    # Characterization and Representation of Weighted Core Inverse of Matrices

In this paper, we introduce new representation and characterization of the weighted core inverse of matrices. Then, we study some properties of the one-sided core and dual-core inverse of matrices along with group inverse and weighted Moore-Penrose inverse. Further, by applying the new representation and properties of the weighted core inverse of the matrix we discuss a few new results related to the reverse order law for these inverses.

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## 1 Introduction and preliminaries

Let be the

-dimensional complex vector space and

be the set of matrices over complex field . For a matrix , , and denotes the range space, null space and conjugate transpose of respectively. The index of is defined as the smallest non-negative integer , for which and denoted by . In particular, if , then the matrix is called index one matrix or core matrix or group matrix. We now recall the Moore-Penrose inverse Rao and Mitra (1971); Ben-Israel and Greville (1974) of a matrix . The unique matrix , satisfying

 (1) AXA=A,  (2) XAX=X,  (3) (AX)∗=AX,  (4) (XA)∗=XA

is called the Moore-Penrose inverse of and denoted by . Apart from the generalized inverses, the weighted Moore-Penrose inverse is also important, as it can be simplified to a Moore-Penrose, as well as an ordinary matrix inverse. We recall this definition below.

###### Definition 1.1.

Let , and be two hermitian invertible matrices. A matrix satisfying

 (1) AXA=A,  (2) XAX=X  (3M) (MAX)∗=MAX,  (4N) (NXA)∗=NXA,

is called generalized weighted Moore-Penrose inverse of and denoted by .

Note that generalized weighted Moore-Penrose inverse does not exist always Sheng and Chen (2007). But if we replace the invertibility of and as positive definite then always exists and we call weighted Moore-Penrose inverse Ben-Israel and Greville (1974) of the matrix . Further, consider with . A matrix satisfying

 (1k) XAk+1=Ak,  (2) XAX=X,  (5) AX=XA,

is called the Drazin inverse of and denoted by , which was introduced Drazin (1958) in the context of associative rings and semigroups in 1958. In particular, for with , if a matrix satisfies

 (1) AXA=A,  (2) XAX=X,  (5) AX=XA,

then is called the group inverse of and denoted by . We use the notation for an element of -inverses of and for the class of -inverses of , where is the nonempty subset of . For instance, a matrix is called a -inverse of if satisfies equation for each .

Baksalary and Trenkler Baksalary and Trenkler (2010, 2014) introduced the core inverse of a square matrix and discussed the existence of these matrices. However, the right weak generalized inverse was proposed Ben-Israel and Greville (1974); Cline (1968) earlier, which renamed as core inverse. In the literature, many authors have been discussed the core inverse Zhou et al. (2019); Gao and Chen (2018); Wang et al. (2019); Ma and Li (2019), and achieved much in the aspect of representations of other generalized inverses Kurata (2018); Rakić et al. (2014); Wang and Liu (2015). Prasad and Mohana Manjunatha Prasad and Mohana (2014) discussed Core EP inverse of a square matrix of arbitrary index. Then the authors Ferreyra et al. (2018) extend the notion of core EP inverse to weighted core EP inverse and study some properties of weighted core EP inverses. Further, the authors Gao and Chen (2018) discussed the characterization and representations of the core-EP inverse. We present these equations Gao et al. (2018) as the following definition.

###### Definition 1.2.

Gao and Chen (2018) Let with . A matrix satisfying

 (6k) XAk+1=Ak,  (7) AX2=X,  (3) (AX)∗=AX,

is called the core-EP inverse of and denoted by

In particular, for with , if a matrix satisfies

 (6) XA2=A,  (7) AX2=X,  (3) (AX)∗=AX,

then is called the core inverse of and denoted by . One can notice that, the core-EP inverse is unique and . From core-EP inverse of a square matrix to a rectangular matrix was extended recently in Ferreyra et al. (2018), which is recalled next.

###### Definition 1.3.

Ferreyra et al. (2018) Let , and . A matrix satisfying

 WAWX=(WA)k[(WA)k]†,  R(X)⊆R((AW)k),

is called the -weighted core-EP inverse of and denoted by .

Further, the authors Gao et al. (2018) in Theorem 2.2 discussed the new expression of the -weighted core inverse, as follows.

