Determinantal Representations of General and ( Skew-) Hermitian Solutions to the Generalized Sylvester-Type Quaternion Matrix Equation

and Applied Analysis 3 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 rowcolumn determinants. Suppose S푛 is the symmetric group on the set I푛 = {1, . . . , n}. Let A ∈ H. Row determinants of A along each row can be defined as follows. Definition 1 (see [32]). The ith row determinant ofA = (a푖푗) ∈ H is defined for all i = 1, . . . , n by putting r det푖A = ∑ 휎∈푆푛 (−1)푛−푟 ⋅ (a푖푖푘1 a푖푘1 푖푘1+1 . . . a푖푘1+푙1 푖) . . . (a푖푘푟 푖푘푟+1 . . . a푖푘푟+푙푟 푖푘푟 ) , σ = (ii푘1i푘1+1 . . . i푘1+푙1) (i푘2i푘2+1 . . . i푘2+푙2) . . . (i푘푟 i푘푟+1 . . . i푘푟+푙푟) , (5) where i푘2 < i푘3 < ⋅ ⋅ ⋅ < i푘푟 and i푘푡 < i푘푡+푠 for all t = 2, . . . , r and s = 1, . . . , l푡. Similarly, for a column determinant along an arbitrary column, we have the following definition. Definition 2 (see [32]). The jth column determinant of A = (a푖푗) ∈ H is defined for all j = 1, . . . , n by putting c det푗A = ∑ 휏∈푆푛 (−1)푛−푟 ⋅ (a푗푘푟 푗푘r+푙푟 . . . a푗푘푟+1푗푘푟 ) . . . (a푗푗푘1+푙1 . . . a푗푘1+1푗푘1a푗푘1 푗) , τ = (j푘푟+푙푟 . . . j푘푟+1j푘푟) . . . (j푘2+푙2 . . . j푘2+1j푘2) ⋅ (j푘1+푙1 . . . j푘1+1j푘1j) , (6) where j푘2 < j푘3 < ⋅ ⋅ ⋅ < j푘푟 and j푘푡 < j푘푡+푠 for t = 2, . . . , r and s = 1, . . . , l푡. So an arbitrary n × n quaternion matrix inducts a set from n row determinants and n column determinants that are different in general. Only for Hermitian A, we have [32], r det1A = ⋅ ⋅ ⋅ = r det푛A = c det1A = ⋅ ⋅ ⋅ = c det푛A ∈ R, (7) which enables defining the determinant of aHermitianmatrix by putting detA fl r det푖A = c det푖A (8) for all i = 1, . . . , n. Its properties are similar to the properties of an usual (commutative) determinant and they have been completely explored in [32] by using row and column determinants that are so defined only by construction. We note the following that will be required below. Lemma 3. LetA ∈ H.Then c det푖A∗ = r det푖A, r det푖A∗ = c det푖A. 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 α fl {α1, . . . , α푘} ⊆ {1, . . . , m} and β fl {β1, . . . , β푘} ⊆ {1, . . . , n} be subsets of the order 1 ≤ k ≤ min{m, n}. A훼훽 denotes a submatrix ofAwhose rows are indexed byα and the columns indexed by β. So A훼훼 denotes a principal submatrix of A with rows and columns indexed by α. If A ∈ H is Hermitian, then |A|훼훼 denotes the corresponding principal minor of detA. Let L푘,푛 fl {α : α = (α1, . . . , α푘), 1 ≤ α1 < ⋅ ⋅ ⋅ < α푘 ≤ n} denote the collection of strictly increasing sequences of k integers chosen from {1, . . . , n} for all 1 ≤ k ≤ n. Then, for fixed i ∈ α and j ∈ β, the collection of sequences of row indexes that contain the index i is denoted by I푟,푚{i} fl {α : α ∈ L푟,푚, i ∈ α}; similarly, the collection of sequences of column indexes that contain the index j is denote by J푟,푛{j} fl {β : β ∈ L푟,푛, j ∈ β}. Let a.푗 be the jth column and a푖. be the ith row of A. Suppose A.푗(b) denote the matrix obtained from A by replacing its jth column with the column b andA푖.(b) denote the matrix obtained from A by replacing its ith row with the row b. Denote by a∗.