Determinantal Representations of Solutions and Hermitian Solutions to Some System of Two-Sided Quaternion Matrix Equations

Within the framework of the theory of quaternion row-column determinants previously introduced by the author, we derive determinantal representations (analogs ofCramer’s rule) of solutions andHermitian solutions to the systemof two-sided quaternion matrix equationsA1XA ∗ 1 = C1 andA2XA∗2 = C2. Since theMoore-Penrose inverse is a necessary tool to solve matrix equations, we use determinantal representations of the Moore-Penrose inverse previously obtained by the theory of row-column determinants.


Introduction
The study of matrix equations and systems of matrix equations is an active research topic in matrix theory and its applications.Research on the classical system of two-sided matrix equations over the complex field, a principle domain, and the quaternion skew field has been actively ongoing for many years.For instance, Mitra [1, 2] gave necessary and sufficient conditions of system (3) over the complex field and the expression for its general solution.Navarra et al. [3] derived a new necessary and sufficient condition for the existence and a new representation of the general solution to (3) over the complex field and used the results to give a simple representation.Özgüler et al. [4] gave solutions to (3) over a principle domain.Wang [5] got solvability conditions of system (3) over the quaternion skew field and represented its general solution in terms of generalized inverses.
Since quaternion matrices play an important role in quantum mechanics, signal processing, and control theory, research on quaternion matrix equations and systems of quaternion matrix equations, their general solutions, especially Hermitian solutions, has been actively developing for more recent years (see, e.g., [6-26]).
Throughout the paper, we denote the real number field by R, the set of all  ×  matrices over the quaternion algebra by H 푚×푛 , and by H 푚×푛 푟 its subset of matrices of a rank .Let M(, H) be the ring of  ×  quaternion matrices.For A ∈ H 푛×푚 , the symbol A * stands for the conjugate transpose (Hermitian adjoint) matrix of A. The matrix A = ( 푖 푗 ) ∈ H 푛×푛 is Hermitian if A * = A.
Motivated by the wide application of quaternion matrix equations and in order to improve the theoretical development of solutions and Hermitian solutions to quaternion matrix equations, we consider a special case of (3), more specifically, Generalized inverses are useful tools used to solve matrix equations.The definition of the Moore-Penrose inverse matrix has been extended to quaternion matrices as follows.(4)   The main goal of this paper is to derive determinantal representations of solutions and Hermitian solutions to system (1) over the quaternion skew field using previously obtained determinantal representations of the Moore-Penrose inverse.Evidently, 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 [27].
Through the noncommutativity of the quaternion algebra when difficulties arise already in determining the quaternion determinant, the problem of the determinantal representation of generalized inverses only now can be solved due to the theory of column-row determinants introduced in [28, 29].Within the framework of the theory of columnrow determinants, determinantal representations of various kinds of generalized inverses (generalized inverses) solutions of quaternion matrix equations have been derived by the author (see, e.g., [30-39]) and by other researchers (see, e.g., [40-43]).
The paper is organized as follows.In Section 2, we start with preliminary introduction of row-column determinants, determinantal representations of the Moore-Penrose inverse previously obtained within the framework of the theory of row-column determinants, and Cramer's rules for the twosided matrix equation and of its special cases, left-and rightsided equations.We derive some simplified expressions of general and partial solutions to (3), a solvability criterion and expressions of general and partial solutions to system (1), and determinantal representations (analogs of Cramer's rule) of its solution and Hermitian solution.A numerical example to illustrate the main results is considered in Section 4. Finally, the conclusion is drawn in Section 5.
Since [28] for Hermitian A we have the determinant of a Hermitian matrix is defined by putting det A fl rdet 푖 A = cdet 푖 A for all  = 1, . . ., .
The properties of row and column determinants are completely explored in [29].We note the following that will be required below.

Lemma 4. Let
We shall use the following notations.Let  fl { 1 , . . .,  푘 } ⊆ {1, . . ., } and  fl { 1 , . . .,  푘 } ⊆ {1, . . ., } be subsets of the order 1 ≤  ≤ min{, }.Let A 훼 훽 be a submatrix of A whose rows are indexed by  and columns indexed by .Similarly, let A 훼 훼 be a principal submatrix of A whose rows and columns are indexed by .If A ∈ M(, H) is Hermitian, then |A| 훼 훼 is the corresponding principal minor of det A. For 1 ≤  ≤ , the collection of strictly increasing sequences of  integers chosen from {1, . . ., } is denoted by  푘,푛 fl { : Let a .푗be the th column and a 푖.be the th row of A. Suppose A .푗(b) denotes the matrix obtained from A by replacing its th column with column b, and A 푖. (b) denotes 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 5 (see [30]).If A ∈ H 푚×푛 푟 , then the Moore-Penrose inverse A † = ( † 푖 푗 ) ∈ H 푛×푚 has the following determinantal representations, and Remark .For an arbitrary full-rank matrix where a column vector d .푗and a row vector d 푖. have appropriate sizes.
The orthogonal projectors L 퐴 fl I − A † A and R 퐴 fl I − AA † induced by A will be used below.
Theorem 10 (see [44]).Let A ∈ H 푚×푛 , B ∈ H 푟×푠 , C ∈ H 푚×푠 be known and X ∈ H 푛×푟 be unknown.en 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 appropriate dimensions.
Theorem 11 (see [31]). 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 12.
Let A ∈ H 푚×푛 푘 , C ∈ H 푚×푠 be known and X ∈ H 푛×푠 be unknown.en 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 appropriate dimensions.e partial solution X 0 = A † C has the following determinantal representation, where ĉ.푗 is the th column of Ĉ = A * C.
Corollary 13.Let B ∈ H 푟×푠 푘 , C ∈ H 푛×푠 be given and X ∈ H 푛×푟 be unknown.en 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 conformable dimension.Moreover, its partial solution X = CB † has the determinantal representation, where ĉ푖 . is the th row of Ĉ = CB * .

