Determinantal representations of the weighted core-EP, DMP, MPD, and CMP inverses of matrices with quaternion and complex elements

In this paper, we extend notions of the weighted core-EP right and left inverses, the weighted DMP and MPD inverses, and the CMP inverse to matrices over the quaternion skew field H that have some features in comparison to these inverses over the complex field. We give the direct methods of their computing, namely, their determinantal representations by using noncommutative column and row determinants previously introduced by the author. As the special cases, by using the usual determinant, we give their determinantal representations for matrices with complex entries as well. A numerical example to illustrate the main result is given.


Introduction
In the whole article, the notations R and C are reserved for fields of the real and complex numbers, respectively. H m×n stands for the set of all m × n matrices over the quaternion skew field H m×n r determines its subset of matrices with a rank r. For given h = h 0 + h 1 i + h 2 j + h 3 k ∈ H, the conjugate of h is h = a 0 − h 1 i − h 2 j − h 3 k. For A ∈ H m×n , the symbols A * and rk(A) specify the conjugate transpose and the rank of A, respectively. A matrix A ∈ H n×n is Hermitian if A * = A. The index of A ∈ H n×n , denoted Ind A = k, is the smallest positive number such that rk(A k+1 ) = rk(A k ).
Due to [7] the definition of the weighted Drazin inverse can be generalized over H as follows.
respectively. The concepts of the complex weighted DMP and CMP inverses were introduced in [36] and [39], respectively.
The main goals of this paper are extended the notions of the weighted core-EP inverses, and the weighted DMP and CMP inverses over the quaternion skew-field H, and get their determinantal representations that are the direct methods of their obtaining by using determinants.
The determinantal representation of the usual inverse is the matrix with cofactors in entries that suggests a direct method of finding the inverse of a matrix. The same is desirable for the generalized inverses. But, there are various expressions of determinantal representations of generalized inverses even for matrices with complex or real entries, (see, e.g. [4, 5, 14-16, 46, 47]). In view of the noncommutativity of quaternions, the problem of the determinantal representation of quaternion generalized inverses is evidently dependent on complexities related with definition of the determinant with noncommutative entries (it is also called a noncommutative determinant).
The majorities of the previous defined noncommutative determinants are derived by transforming the quaternion matrix to an equivalent complex or real matrix (see, e.g. [1,8]). However, by this way it is impossible for us to give determinantal representations of quaternionic generalized inverses. Only now it can be done thanks to the theory of column-row determinants introduced by the author in [17,18]. Currently, by using of row-column determinants, determinantal representations of various generalized inverses have been derived and applied to solutions of quaternion matrix equations by the author (see, e.g. [19][20][21][22][23][24][25][26][27][28][29]), (among them the core inverse and its generalizations in the quaternion [30] and complex [31] cases), and by other researchers (see, e.g. [48][49][50]).
The paper is organized as follows. In Section 2, we start with preliminary introduction of the theory of row-column determinants and the determinantal representations of the Moore-Penrose inverse, of the Drazin and weighted Drazin inverses, and of the core inverse and its generalizations over the quaternion skew field previously obtained by using row-column determinants. In Section 3, we introduce the concepts of the left and right weighted core-EP inverses over the quaternion skew field and give their determinantal representations. In Section 4, the quaternion weighted DMP and MPD inverses are established and their determinantal representations are obtained. Determinantal representations of the quaternion CMP inverse are get in Section 5. A numerical example to illustrate the main results is considered in Section 6. Finally, in Section 7, the conclusions are drawn.

