Cramer’s rules for the system of quaternion matrix equations with η-Hermicity

The system of two-sided quaternion matrix equations with g-Hermicity, A1XA 1 1⁄4 C1, A2XA 2 1⁄4 C2 is considered in the paper. Using noncommutative row-column determinants previously introduced by the author, determinantal representations (analogs of Cramer’s rule) of a general solution to the system are obtained. As special cases, Cramer’s rules for an g-Hermitian solution when C1 1⁄4 C 1 and C2 1⁄4 C 2 and for an g-skew-Hermitian solution when C1 1⁄4 C 1 and C2 1⁄4 C 2 are also explored.


Introduction
In the whole article, the notation R is reserved for the real number field and H mÂn stands for the set of all m Â n matrices over the quaternion skew field H mÂn r specifies its subset of matrices with a rank r. For given h ¼ h 0 þ h 1 i þ h 2 j þ h 3 k 2 H, the conjugate of h is h ¼ a 0 À h 1 i À h 2 j À h 3 k. For given A 2 H nÂm , A* represents the conjugate transpose (Hermitian adjoint) matrix of A. The matrix A 2 H nÂn is Hermitian if A* = A. A y means the Moore-Penrose inverse of A 2 H nÂm , i.e. the exclusive matrix X satisfying the following four equations ð1Þ AXA ¼ A; ð2Þ XAX ¼ X; ð3Þ ðAXÞ Ã ¼ AX; ð4Þ ðXAÞ Ã ¼ XA: Quaternions have ample use in diverse areas such, such as color imaging and computer science [1][2][3][4][5], fluid mechanics [6,7], quantum mechanics [8,9], the attitude orientation and spatial rigid body dynamics [10][11][12], signal processing [13][14][15], etc.
In this paper, we construct novel explicit determinantal representation formulas (an analog of Cramer's rule) of the general and g-(skew-)Hermitian solutions to the system (2), by using determinantal representations of the Moore-Penrose matrix that was obtained within in the framework of the theory of row-column noncommutative determinants. According to our best of knowledge, our Cramer's rule proposed is a unique direct method to compute the g-(skew-)Hermitian solutions to quaternion matrix equations unlike other similar works (see, e.g. [24-26, 29, 32]), where obtained explicit forms of solutions have mostly only theoretical significance.
In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, for generalized inverse matrices, in particular, Moore-Penrose matrices, there exist different determinantal representations even for matrices with real or complex entries as a result of the search of their more applicable explicit expressions (for the Moore-Penrose matrix, see, e.g., [39][40][41]). For quaternion matrices, in view of the noncommutativity of quaternions, the problem of the determinantal representation of generalized inverse matrices remained open for a long time and only now can be solved due to the theory of row-column determinants which were introduced in [42,43].
The remainder of the paper is directed as follows. In Section 2, we start with preliminaries in general properties generalized inverses, projectors, and g-matrices in Section 2.1, and in the theory of row-column determinants and determinantal representations of the Moore-Penrose inverses of a quaternion matrix, its Hermitian adjoint and g-Hermitian adjoint matrices in Section 2.2. Determinantal representations of a general, g-Hermitian and g-skew-Hermitian solutions to the system (2) are derived in Section 3. Finally, the conclusion is drawn in Section 4.

Preliminaries: Determinantal representations of solutions to quaternion matrix equations
General properties generalized inverses, projectors, and g-matrices We begin with some famous results on generalized inverses and projectors inducted by them which will be used in the remaining part of this paper.
(3) ðA y AÞ gÃ ¼ A gÃ ðA y Þ gÃ ¼ ðA y AÞ g ¼ ðA y Þ g A g : (4) ðAA y Þ gÃ ¼ ðA y Þ gÃ A gÃ ¼ ðAA y Þ g ¼ A g ðA y Þ g : [72] Let A, B and C be given matrices with right sizes over H. Then Remark 2.1. For any g l 2 fi; j; kg for all l = 1, 2, 3, and q = q 0 + q 1 g 1 + q 2 g 2 + q 3 g 3 , we denote So, elements of the main diagonal of an g 1 -Hermitian matrix 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 g 1 -skew-Hermitian matrix A ¼ ÀA and a pair of elements which are symmetric with respect to the main diagonal can be represented as where a l 2 R for all l = 0,. . ., 3.