푗 and a ∗ 푖. the jth column and the ith row of A, respectively. Theorem 4 (see [34]). If A ∈ H 푟 , then the Moore-Penrose inverse A = (a† 푖푗) ∈ H have the following determinantal representations, a† 푖푗 = ∑훽∈퐽푟,푛{푖} c det푖 ((A∗A).푖 (a.푗))훽 ∑훽∈퐽푟,푛 |A∗A|훽 , (9) a† 푖푗 = ∑훼∈퐼푟,푚{푗} r det푗 ((AA ∗)푗. (a푖.))훼 ∑훼∈퐼푟,푚 |AA∗|훼훼 . (10) Remark 5. For an arbitrary full-rank matrix A ∈ H 푟 , a column vector b ∈ H and a row vector c ∈ H we put c det푖 ((A∗A).푖 (b)) = ∑ 훽∈퐽푛,푛{푖} c det푖 ((A∗A).푖 (b))훽 , det (A∗A) = ∑ 훽∈퐽푛,푛 󵄨󵄨󵄨󵄨A∗A󵄨󵄨󵄨󵄨훽훽 when r = n, r det푗 ((AA∗)푗. (c)) = ∑ 훼∈퐼푚,푚{푗} r det푗 ((AA∗)푗. (c))훼 , det (AA∗) = ∑ 훼∈퐼푚,푚 󵄨󵄨󵄨󵄨AA∗󵄨󵄨󵄨󵄨훼훼 when r = m. (11) 4 Abstract and Applied Analysis Remark 6. First note that (A∗)† = (A†)∗. Because of symbol equivalence, we shall use the denotation A fl (A∗)† aswell. So by Lemma 3, for the Hermitian adjoint matrix A ∈ H 푟 determinantal representations of its Moore-Penrose inverse (A∗)† = ((a∗ 푖푗)†) ∈ H are (a∗ 푖푗) = (a푗푖) = ∑훼∈퐼푟,푛{푗} r det푗 ((A ∗A)푗. (a푖.))훼 ∑훼∈퐼푟,푛 |A∗A|훼훼 , (12) (a∗ 푖푗) = ∑훽∈퐽푟,푚{푖} c det푖 ((AA∗).푖 (a.푗))훽 ∑훽∈퐽푟,푚 |AA∗|훽 . (13) Since the projection matrices AA š P퐴 = (p푖푗) and AA š Q퐴 = (q푖푗) are Hermitian, then p푖푗 = p푗푖 and q푖푗 = q푗푖 for all i ̸ = j. So due to Theorem 4 and Remark 6 we have evidently the following corollaries. Corollary 7. If A ∈ H 푟 , then the projection matrix P퐴 = (p푖푗)푛×푛 has the determinantal representations p푖푗 = ∑훽∈퐽푟,푛{푖} c det푖 ((A ∗A).푖 (ȧ.푗))훽 ∑훽∈퐽푟,푛 |A∗A|훽 = ∑훼∈퐼푟,푛{푗} r det푗 ((A ∗A)푗. (ȧ푖.))훼 ∑훼∈퐼푟,푛 |A∗A|훼훼 , (14) where ȧ.푗 and ȧ푖. are the jth column and ith row ofAA ∈ H, respectively. Corollary 8. If A ∈ H 푟 , then the projection matrix AA š Q퐴 = (q푖푗)푚×푚 has the determinantal representation q푖푗 = ∑훼∈퐼푟,푚{푗} r det푗 ((AA ∗)푗. (ä푖.))훼 ∑훼∈퐼푟,푚 |AA∗|훼훼 = ∑훽∈퐽푟,푚{푖} c det푖 ((AA ∗).푖 (ä.푗))훽 ∑훼∈퐽푟,푚 |AA∗|훽 , (15) where ä푖. and ä푗. are the ith row and the jth column of AA ∈ H. Determinantal representations of orthogonal projectors L퐴 fl I − AA and R퐴 fl I − AA induced from A can be derived similarly. Theorem 9 (see [3]). Let A ∈ H, B ∈ H, C ∈ H be known and X ∈ H be unknown. Then the matrix equation AXB = C (16) is consistent if and only if AACBB = C. In this case, its general solution can be expressed as X = ACB + L퐴V +WR퐵, (17) where V,W are arbitrary matrices over H with allowable dimensions. Theorem 10 (see [35]). Let A ∈ H 푟1 , B ∈ H 푟2 . Then the partial solution X = ACB = (x0푖푗) ∈ H to (16) has determinantal representations, x0푖푗 = ∑훽∈퐽푟1 ,푛{푖} c det푖 ((A∗A).푖 (d.푗))훽 ∑훽∈퐽푟1 ,푛 |A∗A|훽 ∑훼∈퐼푟2,푟 |BB∗|훼훼 , (18) or x0푖푗 = ∑훼∈퐼푟2,푟{푗} r det푗 ((BB ∗)푗. (d퐴푖. ))훼 ∑훽∈퐽푟1 ,푛 |A∗A|훽 ∑훼∈퐼푟2,푟 |BB∗|훼훼 , (19) where d퐵.푗 = [ ∑ 훼∈퐼푟2 ,푟{푗} r det푗 ((BB∗)푗. (c̃푘.))훼훼] ∈ H푛×1, k = 1, . . . , n, d퐴푖. = [ ∑ 훽∈퐽푟1 ,푛{푖} c det푖 ((A∗A).푖 (c̃.푙))훽] ∈ H1×푟, l = 1, . . . , r, (20) are the column vector and the row vector, respectively. c̃푖. and c̃.푗 are the ith row and the jth column of C̃ = ACB. Corollary 11. Let A ∈ H 푘 , C ∈ H be known and X ∈ H be unknown. Then the matrix equation AX = C is consistent if and only if AAC = C. In this case, its general solution can be expressed as X = AC + L퐴V, where V is an arbitrary matrix over H with an allowable dimension. The partial solution X = AC has the following determinantal representation, x0푖푗 = ∑훽∈퐽푘,푛{푖} c det푖 ((A∗A).