Elements of the theory of row-column determinants.
Suppose S n is the symmetric group on the set I n = {1, . . . , n}. where σ 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 kt < i kt+s for all t = 2, . . . , r, s = 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 k2 < i k3 < · · · < i kr .
Similarly, for a column determinant along an arbitrary column, we have the following definition.
The jth column determinant of A = (a ij ) ∈ H n×n is defined for any j ∈ I n by setting where τ 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 kt+s for all t = 2, . . . , r, s = 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 k2 < j k3 < · · · < j kr .
The row and column determinants have the following linear properties.
Lemma 2.1. [17] If the ith row of A ∈ H n×n is a left linear combination of some row vectors, i.e. a i. = α 1 b 1 + · · · + α k b k , where α l ∈ H and b l ∈ H 1×n for all l = 1, . . . , k and i = 1, . . . , n, then Lemma 2.2. [17] If the jth column of A ∈ H m×n is a right linear combination of other column vectors, i.e. a .j = c 1 α 1 + · · · + c k α k , where α l ∈ H and c l ∈ H n×1 for all l = 1, . . . , k and j = 1, . . . , n, then So, an arbitrary n × n quaternion matrix inducts n row determinants and n column determinants that are different in general. Only for a Hermitian matrix A, we have [17], that enables to define the determinant of a Hermitian matrix by setting det A := rdet i A = cdet i A for all i = 1, . . . , n. Its properties have been completely studied in [18]. 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. rk A = rk(A * A) = rk(AA * ).
Remark 2.4. For an arbitrary full-rank matrix A ∈ H m×n r , a row-vector b ∈ H 1×m , and a column-vector c ∈ H n×1 , we put, respectively, whereȧ .j andȧ .i ,ä i. andä .j are the i-th rows and the j-th columns of A * A ∈ H n×n and AA * ∈ H m×m , respectively.
The following corollary gives determinantal representations of the Moore-Penrose inverse and of both projectors in complex matrices.
Corollary 2.2. [19] Let A ∈ C m×n r . Then the following determinantal representations are obtained (ii) for the projector Q A = (q ij ) n×n , whereȧ .j is the jth column of A * A; (iii) for the projector P A = (p ij ) m×m , There are two case for determinantal representations of the W-weighted Drazin inverse over the quaternion skew field.
(ii) if WA ∈ H n×n is Hermitian and rk(WA) k+1 = rk(WA) k = r, then The following corollary gives determinantal representations of the W-weighted Drazin inverse in complex matrices.
Quaternion column-vectors form a right vector H-space with quaternionscalar right-multiplying, and quaternion row-vectors form a left vector H-space with quaternion-scalar left-multiplying. We denote them by H r and H l , respectively. Moreover, H r and H l possess corresponding H-valued inner products by putting x, y r =y 1 x 1 + · · · + y n x n for x = (x i ) n i=1 , y = (y i ) n i=1 ∈ H r , x, y l =x 1 y 1 + · · · + x n y n for x, y ∈ H l , that satisfy the inner product relations, namely, conjugate symmetry, linearity, and positive-definiteness but with specialties xα + yβ, z = x, z α + y, z β when x, y, z ∈ H r , αx + βy, z = α x, z + β y, z when x, y, z ∈ H l , for any α, β ∈ H.
So, an arbitrary quaternion matrix induct right and left vector H-spaces that introduced by the following definition.
Definition 2.5. For an arbitrary matrix over the quaternion skew field, A ∈ H m×n , we denote by • C r (A) = {y ∈ H m×1 : y = Ax, x ∈ H n×1 }, the right column space of A, • N l (A) = {x ∈ H 1×m : xA = 0}, the left null space of A.

