is the th row of .
(iii)where is the th column of , the matrix such that where is the th column of and is Hermitian adjoint to from (51).
(iv)orwhere are the column vector and the row vector, respectively. and are the th row and the th column of .
(v)where and such thatare the column vector and the row vector, respectively. and are the th row and the th column of .
Proof. The proof evidently follows from the proof of Theorem 15 by substitution corresponding matrices. For a better understanding more complete proof will be made in some points, and a few comments will be done in others.
(i) For the first term of (43), , we have By using determinantal representations (9) and (13) of the Moore-Penrose inverses and , respectively, we obtain Suppose and are the unit row vector and the unit column vector, respectively, such that all their components are , except the th components, which are . Denote . Since , then If we denote by the th component of a row vector , then