푖 (ĉ.푗))훽 ∑훽∈퐽푘,푛 |A∗A|훽 . (21) where ĉ.푗 is the jth column of Ĉ = AC. Corollary 12. Let B ∈ H 푘 , C ∈ H be given and X ∈ H be unknown. Then the equation XB = C is solvable if and only if C = CBB and its general solution isX = CB†+WR퐵, where W is any matrix with an allowable dimension. Moreover, its partial solutionX = CB has the determinantal representation, x푖푗 = ∑훼∈퐼푘,푟{푗} r det푗 ((BB ∗)푗. (ĉ푖.))훼 ∑훼∈퐼푘,푟 |BB∗|훼훼 . (22) where ĉ푖. is the ith row of Ĉ = CB. Abstract and Applied Analysis 5and Applied Analysis 5 2.3. Determinantal Representations of the General Solution to the Sylvester Matrix Equation (3) Lemma 13 (see [3]). Let A ∈ H, B ∈ H, C ∈ H, D ∈ H, E ∈ H. Put M = R퐴C, N = DL퐵, S = CL푀. Then the following results are equivalent. (i) Eq. (3) has a pair solution (X,Y), where X ∈ H, Y ∈ H. (ii) R푀R퐴E = 0, R퐴EL퐷 = 0, EL퐷L푁 = 0, R퐶EL퐵 = 0. (iii) Q푀R퐴EP퐷 = R퐴E, Q퐶EL퐵P푁 = EL퐵. (iv) rank [A C E] = rank [A C], rank [B∗ D E∗] = rank [B∗ D∗], rank [ A E 0 D ] = rank [ A 0 0 D ], rank [ C E 0 B ] = rank [ C 0 0 B ]. In that case, the general solution to (3) can be expressed as X = AEB − A†CM†R퐴EB† − A†SC†EL퐵N†DB† − −A†SVR푁DB† + L퐴U + ZR퐵, (23) Y = M†R퐴ED† + L푀S†SC†EL퐵N† + L푀 (V − S†SVNN†) +WR퐷, (24) where U, V, Z, and W are arbitrary matrices over H obeying agreeable dimensions. Some simplifications of (23) and (24) can be derived due to the quaternionic analog of the following proposition. Lemma 14 (see [57]). If A ∈ H is Hermitian and idempotent, then for any matrix B ∈ H the following equations hold A (BA)† = (BA)† , (AB)† A = (AB)† . (25) Since R퐴, L퐵, and L푀 are projectors, then by Lemma 14 the simplifications of (23) and (24) are as follows: X = AEB − ACMEB − ASCENDB − A†SVR푁DB† + +L퐴U + ZR퐵, Y = MED + P푆C†EN† + L푀 (V − P푆VQ푁) +WR퐷. (26) By putting U,V,Z, and W as zero-matrices, we obtain the partial solution to (3), X = AEB − ACMEB − A†SC†EN†DB†, (27) Y = MED + P푆C†EN†. (28) The following theoremgives determinantal representations of (27)-(28). Theorem 15 (see [45]). Let A ∈ H 푟1 , B ∈ H 푟2 , C ∈ H 푟3 , D ∈ H 푟4 , rankM = r5, rankN = r6, rank S = r7. Then the pair solution (27)-(28), X = (x푖푗) ∈ H, Y = (x푔푓) ∈ H, to (3) by the components x푖푗 = x(1) 푖푗 − x(2) 푖푗 − x(3) 푖푗 , y푔푓 = y(1) 푔푓 + y(2) 푔푓 , (29) has the determinantal representation, as follows. (i) x(1) 푖푗 = ∑훽∈퐽푟1 ,푛{푖} c det푖 ((A∗A).푖 (d.푗))훽 ∑훽∈퐽푟1 ,푛 |A∗A|훽 ∑훼∈퐼푟2 ,푟 |BB∗|훼훼 , (30) or x(1) 푖푗 = ∑훼∈퐼푟2 ,푟{푗} r det푗 ((BB ∗)푗. (d퐴푖. ))훼 ∑훽∈퐽푟1 ,푛 |A∗A|훽 ∑훼∈퐼푟2,푟 |(BB∗ |훼훼 , (31)


Introduction
Let H 푚×푛 and H 푚×푛 푟 stand for the set of all  ×  matrices and for its subset of matrices with rank , respectively, over the quaternion skew field where R is the real number field.For A ∈ H 푚×푛 , the symbol A * stands for conjugate transpose (Hermitian adjoint) of A.