Determinantal representations of the core inverses and the core-EP inverses
Because of quaternion noncommutativity, Definition 1.3 can be expand to matrices over H as follows.
Definition 2.6. A matrix X ∈ H n×n is said to be the right core inverse of A ∈ H n×n if it satisfies the conditions When such matrix X exists, it is denoted A # .
When such matrix X exists, it is denoted A # .
Similar as in [42], we introduce two core-EP inverses over quaternion skew field.
The lemma below give characterization of the right core-EP inverse. Due to [42], the right weighted core-EP inverse is characterized in terms of three equations.
Lemma 2.5. Let A, X ∈ H n×n be such that Ind(A) = k. Then X is the right core-EP inverse of A if and only if X satisfies the conditions: Taking to account ( [9], Theorem 2.3), the following expression can be extend to quaternion matrices. Lemma 2.6. Let A ∈ H n×n and let l be a non-negative integer such that , then the left core inverse A † of A ∈ C n×n is similar to the * core inverse introduced in [42], and the dual core-EP inverse introduced in [55].
Similarly, we have the following characterization of the left core-EP inverse.
Theorem 2.11. Let X, A ∈ H n×n and let l be a non-negative integer such that l ≥ k = Ind(A). The following statements are equivalent: Thanks to [9], there exists the simple relation between the left and right core-EP inverses, (A † ) * = (A * ) † . So, it is enough to investigate the left core-EP inverse, and right core-EP inverse case can be investigated analogously. But in [30], we gave separately determinantal representations of both core-EP inverses.
The following corollary gives determinantal representations of the right and left, core and core-EP inverses for complex matrices.

Concepts of quaternion W-weighted core-EP inverses and their determinantal representations
The concept of the W-weighted core-EP inverse in complex matrices was introduced by Ferreyra et al. [13] that can be expended to quaternion matrices as follows.
It is denoted A † ,W,r .
Due to [42], the right weighted core-EP inverse over the quaternion skew field can be determined as follows.
Theorem 3.2. Let A, X ∈ H m×n , W ∈ H n×m , and k = max{Ind(WA), Ind(AW)}. The following statements are equivalent: We propose to introduce a left weighted core-EP inverse as well.
The following statements are equivalent: (ii) X is the left weighted core-EP inverse of A.
(iii) X is the unique solution to the three equations: Indeed, and due to Theorem 2.12, By Theorem 2.12, taking into account (AW) Finally, Suppose there also exists the left weighted core-EP inverse Y such that Finally, from the uniqueness of X follows (iii) → (i). Now, we give determinantal representations of the quaternion W-weighted core-EP inverses.
Using the determinantal representation (2.9) of U † gives Similarly, it can be proved theorem on the determinantal representation of the quaternion W-weighted left core-EP inverse.
It's evident that the following corollaries can be get in the complex case.
The W-weighted DMP and MPD inverses and their determinantal representations.
Similar as for complex matrices [34], if a quaternion matrix satisfies the system of equations (4.1), then it is unique and has the representation, In [30], we also introduce the MPD inverse. It is denoted A †,d .
It is not difficult to show that A †,d is unique and it can be represented as In [30], we gave the determinantal representations of the DMP and MPD inverses over the quaternion skew field.
Recently in [36], the definition of the DMP inverse of a square matrix with complex elements was extended to rectangular matrices. We extend it over the quaternion skew field. Definition 4.3. Let A ∈ H m×n and W ∈ H n×m be a nonzero matrix. The W-weighted DMP (WDMP) inverse of A with respect to W is defined as Similarly to complex matrices can be proved the next lemma. We propose to introduce the weighted MPD inverse as well.
It means that X = A † AWA d,W W is the solution to the equations (4.3).
To prove uniqueness, suppose both X 1 and X 2 are two solutions to (4.3). Using repeated applications of the equations in (4.3) and in Definition 1.1, we obtain It completes the proof.
The following corollary can be get in the complex case. where ω i. is the i-th row of Ω = ΩWAA * . The matrix Ω = (ω is ) is such that For the weighted MPD inverse, we have similarly.
where ψ .j is the j-th column of Ψ := (V 2k+1 ) * V 2k ΨAW ∈ H m×n . Here the matrix Ψ = (ψ s l ) ∈ H m×m is such that wherev .l is the l-th column of (V 2k+1 ) * V k =:V ∈ H m×m . (ii) If the matrix AW is Hermitian, then where v .j is the j-th column of V = (AW) k+1 .
Theorems 4.5 and 4.6 give the determinantal representations of the weighted DMP and DMP inverses over the quaternion skew field. For better understanding, we present the algorithm of finding one of them, for example WDMP from Theorem 4.5 in the case (i). 1. Compute the matrixǓ = U k (U 2k+1 ) * .

