W - MPD – N - DMP - solutions of constrained quaternion matrix equations

: The solvability of several new constrained quaternion matrix equations is investigated, and their unique solutions are presented in terms of the weighted MPD inverse and weighted DMP inverse of suitable matrices. It is interesting to consider some exceptional cases of these new equations and corresponding solutions. Determinantal representations for the solutions of the equations as mentioned earlier are estab - lished as sums of appropriate minors. In order to illustrate the obtained results, a numerical example is shown.

Recent research on the core inverse [12] and the core-EP inverse [13,14] have revived increased interest in studying new generalized inverses, in particular, obtained by combining them.The notion of the DMP inverse, that is combining the Drazin inverse and Moore-Penrose inverse, was introduced initially for complex square matrices in [15].Then, it was extended as weighted DMP inverse to rectangular matrices in [16], to bounded linear Hilbert space operators in [17], and to quaternion matrices in [18].
, the W -weighted DMP (W -DMP) inverse of A is expressed as follows: and uniquely determined by the equations As the dual W -DMP inverse, the W -weighted MPD (W -MPD) inverse of A is defined as follows: Note that the DMP inverse = A AA A D, † D † and the MPD inverse = A AA A †,D † D are special cases of the W -DMP inverse and the W -MPD inverse, respectively, when = m n and = W I n .Recently, many authors considered the properties of DMP and W -DMP inverses.In particular, char- acterizations and representations for DMP and W -DMP inverses can be found in [19][20][21][22][23]. Especially, integral representations of the DMP inverse were established in [24], and determinantal (D-) representations for DMP and W -DMP inverses in [18,25].In [26], one can see an iterative method for finding the DMP inverse.There are extensions of DMP inverse for operators [27], for elements of rings [28], and for tensors in [29].
Many problems in engineering, mathematics, and so on were converted into some linear matrix equations.Many authors considered constrained quaternion matrix equations (CQMEs) since they have applications in quantum mechanics, quantum physics, computer science, signal and image processing, rigid mechanics, statistic, control theory, field theory, and other fields.The solvability of CQMEs is investigated by various techniques, such as the usage of different representations and characterization of matrices with quaternion entries, the Kronecker product, and utilization of the Moore-Penrose inverse and other kinds of generalized inverses.Some interesting results can be found in [30][31][32][33][34][35].
In [2], the solvability of some CQMEs is studied.It is proved that the unique solutions of these equations can be expressed by the corresponding DMP and MPD inverses and adequate element-wise D-expressions.A characterization of the best-approximate solution = X ACB † † as well as a characterization of the Drazin- inverse solution = X A CB D D of the general linear equation = AXB C is also given [1,2].

Preliminaries and detailed motivation
In order to further generalize the best-approximation solutions and results proved in [2], we study several new CQMEs.The solutions of new CQMEs are uniquely determined and presented in terms of the weighted MPD inverse and weighted DMP inverse of corresponding matrices.Considering important particular cases of these CQMEs, the best-approximation solution is obtained as a particular case.It is interesting to develop D-representations for the solutions of examined CQMEs.
The main research directions of this article are briefly described as follows: (1) The first result is related to solvability of the CQME where ) To this end, we prove that its unique solution is expressed by the W -DMP inverse of A and N -MPD inverse of B. (3) Using the W -DMP inverse of A and N -DMP inverse of B, we verify solvability of the two-sided CQME and ∈ × C m q .(4) Utilizing the W -MPD inverse and N -MPD inverse of A and B, respectively, we solve the equation The content of this research is systemized as follows.Preliminaries, detailed motivation, and explanation of considered problems are presented in Section 2. Solvability of equations (2.1)-(2.4) is studied in Section 3. D-representations of solutions to the aforementioned equations are developed in Section 4. A numerical example is given in Section 5.