The Moore-Penrose inverse of A ∈ H 푚×푛 is called the unique matrix X ∈ H 푛×푚 satisfying the following four equa- ( It is denoted by A † .The two-sided generalized Sylvester matrix equation has been well studied in matrix theory.For instance, Huang [1] obtained necessary and sufficient conditions for the existence of solutions to (3) with X = Y over the quaternion skew field.Baksalary and Kala [2] 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 [5].In [6] 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. [7] 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 [8] and to solve the minimum Frobenius norm residual problem for a system of Sylvester-type matrix equations over generalized reflexive matrix in [9].Systems of periodic discrete-time coupled Sylvester quaternion matrix equations [10], systems of quaternary coupled Sylvester-type real quaternion matrix equations [11], and optimal pole assignment of linear systems by the Sylvester matrix equations [12] 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 [15].S ¸ims ¸ek et al. [16] established the precise solutions on the minimum residual and matrix nearness problems of the quaternion matrix equation (AXB, DXE) = (C, F) 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 [19] and Hermitian solutions over a split quaternion algebra [20] 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 Sylvestertype matrix equation involving * -Hermicity Chang and Wang [23] 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. [24] have given a representation of the least-squares Hermitian (skew-Hermitian) solution to the matrix equation (4).Zhang [25] obtained a representation of the general Hermitian nonnegative-definite (respectively positive-definite) solution to (4) within the complex settings.Yuan et al. [26] derived the expression of Hermitian solution for the matrix nearness problem associated with the quaternion matrix equation (4).Wang et al. [27] gave a necessary and sufficient condition for the existence and an expression for the re-nonnegative definite solution to (4) over H by using the decomposition of pairwise matrices.Wang et al. [28] established the extreme ranks for the general (skew-)Hermitian solution to (4) over H. 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][30][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 [34], the Drazin inverse [35], the W-weighted Drazin inverse [36], and the weighted Moore-Penrose inverse [37], have been derived by the author.These determinantal representations were used to obtain explicit representation formulas for the minimum norm least squares solutions [38] and weighted Moore-Penrose inverse solutions [39] 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][41][42].Recently, determinantal representations of solutions to some systems of quaternion matrix equations [43,44] and, in [45], 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 A 2 푇,푆 [46] and the Bott-Duffin inverse [47].Song et al. obtained the Cramer rules for the solutions of restricted matrix equations [48] and for the generalized Stein quaternion matrix equation [49], and so forth.Moreover, Song et al. [50] have just recently considered determinantal representations of the general solution to the generalized Sylvester matrix equation (3) over H 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 [50] 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 AXB = C 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.

