Explicit Determinantal Representation Formulas of W-Weighted Drazin Inverse Solutions of Some Matrix Equations over the Quaternion Skew Field

By using determinantal representations of the -weighted Drazin inverse previously obtained by the author within the framework of the theory of the column-row determinants, we get explicit formulas for determinantal representations of the -weighted Drazin inverse solutions (analogs of Cramer’s rule) of the quaternion matrix equations , , and .


Introduction
Throughout the paper, we denote the real number field by R, the set of all  ×  matrices over the quaternion algebra by H × , and the set of all  ×  matrices over H with a rank  by H ×  .Let (, H) be the ring of × quaternion matrices.For A ∈ H × , the symbol A * stands for the conjugate transpose (Hermitian adjoint) matrix of A. The matrix A = (  ) ∈ H × is Hermitian if A * = A.
For A ∈ H × , we denote by In the past, researches into the quaternion skew field had a more theoretical importance, but now a growing number of investigations give wide practical applications of quaternions.In particular through their attitude orientation, the quaternions arise in various fields such as quaternionic quantum theory [1], fluid mechanics and particle dynamics [2,3], computer graphics [4], aircraft orientation [5], robotic systems [6], and life science [7,8].
Research on quaternion matrix equations and generalized inverses, which are usefulness tools used to solve matrix equations, has been actively ongoing for more recent years.We mention only some recent papers.Yuan et al. [9] derived solutions of the quaternion matrix equation  =  and their applications in color image restoration.Wang et al. [10] studied extreme ranks of real matrices in solution of the quaternion matrix equation  = .Yuan et al. [11] obtained the expressions of least squares Hermitian solution with minimum norm of the quaternion matrix equation (, ) = (, ).Feng and Cheng [12] gave a clear description of the solution set to the quaternion matrix equation − = 0. Jiang and Wei [13] derived the explicit solution of the quaternion matrix equation  −  X = .Caiqin et al. [14] obtained the expressions of the explicit solutions of quaternion matrix equations  −  =  and  −  X = .Yuan and Wang [15] gave the expressions of the least squares -Hermitian solution with the least norm of the quaternion matrix equation  +  = .Zhang et al. derived [16] the expressions of the minimal norm least squares solution, the pure imaginary least squares solution, and the real least squares solution for the quaternion matrix equation  = .
The definitions of the generalized inverse matrices have been extended to quaternion matrices as follows.
The Moore-Penrose inverse of A ∈ H × , denoted by A † , is the unique matrix X ∈ H × satisfying the following equations: (2) For A ∈ H × with  = Ind A being the smallest positive number such that rank A +1 = rank A  , the Drazin inverse of A, denoted by A  , is defined to be the unique matrix X that satisfies (3) and the equations In particular, when Ind A = 1, then X is called the group inverse of A and is denoted by X = A  .If Ind A = 0, then A is nonsingular, and Cline and Greville [17] extended the Drazin inverse of square matrix to rectangular matrix that has been generalized to the quaternion algebra as follows.
For A ∈ H × and W ∈ H × , the -weighted Drazin inverse of A with respect to W is the unique solution to the following equations: where  = max{Ind(AW), Ind(WA)}.
Cramer's rule for the -weighted Drazin inverse solutions, in particular, has been derived in [27] for singular linear equations and in [26] for a class of restricted matrix equations.Recently, within the framework of the theory of the column-row determinants, Song [28] has first obtained a determinantal representation of the -weighted Drazin inverse and Cramer's rule of a class of restricted matrix equations over the quaternion algebra.But in obtaining, he has used auxiliary matrices other than that are given.In [29], we have obtained new determinantal representations of the -weighted Drazin inverse over the quaternion skew field without any auxiliary matrices.
An important application of determinantal representations of generalized inverses is the Cramer rule for generalized inverse solutions of matrix equations.
But when is there a need for a -weighted Drazin inverse solution?Consider, for example, the following matrix equation: A 1 X = D. Let A 1 be rectangular and we can represent it as A 1 = WAW, where WA and AW are quadratic and singular.Furthermore, we have the following restrictions: R  (X) ⊂ R  ((AW)  ), N  (X) ⊃ N  ((WA)  ).Then its weighted Drazin inverse solution is needed.
In the paper we investigate analogs of Cramer's rule for -weighted Drazin inverse solutions of the following quaternion matrix equations: The paper is organized as follows.We start with introducing of the row-column determinants and determinantal representations of the Moore-Penrose and Drazin inverses for a quaternion matrix obtained by them in Section 2.1.Determinantal representations of the -weighted Drazin inverse and its properties were considered in Section 2.2.In Section 3.1, we give the background of the problem of Cramer's rule for the -weighted Drazin inverse solution.In Section 3.2 we obtain explicit representation formulas of the -weighted Drazin inverse solutions (analogs of Cramer's rule) of the quaternion matrix equation (10).Consequently, we get both similar and special determinantal representation formulas of the -weighted Drazin inverse solutions of ( 8) and (9).In Section 4, we give numerical examples to illustrate the main result.