W -MPD-N-DMP-solutions of constrained equations
Two-sided CQMEs (2.1)-(2.4)are solved in this section.Based on the corresponding weighted MPD and DMP inverses and the matrix C, the solutions of these equations are characterized and represented.
and ∈ × C m q , the unique solver of (2.1) is equal to Proof.For X presented as in (3.1), by (1.2) and (1.1), one can see we deduce that X solves (2.1).
To prove the uniqueness of solution (3.1), we set Y and X for two solutions of equation Based on the previous, ( ) ) and the corre- sponding equations can be solved.
When we add the assumption solves in a unique way the following CQME: . Using Theorem 3.1 and we complete the proof.□ Consider that (3.1) is a solution for one more kind of CQME.
and ∈ × C m n , the matrix X defined in (3.1) presents the unique solution to Proof.According to Theorem 3.1, X given by (3.1) satisfies  The proof can be completed as in the proof of Theorem 3.1.□ As a specific input of Theorems 3.1 and 3.2, for = m n, = p q, = W I n , and = N I p , the result [2, Theorem 3.1] can be concluded.
We solve the next one-sided equations in a particular choice = p q and = = N I B and (3.4) presents the unique solution to CQMEs and B N †,D, , the solution to (2.2) is obtained as a continuation of results verified in Theorem 3.1. and Note that the unique solution to the following equation is also represented by (3.1).
and ∈ × C m n , the matrix X defined in (3.1) presents the unique solution to We now state one-sided equations whose solvability arises from Theorems 3.3 and 3.4. and presents the unique solution to CQMEs and presents the unique solution to CQMEs To prove the solvability of (2.3), we use the W -DMP inverse and N -DMP inverse of A and B, respectively. and One more equation can be studied by applying the W -DMP inverse and N -DMP inverse of A and B, respectively.
and ∈ × C m n , (3.1) presents the unique solution to According to Theorems 3.5 and 3.6, we can obtain solutions of the next equations. and presents the unique solution to CQMEs and The two-sided equation (2.4) can be solved in terms of the W -MPD inverse and N -MPD inverse of A and B, respectively.
and ∈ × C m n , the unique solver of equation (2.4) is as follows: The following result can be checked, too. and Theorems 3.7 and 3.8 imply solvability of the next one-sided equations. and presents the unique solution to CQMEs and (3.13) presents the unique solution to CQMEs

Cramer's representations of obtained solutions
The classical Cramer's rule in the case of solving complex matrix equations is based on the applications of the usual determinant, which could not be used for quaternion matrix equations.In the noncommutative quaternion case, the row (R-) and column (C-) determinants (recently introduced in [36,37]) demonstrated their usability in D-representations of generalized inverses and solutions to appropriate matrix equations (see, e.g., [38][39][40][41]).
The following notations are used to express D-representations of generalized inverses.Primarily, ).For a Hermitian A, the notation | | A λ λ stands for the corresponding principal . Denote by a j .(resp.* a j .) the jth column, and by a i. (resp.* a i. ) the ith row of A (resp.* A ). Suppose that 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.
Based on D-representations of the Moore-Penrose inverse [38] and the W -weighted Drazin inverse [42], D-representations of the W -DMP and W -MPD inverses are given in the following lemmas.
is expressed as follows: U WA is an arbitrary matrix, then where where , and Here, u ˜t . is the t-th row of ( where is is expressed as follows: and ( ) The weighted MPD inverse is represented similarly.
are defined as follows: (i) In the case of arbitrary = V AW, .
where ͠υ j . is the jth column in where  ψ j .denotes the jth column of  ( ) . Then, where v ˜l .stands for the l-th column of ( where ͠ υ j .represents the j-th column of the expression with ( ) Furthermore, we develop the D-representations of solutions to CQMEs from Section 3. We start with the D-representations of (3.1) and its special appearances (3.3), (3.4).
) is represented in one of subsequent element-wise representations. (1) where and ͠ υ f .denotes the f th column of , where Φ is of an appropriate size and index. (2) where Proof.According to (3.1) and the representations .
where ͠ υ f .stands for the f th column of the matrix expression and ͠ ω g. is the g th row of and Φ is of appropriate dimension and index.To acquire demonstrative formulae, we make convolutions of (4.15).Since the column vector can be multiplied from the right and the row vectorfrom the left, the subsequent two cases are used.
(1) If we carry out a series of the following successive designations, namely, r n , then equality (4.11) follows.
(2) A series of successive designations give the equality (4.13).□ If the matrices = V AW and = U NB are Hermitian, then it can be given some simplifications of the obtained expressions (4.11) and (4.13).In these cases, we use D-representations of W -DMP and W -MPD inverses, (4.4) and (4.9), respectively.The following corollaries can be proved similarly to the proof of Theorem 4.1.
1 , and ∈ × C m q .Suppose that = V AW is an arbitrary matrix and = U NB is Hermitian.The matrix determined in (3.1) is defined in one of the subsequent element-wise representations: Here, ( ) ω i.