Determinantal representations of the weighted CMP inverse
In [35] by M. Mehdipour and A. Salemi the CMP inverse was investigated that can be extended to quaternion matrices as follows.
Definition 5.1. [30] Suppose A ∈ H n×n has the core-nilpotent decomposition Similarly to complex matrices can be proved the next lemma.
Lemma 5.1. Let A ∈ H n×n . The matrix X = A c, † is the unique matrix that satisfies the following system of equations: Determinantal representations of the CMP inverse over the quaternion skew field within the framework of the theory of row-column determinants are derived in [30].
Recently, Mosic [40] introduced the weighted CMP inverse of a rectangular matrix that can be extended over the quaternion skew field without any changes. where ω i. is the i-th row of Ω = Ω(WA) k+1 A * . The matrix Ω = (ω iz ) is such that ω iz is determined by .z β β .
(ii) By applying the determinantal representations (2.6) for A d,W and the same as in the above point for Q A and P A , we get s. ) .z is the z-th column of W 3 := A * (AW) k+1 and w (2) s. is the s-th row of W 2 := WAA * = (w z. is the z-th row of Ψ (1) = ΨW 2 = ΨWAA * and construct the matrix Υ = (υ zj ). Taking into account that n z=1 β∈Jr,n{i} where υ .j is the j-th column of Υ = W 3 Υ = A * (AW) k+1 Υ, finally from (5.11), it follows (5.6).
Simpler expressions of determinantal representations of the WCMP inverse can be obtained in the cases having Hermicity. where ω i. is the i-th row of Ω = ΩWAA * . The matrix Ω = (ω is ) is such that Here φ (1) i. is the i-th row of Φ 1 = ΦA (WA) k and the matrix Φ = (φ it ) is such that .t is the t-th column of W 1 = A * AW. where υ .j is the j-th column of Υ = A * AWΥ. The matrix Υ = (υ tj ) is determined by .j is the j-th column of Ψ (1) = (AW) k AΨ. Here Ψ = (ψ sj ) is such that s. is the s-th row of W 2 = WAA * .
(ii) By applying (2.8) for the determinantal representation of A d,W and the same determinantal representations of Q A and P A as in the point (i), we get where w (1) .t is the t-th column of W 1 := A * AW,v .s is the s-th column of V = (AW) k A, and w (2) s. is the s-th row of W 2 := WAA * . Denote ψ sj := α∈Ir,m{j} rdet j (AA * ) j. (w (2) s. ) α α and construct the matrix Ψ = (ψ sj ). Then, introduce .j is the j-th column of Ψ (1) = (AW) k AΨ and construct the matrix Υ = (υ tj ). Taking into account that n t=1 β∈Jr,n{i} where υ .j is the j-th column of Υ = A * AWΥ, finally from (5.15), it follows (5.13).
.j is the j-th column of Ψ (1) = (AW) k AΨ. Here Ψ = (ψ sj ) is such that ψ sj := α∈Ir,m{j} where w (2) s. is the s-th row of W 2 = WAA * . Theorems 5.3 and 5.4 give determinantal representations of the WCMP inverse over the quaternion skew field. For better understanding, we present the algorithm of its finding, for example, in Theorem 5.3 the case (i). Other algorithms can be construct similarly.
We shall find the weighted DMP inverse due to Algorithm 4.7.

Conclusions
Notions of the weighted core-EP right and left inverses, the weighted DMP and MPD inverses, and the weighted CMP inverse have been extended to quaternion matrices in this paper. Due to noncommutativity of quaternions, these generalized inverses in quaternion matrices have some features in comparison to complex matrices. We have obtained their determinantal representations within the framework of the theory of column-row determinants previously introduced by the author. As the special cases, their determinantal representations in complex matrices have been obtained as well.