Cramer's Rules for the Solution and Hermitian Solution to System (3)
First, consider the general system (1).

en system ( ) is consistent if and only if
In that case, the general solution to ( ) can be expressed as the following: where Z and W are arbitrary matrices over H with compatible dimensions.
Some simplification of ( 25) can be derived due to the quaternionic analogue of the following proposition.
Lemma 15 (see [45]).If A ∈ H 푛×푛 is Hermitian and idempotent, then the following equation holds for any matrix It is evident that if A ∈ H 푛×푛 is Hermitian and idempotent, then the following equation is true as well: Since L 퐴 1 , R 퐵 1 , and R 퐻 are projectors, then by ( 26) and ( 27), we have, respectively, Using ( 28) and ( 23), we obtain the following expressions of (25): By putting Z, W as zero-matrices in (29), we obtain the following partial solution of (25): Now consider system (1).Since so Due to the above, we obtain the following analog of Lemma 14.

Lemma 16. Let
H 푘×푘 be given and X ∈ H 푛×푛 is to be determined.en system ( ) is consistent if and only if In that case, the general solution to ( ) can be expressed as follows: Z and W are arbitrary matrices over H with compatible dimensions.
and c (21) .푙 are the th row and the th column of The third term of ( 34) can be obtained similarly as well.So, or where cdet 푖 ((T * T) .푖(c (22) .푙)) are the column vector and the row vector, respectively.c (22) 푞.
are the th row, and c (22) .푙 are the th column of C 22 = T * C 2 H.The following expressions give some simplifications in computing.Since (9) for determinantal representations of H † and T † in the fourth term of (34), we obtain where a (2,퐻) .푖 , a (2,푇) .푖 are the th columns of the matrices H * A 2 and T * A 2 , respectively;  (1) 푧푓 is the first term;  푓푗 is the ()th element of P 퐴 2 with determinantal representation by (14) as where ȧ (2)  .푗 is the th column of A * 2 A 2 .Note that Similarly to the previous case,  (5) )) (vi) Consider the sixth term by analogy to the fourth term.So, where and cdet 푞 ((T * T) .푞(c (23) .푙)) are the column vector and the row vector, respectively.c (23) 푞.
are the th column of T * A 2 and the th row of A * 2 H = A * 2 A 2 L 퐴 1 , respectively.Hence, we prove the following theorem.
Due to Khatri and Mitra [46], the next lemma can be generalized to H.

Lemma 18.
Let A ∈ H 푚×푛 and B ∈ H 푚×푚 and B = B * be known and X ∈ H 푛×푛 be unknown.en the matrix equation has a Hermitian solution if and only if Q 퐴 B = B.In that case, the general Hermitian solution of ( ) is where V ∈ H 푛×푛 is any matrix.
As it follows from the above, if rank A = , then the determinantal representation of the partial Hermitian solution where are the row vector and the column vector and b (1) .푠and b (1) 푓. are the th column and the th row of B 1 fl A * BA, respectively.
, a (2,퐻) .푓 are the th row of A * 2 T and the th column of H * A 2 , respectively.

An Example
In this section, we give an example to illustrate our results.Let us consider the system of matrix equations where then rank A 1 = 2, rank A 2 = 2.By Theorem 5, one can find Since 1 T * , then, by Lemma 16, system (75) is consistent.
First, we can find the solution of (75) by direct calculation.Since X = ∑ 훿 X 훿 , where , then by ( 41) cdet 1 ((A * 1 A 1 ) .1 (c (11) .Similarly, we can obtain for all the remainder solutions.Note that we used Maple with the package CLIFFORD in the calculations.

Conclusions
Within the framework of the theory of quaternion rowcolumn determinants previously introduced by the author, we have derived determinantal representations (analogs of Cramer's rule) of the general and Hermitian solutions to the system of two-sided quaternion matrix equations A 1 XA * 1 = C 1 and A 2 XA * 2 = C 2 .Since the Hermitian solution is Y = (1/2)(X + X * ), where X is an arbitrary solution, the determinantal representation of X * is derived as well.To accomplish that goal, we have used the determinantal representations of the Moore-Penrose matrix inverse which were previously introduced by the author.

1 C 1 A 1 =
find the solution of (75) by its determinantal representation by Theorem 17.Since C 11 = A *