Calculate the matrices
2 and Ψ by (4.8) and  ( ) , then generate the matrix ϒ by (4.7) and ͠ = * + ϒ A V ϒ k 1 .4. Compute the minor sums  (iv) Compute and return X by Algorithms inducted by Corollaries 4.1-4.3can be proposed in a similar way.Furthermore, we note that the D-representations for the solution (3.2) were explored completely in [43].

Calculate the matrices
In a particular case that = p q and = = N I B p in accordance to Corollary 3.2, it follows.
defined by (3.3) is expressible element-wise in one of the following two possible cases: with ( ) υ j .
1 denoting the j-th column of and ϒ is expressed by (4.7).
In a particular case that = m n and = = W I A n , we have the next result.(1) If = NB U is arbitrary, then where ( ) ω i. 1 is the i-th row of = + *

Ω CΩU B
l 1 1 , and Ω can be found by (4.2) given that  , where Φ is an appropriate size and index.

Ω CΩU B
l 1 1 .The matrix Ω can be found by (4.5) with an appropriate size and index.Now, we derive D-representations of (3.5) and its particular cases (3.6), (3.7). where Here, the matrices where ͠ and ͠ υ j .denotes the jth column covered in Here, the matrices Ω and ϒ are determined by (4.2) and (4.7) with appropriate sizes and indexes.
(2) When the both matrices = U WA and = V BN are Hermitian, then where the matrix is defined as in the case (1) with the difference that Ω and ϒ are determined by (4.5) and (4.10), respectively, with appropriate sizes and indexes.
(3) When the matrix = U WA is Hermitian and = V BN is arbitrary, then Here, the matrices Ω and ϒ are determined by (4.5) and (4.7), respectively, with appropriate sizes and indexes.
(4) When the matrix = U WA is arbitrary and = V BN is Hermitian, then Here, the matrices Ω and ϒ are determined by (4.2) and (4.10), respectively, with appropriate sizes and indexes. Proof.
(1) According to (3.5) and the representations (4.1) of the W -DMP inverse and . where ).The matrices ϒ and Ω are determined by (4.7) and (4.2), respectively, with appropriate sizes and indices.The row determinant cannot be multiplied from the right and the column determinant from the left by c fg in the equality (4.20).So, the convolutions similar as those made in Theorem 4.2 cannot be used.Denote and construct the matrices given by (3.6) is defined in one of the following representations: where   ( ) . Here, the matrix if is determined by equality (4.17) with Ω that is defined by (4.2).
(2) When = U WA is Hermitian, then In this case, Ω is determined by (4.5).
Let now = m n and = = W I A n .Then, in accordance to Corollary 3.5, it follows.
Corollary 4.7.Let ( ) ) is determined in one of the following representations: (1) When = V BN is arbitrary, then where   ( ) . Here, ) and is defined by (4.7) with an appropriate size and index.
(2) When = V BN is Hermitian, then In this case, ϒ is determined by (4.10).3. Find the matrices