###### Definition 1.4.

Let , and . A matrix satisfying

 XW(AW)k+1=(AW)k, A(WX)2=X, (WAWX)∗=WAWX, (1.1)

is called the -weighted core-EP inverse of .

The authors of Gao et al. (2018) also discussed, the unique matrix satisfies the equations (1.1). In addition to this, several characterization on the -weighted core-EP inverse are also discussed in Gao et al. (2018). Thus, it is observable that the weighted inverses of matrices have been frequently investigated recently Stanimirović et al. (2017, 2020); Zhang et al. (2016). However, many results are also available in elements of rings with involution Mosić et al. (2018). In addition to these, the authors of Gao et al. (2018) provided a characterization of the weighted core-EP inverse Ma (2019) in terms of matrix decomposition. Tian and Wang Tian and Wang (2011) discussed weighted-EP matrices as a generalization of EP matrices. The idea of the following definition is borrowed from Tian and Wang (2011) where the authors proved it in the Theorem 3.5.

###### Definition 1.5.

Let , be two hermitian invertible matrices and be a core matrix. Then is called weighted-EP with respect to if exists and .

On the other hand, the reverse order law for generalized inverse plays an important role in the theoretic research and numerical computations in many areas Rao and Mitra (1971); Wang et al. (2018); Sun and Wei (2002). Consider two invartible matrices and . The equality is called the reverse order law, which is always true for invertible matrices. When the ordinary inverse is replaced by generalized inverse this reverse order law is not true in general Ben-Israel and Greville (1974). In this context, Greville Greville (1966) first discussed a sufficient and necessary condition of this equality in the framework of the Moore-Penrose inverse in 1966. Since then, a significant number of papers investigated the reverse order law for various classes of generalized inverses Sun and Wei (1998); Panigrahy et al. (2020); Sahoo and Behera ; Mosić and Djordjević (2012, 2011).

In the sprite of the core inverse inverse of matrices, Sahoo, et al. recently introduced the core and core EP inverse of tensor

Sahoo and Behera ; Sahoo et al. (2020) via the Einstein product. Indeed, tensors are multidimensional generalizations of vectors and matrices. Now, we recall the following two results from Sahoo and Behera and write in form of matrix.

###### Lemma 1.6 (Lemma 2.10 Sahoo and Behera ).

Let . Then the group inverse of is exist if and only if for some .

###### Corollary 1.7 (Corollary 2.11 Sahoo and Behera ).

Let be a core matric. Then for some some .

The main objective of this paper is to study the weighted core inverse of matrices and discuss its new representations. The paper is carried out as follows. In Section 2, we introduce a new expression of weighted core and the dual-core inverse of a matrix. In addition to this, we discuss a few representation and characterization of weighted core inverse of matrices. Further a few results on reveres order law are discussed in section 3. Finally, the proposed new representations of weighted core inverse has been summarized in Section 5 with a brief discussion on further research in this direction.

## 2 One sided weighted core and dual core inverse

In this section, we first introduce -weighted core and -weighted dual core inverse. Then we discuss a few new representation and characterization of these inverses.

###### Definition 2.1.

Let

be a hermitian invertible matrix and

be a core matrix. A matrix satisfying

 (3M) (MAX)∗=MAX,  (6) XA2=A,  (7) AX2=X,

is called the -weighted core inverse of of and denoted by .

It is easy to verify the following result.

###### Proposition 2.2.

For , if , then .

Using Proposition 2.2, we can show the uniqueness of -weighted core inverse which stated in the next theorem.

###### Theorem 2.3.

Let be a hermitian invertible matrix and with . Then the -weighted core inverse of of is unique.

###### Proof.

Suppose there exist two -weighted core inverses say and . Then applying Proposition 2.2, we obtain

 Y = YAY=YAXAY=YAXM−1MAY=YAXM−1(MAY)∗ = YAXM−1Y∗A∗M=YM−1MAXM−1Y∗A∗M = YM−1X∗A∗Y∗A∗M=YM−1X∗A∗M=YAX = YA2X2=AX2=X.

###### Definition 2.4.

Let be a hermitian invertible matrix and be a core matrix. Then a matrix satisfying

 (4N) (NXA)∗=NXA,  (8) A2X=A,  (9) X2A=X,

is called -weighted dual core inverse of of and denoted by .