Determinantal Representations of the Moore-Penrose and
Drazin Inverses by the Column and Row Determinants.The theory of the row-column determinants over the quaternion skew field has been introduced in [30][31][32], and later it has been applied to research generalized inverses and generalized inverse solutions of matrix equations.In particular, determinantal representations of the Moore-Penrose [33,34] and explicit representation formulas for the minimum norm least squares solutions of some quaternion matrix equations [35] and determinantal representations of the Drazin [36] and -weighted Drazin inverses [29] have been obtained by the author.Song derived determinantal representation of the generalized inverse  2 , [37], the Bott-Duffin inverse [38], the Cramer rule for the solutions of restricted matrix equations [39], the generalized Stein quaternion matrix equation [40], and so forth.
For A = (  ) ∈ (, H) we define  row determinants and  column determinants as follows.
Suppose that   is the symmetric group on the set   = {1, . . ., }.Definition 1.The th row determinant of A = (  ) ∈ (, H) is defined for all  = 1,  by putting with conditions Suppose that A  denotes the submatrix of A obtained by deleting both the th row and the th column.Let a ⋅ be the th column and let a ⋅ be the th row of A. Suppose that A ⋅ (b) denotes the matrix obtained from A by replacing its th column with the column b and that A ⋅ (b) denotes the matrix obtained from A by replacing its th row with the row b.
The following theorem has a key value in the theory of the column and row determinants.
Since all column and row determinants of a Hermitian matrix over H are equal, we can define the determinant of a Hermitian matrix A ∈ (, H).By definition, we put det A fl rdet  A = cdet  A, for all  = 1, .The determinant of a Hermitian matrix has properties similar to a usual determinant.They are completely explored in [30,31] by its row and column determinants.In particular, within the framework of the theory of the column-row determinants, we have the determinantal representation of the inverse matrix over H by analogs of classical adjoint matrix.Further, we consider determinantal representations of generalized inverses obtained by the column-row determinants.
Denote by a * ⋅ and a * ⋅ the th column and the th row of A * and by a ()  ⋅ and a () ⋅ the th column and the th row of A  , respectively.
Theorem 6 (see [33]).If A ∈ (, H) with Ind A =  and rank A +1 = rank A  = , then the Drazin inverse A  possesses the following determinantal representations: In the special case, when A ∈ (, H) is Hermitian, we can obtain simpler determinantal representations of the Drazin inverse.

Determinantal Representations of the 𝑊-Weighted Drazin
Inverse.We introduce some mathematical background from the theory of the -weighted Drazin inverse [27,41,42] that can be generalized to H.
where the first factor is one of the following four possible equations: for all ,  = 1, 2, and an entry of the Drazin inverse V  is denoted by or where k⋅ and ǩ ⋅ are the th column of The point (c) of Lemma 8 due to [23] has been generalized to H in [33].Using this proposition, we have obtained the following two determinantal representations of the weighted Drazin inverse.
Theorem 9 (see [29]).Let A ∈ H × and W ∈ H ×  1 with  = Ind(AW) and  = rank(AW) +1 = rank(AW)  .Then the -weighted Drazin inverse of A with respect to W possesses the following determinantal representations: where In the special cases, when AW ∈ H × and WA ∈ H × are Hermitian, we can obtain simpler determinantal representations of the -weighted Drazin inverse.