Compute values of minor sums
.
In this case, the matrices Ω 1 and Ω 2 are determined by (4.2) and (4.5), respectively, with appropriate sizes and indexes.
To obtain expressive formulas, we make convolutions of (4.24).Using a series of the successive designations,   (3.11) is expressed in one of the following representations:
To derive meaningful representations, we make convolutions of (4.27).We carry out the following consecutive designations: Determinantal representations of solutions to novel CQMEs and solutions to several particular cases are developed.An illustrative numerical example is presented.
the set of all × m n matrices over .Denote by * A and ( ) A rank , respec- tively, the conjugate-transpose and rank of A. The right/left range space and the right/left null space are defined as for which = * A A holds, is Hermitian.Generalized inverses play an essential role in the study of the solvability of various equations and systems of equations[1].The most well-known generalized inverses are the Moore-Penrose and Drazin inverses.
It has been shown that the solution to (2.1) is uniquely determined by W -MPD inverse of A and N -DMP inverse of B.(2) Furthermore, we study the CQME

) 6 )
We investigate special cases of the CQMEs (2.1)-(2.4)as well as equations with the same solutions.(For the CQMEs (2.1)-(2.4)and their special cases, we provide the D-representations of their solutions.(7) Developed D-representations are described by a numerical example.

1 )
is defined in one of the subsequent representations:

1
, and ∈ × C m q .Furthermore, suppose that = V AW is Hermitian and = U NB is an arbitrary matrix.The matrix [ ] the form (3.1) is defined as follows:

. 1 l 1 .
is determined by(4.14),where ͠ υ f. means the f-th column of ͠ = * + The matrix ( ) = υ ϒ tj is determined by (4.7), and Ω can be found by (4.5) with an appropriate size and index.Theorem 4.1 inducts the following algorithm for computing ∈ × X n p by (3.1).

4 )
can be expressed as element-wise in one of the following two possible cases:

Theorem 4 . 2 . 1 )
Let us assume ( the form (3.5) is given in one of the following representations: (In arbitrary case, it follows the equality (4.16) holds.(2) If the both matrices = U WA and = V BN are Hermitian, the D-representations of A D W , †, and B D N †, , are expressed by (4.4) and (4.9), respectively.Doing so gives equation (4.19).The mixed cases (3) and (4) can be obtained in the similar way.□ From Theorem 4.2, the particular one-side cases of (3.5) obviously follow.Let = p q and = = N I B p .Then, in accordance to Corollary 3.4, we have the next corollary.

Theorem 4 .
2 initiates the next algorithm for computing ∈

2 1
and Φ by (4.3) and  Compute and return X by (4.16).Algorithms inducted by Corollaries 4.1-4.3can be proposed by the similar way.Now, we obtain the D-representations for the solution (3.8).
possessing the form (3.8) is expressed element-wise in one of the subsequent possible representations:

)
When the both matrices =

)
When both matrices =

2 ) 2 are( 3 . 1 )( 2 )( 3 ) 3 ( 4 ) 1 ( 7 )
this, equality (4.25) holds.(If both matrices = Hermitian, then D-representations of A D W †, , and B D N †, , are expressed by (4.9).Making that gives equality (4.26).The mixed cases (3) and (4) can be obtained in the similar way.□ Theorem 4.4 yields an algorithm similar to Algorithm 4.3.will be found similarly as in Algorithm 4.1.By (4.7) and (4.8), generate, respectively, the matrices On the other hand, by (4.3), generate the matrix Compute the values of minor sums Finally, by (4.11), compute and return The solvability of several novel CQMEs is investigated.Constraints are expressed in terms of the right/left range space and the right/left null space assumptions on solutions relative to input matrices.It is essential to mention that the presented results remain original in the domain of complex matrices.Obtained solutions to considered CQMEs are expressions of the general pattern = , †,D, involving the W-MPD/W-DMP inverse of A and the N-DMP/N-MPD inverse of B. Particular choices of considered solutions lead to the best-approximate solution ACB † † and the Drazin-inverse solution = X A CB D D .Obtained results confirm the theoretical importance of weighted DMP and MPD inverses.
and ͠ ω g. is the gth row of ͠ = and Ω can be found by (4.2) given that  By using Theorem 4.1 and Corollary 4.1, is easy to obtain the following mixed cases.