Determinantal representations of generalized inverses and of solutions to some quaternion matrix equations
Through the non-commutativity of the quaternion skew field, determining of the determinant with noncommutative entries (it is also called a noncommutative determinant) is not so trivial (see, e.g. [62,63]). There are several versions of the definition of noncommutative determinants (see, e.g., [64][65][66][67][68][69]). But, it is proved in [70], if all functional properties of determinant over a ring are satisfied, then it takes on a value in its commutative subset only. In particular, it means that such determinant can not be expanded by cofactors along an arbitrary row or column. To avoid these difficulties, for A 2 H nÂn , we define n row determinants and n column determinants which are not owning of all functional properties that could be inherent to the usual determinant.
Suppose S n is the symmetric group on the set I n ¼ f1; . . . ; ng.
Definition 2.2. [42] The ith row determinant of A ¼ ða ij Þ 2 H nÂn is called by setting for all i = 1, . . ., n, . . a i kr þlr i kr Þ; where r is the left-ordered permutation. It means that its first cycle from the left starts with i, other cycles start from the left with the minimal of all the integers which are contained in it, i k t < i k t þs for all t ¼ 2; . . . ; r; s ¼ 1; . . . ; l t ; and the order of disjoint cycles (except for the first one) is strictly conditioned by increase from left to right of their first elements, i k 2 < i k 3 < Á Á Á < i kr .
The jth column determinant of A ¼ ða ij Þ 2 H nÂn is called by setting for all j = 1, . . ., n, nÀr ða j kr j kr þlr . . . a j kr þ1 j kr Þ . . . ða jj k 1 þl 1 . . . a j k 1 þ1 j k 1 a j k 1 j Þ; where s is the right-ordered permutation. It means that its first cycle from the right starts with j, other cycles start from the right with the minimal of all the integers which are contained in it, j kt < j ktþs for all t ¼ 2; . . . ; r; s ¼ 1; . . . ; l t ; and the order of disjoint cycles (except for the first one) is strictly conditioned by increase from right to left of their first elements, j k 2 < j k 3 < Á Á Á < j kr .
Remark 2.4. So, for a 2Â2-matrix with quaternion settings A ¼ a 11 a 12 a 21 a 22 ! , we have the four (row-column) determinants rdet 1 A ¼ a 11 a 22 À a 12 a 21 ; rdet 2 A ¼ a 22 a 11 À a 21 a 12 ; Since a ij 2 H for all i, j = 1, 2, they are not equal to each others, in general. We state some properties of row-column determinants needed below.
where a l 2 H and b l 2 H 1Ân for all l = 1,. . ., k and i = 1,. . ., n, then If the jth column of A 2 H mÂn is a right linear combination of other column vectors, i.e.
Remark 2.5. Since [42] for Hermitian A we have the determinant of a Hermitian matrix is called by setting det A : Its properties have been completely studied in [43]. In particular, from them it follows the definition of the determinantal rank of a quaternion matrix A as the largest possible size of nonzero principal minors of its corresponding Hermitian matrices, i.e. rankA ¼ rankðA Ã AÞ ¼ rankðAA Ã Þ.
For determinantal representations of the Moore-Penrose inverse, we use the following notations. Let . . ; n f gbe subsets with 1 k min m; n f g. By A a b denote a submatrix of A 2 H mÂn with rows and columns indexed by a and b, respectively. Then, A a a is a principal submatrix of A with rows and columns indexed by a. Moreover, for Hermitian A, jAj a a is the principal minor of det A. Suppose that, stands for the collection of strictly increasing sequences of 1 k n integers chosen from 1; . . . ; n f g . For fixed i 2 a and j 2 b, put I r;m i f g :¼ a : a 2 L r;m ; i 2 a f g , J r;n j f g :¼ b : b 2 L r;n ; j 2 b f g . By a : j and a Ã :j , a i: and a Ã i: denote the jth columns and the ith rows of A and A*, respectively. Suppose A i: b ð Þ and A :j c ð Þ stand for the matrices obtained from A by replacing its ith row with the row b and its jth column with the column c, respectively.