Preliminaries
We commence with the following preliminaries which have crucial function in the construction of the chief outcomes of the following sections.

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][52][53][54][55][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 rowcolumn determinants.
Suppose  푛 is the symmetric group on the set  푛 = {1, . . ., }.Let A ∈ H 푛×푛 .Row determinants of A along each row can be defined as follows.
Definition 2 (see [32]).The th column determinant of A = ( 푖 푗 ) ∈ H 푛×푛 is defined for all  = 1, . . .,  by putting where So an arbitrary  ×  quaternion matrix inducts a set from  row determinants and  column determinants that are different in general.Only for Hermitian A, we have [32], which enables defining the determinant of a Hermitian matrix by putting for all  = 1, . . ., .
Its properties are similar to the properties of an usual (commutative) determinant and they have been completely explored in [32] by using row and column determinants that are so defined only by construction.We note the following that will be required below.

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 a .푗be the th column and a 푖.be the th row of A. Suppose A .푗(b) denote the matrix obtained from A by replacing its th column with the column b and A 푖. (b) denote the matrix obtained from A by replacing its th row with the row b.Denote by a * .푗and a * 푖. the th column and the th row of A * , respectively.Theorem 4 (see [34]).
Remark 5.For an arbitrary full-rank matrix A ∈ H 푚×푛 푟 , a column vector b ∈ H 푛×1 and a row vector c ∈ H 1×푚 we put Remark 6.First note that (A * ) † = (A † ) * .Because of symbol equivalence, we shall use the denotation A †, * fl (A * ) † as well.So by Lemma 3, for the Hermitian adjoint matrix A * ∈ H 푛×푚 푟 determinantal representations of its Moore-Penrose inverse Since the projection matrices where ȧ .푗and ȧ 푖. are the th column and th row of A * A ∈ H 푛×푛 , respectively.
where ä 푖. and ä 푗. are the th row and the th column of AA * ∈ H 푚×푚 .
Determinantal representations of orthogonal projectors L 퐴 fl I − A † A and R 퐴 fl I − AA † induced from A can be derived similarly.
Theorem 9 (see [3]).Let A ∈ H 푚×푛 , B ∈ H 푟×푠 , C ∈ H 푚×푠 be known and X ∈ H 푛×푟 be unknown.Then the matrix equation is consistent if and only if AA † CB † B = C.In this case, its general solution can be expressed as where V, W are arbitrary matrices over H with allowable dimensions.
Theorem 10 (see [35]).16) has determinantal representations, or where are the column vector and the row vector, respectively.c푖 .and c.푗 are the th row and the th column of C = A * CB * .

Corollary 11.
Let A ∈ H 푚×푛 푘 , C ∈ H 푚×푠 be known and X ∈ H 푛×푠 be unknown.Then the matrix equation AX = C is consistent if and only if AA † C = C.In this case, its general solution can be expressed as X = A † C + L 퐴 V, where V is an arbitrary matrix over H with an allowable dimension.The partial solution X 0 = A † C has the following determinantal representation, where ĉ.푗 is the th column of Ĉ = A * C.
Corollary 12. Let B ∈ H 푟×푠 푘 , C ∈ H 푛×푠 be given and X ∈ H 푛×푟 be unknown.Then the equation XB = C is solvable if and only if C = CB † B and its general solution is X = CB † +WR 퐵 , where W is any matrix with an allowable dimension.Moreover, its partial solution X = CB † has the determinantal representation, where ĉ푖 . is the th row of Ĉ = CB * .
Then the following results are equivalent.