Cramer's Rule for the 𝑊-Weighted Drazin Inverse Solution
3.1.Background of the Problem.In [27] Wei has established Cramer's rule for solving of a general restricted equation: where A ∈ C × and W ∈ C × with Ind(AW) =  1 , Ind(WA) =  2 and rank(AW and  1 =  2 , then (30) has a unique solution, x = A , b, which can be presented by the following Cramer rule: where are matrices whose columns form bases for N((WA)  2 ) and N((AW)  * 1 ), respectively.
Recently, within the framework of a theory of the column and row determinants, Song [28] has considered the characterization of the -weighted Drazin inverse over the quaternion skew and presented a Cramer rule of the restricted matrix equation: where He proved that if and there exist auxiliary matrices of full column rank, L 1 ∈ with additional terms of their ranges and null spaces, then the restricted matrix equation ( 32) has a unique solution: Using auxiliary matrices, L 1 , M 1 , L 2 , and M 2 , Song presented its Cramer's rule by analogy to (31).
In this paper we have avoided such approach and have obtained explicit formulas for determinantal representations of the -weighted Drazin inverse solutions of matrix equations by using only given matrices.
Theorem 12. Suppose that D ∈ H × , A ∈ H × , and , then the restricted matrix equation ( 32) has a unique solution: which possesses the following determinantal representations for all  = 1,  and  = 1, .
(i) Consider where ) can be obtained by (23) and (   ) = U  is the Drazin inverse of U = W 2 B and (   ) (2) can be obtained by (21).(ii) If AW 1 ∈ H × and W 2 B ∈ H × are Hermitian, then or where are the column vector and the row vector, respectively.d ⋅ and d ⋅ are the th row and the th column of D for all  = 1,  and  = 1, .
Proof.The existence and uniqueness of solution (36) can be proved similarly as in [28], Theorem 5.2.
(i) To derive a Cramer rule (37) we use the point (a) from Lemma 8. Then we obtain Denote ADB š D = ( d ) ∈ H × , V fl AW 1 , and U fl W 2 B. Then (42) will be written componentwise as follows: By changing the order of summation, from here it follows (37). ) ∈ H × possess the following determinantal representations, respectively: where k ⋅ is the th column of V = (AW 1 )  1 A for all  = 1,  and where u ⋅ is the th row of U = B(W 2 B)  2 for all  = 1, .By componentwise writing of (36) we obtain Denote by d⋅ the th column of VD = (AW 1 ) Suppose that e ⋅ and e ⋅ are, respectively, the unit row vector and the unit column vector whose components are 0, except the th components, which are 1.Substituting (47) and ( 45) into (46), we obtain Remark 13.To establish a Cramer rule of (32) we will not use the determinantal representations ( 28) and ( 28) for (36) because the corresponding determinantal representations of its solution will be too cumbersome.But they are suitable in the following corollaries.
Corollary 14. Suppose that the following restricted matrix equation is given: where A ∈ H × and W ∈ H ×  55)-( 56) has a unique solution: which possess the following determinantal representations for all  = 1,  and  = 1, .
(i) Consider where (V   ) (2) can be obtained by (23) and where f ⋅ is the th column of F = VD = (AW)  AD.
Proof.To derive a Cramer rule (58), we use the determinantal representation (27) for A , .Then

R
(A) = {y ∈ H  : y = Ax, x ∈ H  } the column right space of A; N  (A) = {y ∈ H  : Ax = 0} the right null space of A; R  (A) = {y ∈ H  : y = xA, x ∈ H  } the column left space of A; N  (A) = {y ∈ H  : xA = 0} the left null space of A.