Remark 2.7. For an arbitrary full-rank matrix A 2 H mÂn r , a row-vector b 2 H 1Âm , and a column-vector c 2 H nÂ1 , we assume that for all i = 1, . . ., m, j = 1, . . ., n, ¼ Àg Since the projection matrices A y A ¼: ; then its inducted projection matrices Q A ¼ q ij À Á nÂn and P A ¼ p ij À Á mÂm are determined as follows where _ a :j and € a i: , _ a i: and € a :j are the jth columns and ith rows of A Ã A 2 H nÂn and AA Ã 2 H mÂm , respectively.
Cramer's rule for the system (2) The next lemma gives the explicit matrix form of a general solution to the system (1).
In that case, the general solution of (1) can be expressed as where Z and W are arbitrary matrices over H with appropriate sizes.
Some simplification of (10) can be derived due to Lemma 2.2. So, we have, Substituting (11) in (10), we get By putting Z = W = 0, we get the following expression of the partial solution Now consider the system (2). We have similarly, P A gÃ i ¼ Q g Ai , and, by Lemma 2.1, L A gÃ i ¼ R g Ai , and R A gÃ i ¼ L g Ai for i = 1, 2. Moreover, by substituting From above, it follows the next analog of Lemma 3.1.
Lemma 3.2. Suppose that A 1 2 H mÂn , A 2 2 H kÂn , C 1 2 H mÂm , C 2 2 H kÂk are known and X 2 H nÂn is unknown. The system (2) is consistent if and only if In that case, the general solution to (2) is expressed as where Z and W are arbitrary matrices over H with appropriate sizes. By putting Z, W as zero-matrices, the partial solution to (2) is Further, we give determinantal representations of (15).  Taking into account (4) and (6) for A y 1 and A gÃ 1 À Á y , respectively, we get Suppose that e l: and e :l are the unit row and column vectors such that all their components are 0 except the lth components which are 1.
fs ; then By v ð1Þ is :¼ denote the sth component of a row-vector v Farther, it's evident that P b2J r 1 ;n Integrating (17) and (18) in (16), the determinantal representation of the first term of (15) can be expressed as where v ð1Þ;g i: Integrating (21) and (22) in (16), we obtain another determinantal representation of the first term where v ð2Þ :j ¼ Àg are the column vector and c ð11Þ;g f : (ii) Similarly above, for the second term ij of (15), we have where (iii) The third term ij of (15) can be obtained similarly as well. So, where d H :j ¼ Àg Here c ð22Þ q: , c ð22Þ :l are the qth row and the lth column of (iv) Now, consider the fourth term ij of (15). Taking into account (4) for determinantal representations of H y and T y , we get Here a zf is the (zf)th element of the first term that is obtained in the point (i). q fj is the (fj)th element of Q A 2 that, by (8), can be expessed as where _ a ð2Þ f : is the fth row of A Ã 2 A 2 . Denote q ð1Þ zj :¼ z: where e x ð1Þ z: is the zth row of e X 1 ¼ X 1 A Ã 2 A 2 for all z, j = 1,. . ., n and X 1 is found in the point (i). Construct the matrix where e t :j is the jth column of e T ¼ T Ã A 2 Q 1 and construct the matrix T 1 ¼ ðt ð1Þ qj Þ 2 H nÂn . Finally, denote e H :¼ H Ã A 2 T 1 . From these denotations and the equation (28), it follows where e h :j is the jth column of e H.
(v) For sj Þ is determined in (29). So, similarly to the previous case, we obtain where b h :j is the jth column of b H.
(vi) Consider the sixth term where and u A 2 :j ¼ Àg is the qth row of C g 23 . Construct the matrix U ¼ / qj À Á such that / qj is determined in (33) and denote e U :¼ H Ã A 2 U. From this denotation and the equation (32), it follows where e / :j is the jth column of e U.
(vii) Finally, consider the seventh term ij of (15). Taking into account (4) for T y and (6) for ðH gÃ Þ y , we get   are the qth column of T Ã A 2 and the fth row of A gÃ 2 H, respectively. Denote such that x qj is determined in (36) and denote b X :¼ T Ã A 2 X. From these denotations and the equation (35), it follows where b x :j is the jth column of b X. Therefore, we proved the following theorem.

Conclusion
Using row-column noncommutative determinants previously introduced by the author, the determinantal representations (analogs of Cramer's rule) of the general, g-Hermitian and g-skew-Hermitian solutions to the system of quaternion matrix equations A 1 XA gÃ 1 ¼ C 1 , A 2 XA gÃ 2 ¼ C 2 have been derived. For these purposes, determinantal representations of the Moore-Penrose inverses of a quaternion matrix, its Hermitian adjoint and g-Hermitian adjoint matrices have been explored and used.