(i) Eq. ( 3) has a pair solution (X, Y), where In that case, the general solution to (3) can be expressed as where U, V, Z, and W are arbitrary matrices over H obeying agreeable dimensions.Some simplifications of ( 23) and ( 24) can be derived due to the quaternionic analog of the following proposition.
Lemma 14 (see [57]).If A ∈ H 푛×푛 is Hermitian and idempotent, then for any matrix B ∈ H 푚×푛 the following equations hold Since R 퐴 , L 퐵 , and L 푀 are projectors, then by Lemma 14 the simplifications of ( 23) and ( 24) are as follows: By putting U, V, Z, and W as zero-matrices, we obtain the partial solution to (3), The following theorem gives determinantal representations of ( 27)- (28).
are the column vector and the row vector, respectively.c (1)  푓. and c (1)  .푠are the th row and the th column of C 1 fl A * CM * .(iii) where s (1)  .푡 is the th column of S 1 fl A * S, e (3)  .푓 is the th column of E 3 fl C * E, and are the row vector and the column vector, respectively.d (1)  .푘and d (1)  푙. are the th column and the th row of D 1 = N * DB * .(iv) or where are the column vector and the row vector, respectively.e (4)  푘. and e (4)  .푙are the th row and the th column of E 4 fl M * ED * .(v) where c det 푡 ((C * C) .푡(e (5)  .푘)) are the row vector and the column vector, respectively.e (5)  .푘and e (5)  푙. are the th column and the th row of E 5 = C * EN * .

Determinantal Representations of the General and (Skew-)Hermitian Solutions to (4)
Now consider (4).Since for an arbitrary matrix A it is evident that (i) Equation ( 4) has a pair solution (X, Y), where In that case, the general solution to (4) can be expressed as follows: where U, V, Z, and W are arbitrary matrices over H with allowable dimensions.
By putting U, V, Z, and W as zero-matrices with compatible dimensions, we obtain the following partial solution to (4), The following theorem gives determinantal representations of ( 43)- (44).
and c (4, * )   .푙 are the th row and the th column of C * 4 fl M * CB.
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), 푖 푗 ), we have By using determinantal representations ( 9) and ( 13) of the Moore-Penrose inverses A † and (A * ) † , respectively, we obtain Suppose e 푙. and e .푙are the unit row vector and the unit column vector, respectively, such that all their components are 0, except the th components, which are 푓푠 , then If we denote by the th component of a row vector and Applied Analysis 9 Further, is evident that , so the first term of (43) has the determinantal representation (46), where k 푖. is (48).
If we denote by 푓푠 the th component of a column vector So another determinantal representation of the first term of ( 43) is (47), where k .푗 is (49).
where φ푖 . is the th row of Φ, then we have (50).
and c (4, * )   .푙 are the th row and the th column of C * 4 fl M * CB.
Due to [24], the following lemma can be generalized to H.
(i)  (1)  푗푖 =  (1)  푖 푗 .(ii) where φ * .푗 is the th column of Φ * = A * CΦ * that is Hermitian adjoint to Φ from the point (i) of Theorem 17, and Abstract and Applied 11 )) are the row vector and the column vector, respectively.b (1, * )   .푓 and b (1)  푠. are the th column and the th row of where υ * 푖. is the th row of Υ * = Υ * S * A, the matrix where c * 푖. is the th row of C * = ΦCB, and Φ is obtained by (51).
are the th column and the th row of C * 4 fl B * CM.
are the row vector and the column vector, respectively.c (4)   .푞 and c (4)  푙. are the th column and the th row of C 4 fl M * CB.

An Example
In this section, we give an example to illustrate our results.Let us consider the matrix equation (97) So  11 obtained by Cramer's rule and the matrix method (94) are equal.Similarly, we can obtain for all  푖 푗 , ,  = 1, 2 and  11 .

Conclusions
Within the framework of the theory of row-column determinants, we have derived explicit determinantal representation formulas (analogs of Cramer's rule) of the general, Hermitian, and skew-Hermitian solutions to the Sylvester-type matrix equation AXA * + BYB * = C over the quaternion skew field.To accomplish that goal, we have used determinantal representations of the Moore-Penrose matrix inverse, which were previously introduced by the author.