Likewise Proposition 2.2, the following result.

###### Proposition 2.5.

For any matrix . If the matrix , then .

###### Proof.

The proof is follows from the following verification:

 AXA=(A2X)XA=A2(X2A)=A2X=A; XAX=(X2A)AX=X2(A2X)=X2A=X.

The next theorem is proved for the uniqueness of the -weighted dual core inverse.

###### Theorem 2.6.

Let be a hermitian invertible matrix and with . Then the -weighted dual core inverse of of is unique.

###### Proof.

Suppose there exist two -weighted dual core inverses say and . Then applying Proposition 2.5, we obtain

 Y = YAY=YAXAY=N−1(NYA)N−1(NXA)Y=N−1A∗Y∗A∗X∗NY = N−1A∗X∗NY=N−1(NXA)∗Y=N−1NXAY=XAY = (X2A)AY=X2(A2Y)=X2A=X.

In view of the Definition 2.1, we have the following theorem which gives an equivalent characterization with other generalized inverses.

###### Theorem 2.7.

Let be an invetible hermitian matrix. If with , then the following five conditions are equivalent:

1. there exist such that , , ;

2. , , , and ;

3. , , and ;

4. exists and ;

5. there exist unique idempotent matrices such that , and .

If any one of the above true, then the -weighted core inverse of is given by . Further , , and .

###### Proof.

Let . Then there exist such that . Using , we obtain

.

Similarly from , we have for some . Which further implies . Thus

since is hermitian and invertible. Now

.

Again from , we have for some . This yields

.

It is trivially true.
It is enough to show exists. Let . From Proposition 2.2, it follows that . Now , , and . Hence .
Let , where If we choose and , then it is easy to verify that and both are idempotent. One can also observe that is a hermitian matrix, . From and , we obtain . Next we will claim the uniqueness of the idempotent matrices and . Suppose there exist another pair and which satisfies . Then and . From and we obtain

.

Similarly, from ) we get and . Which further yields

.

Pre-multiplying by , we obtain .
Let us assume that there exist unique idempotent matrices , such that , and ). From the range condition, we can easily obtain

 A=PA=QA=AQ,  Q=QA(1)A=AA(1)Q,  P=AA(1)P. (2.1)

If we take , then . From and , we obtain . Using equation (2.1), we obtain

and

.

Therefore . The representation of can be verified easily and the other representations follow from the proof of . ∎

The relation between -weighted core and -weighted dual core is given in the next result.

###### Lemma 2.8.

Let, and be an invertible Hermitian matrix. Then exists if and only if exists. Moreover, .

###### Proof.

Let . Then by Theorem 2.7, , , and . Define and . Next we will claim that is -weighted dual core inverse of . From

,

,

, we have . Similarly, we can show the converse part. ∎

In view of the fact of Theorem 2.7 and Lemma 2.8, we have the following equivalent descriptive statement for -weighted dual core inverse

###### Theorem 2.9.

Let be an invetible hermitian matrix and be a core matrix. Then the following five conditions are equivalent:

1. there exist such that , , ;

2. , , , and ;

3. ,, and ;

4. exists and ;

5. there exist unique idempotent matrices such that , and .

If any one of the above is true, then the -weighted dual core inverse of the matrix is given by . Further , , and .

###### Corollary 2.10.

Let be a core matrix and be an invetible hermitian matrices. Further, let with . If satisfies either or , then .

###### Proof.

Let . Then . Thus and hence by Theorem 2.7 , . Further, from , we obtain . Which yields and hence the proof is complete. ∎

The following propositions can be used an equivalent definition for -weighted core inverse.

###### Proposition 2.11.

Let be an invetible hermitian matrices and be a core matrix. If a matrix satisfies

, and

then is the -weighted core inverse of

###### Proof.

Let . Then . Now Therefore, So, by Corollary 2.10, Replacing by we obtain

###### Proposition 2.12.

Let be an invetible hermitian matrices and be a core matrix. If a matrix satisfies

and

then is the -weighted core inverse of

###### Proof.

From and , we get Pre-multiplying both sides, we obtain Now Hence the matrix is the -weighted core inverse of the matrix

###### Theorem 2.13.

Let and be invetible hermitian matrices. Then the following are equivalent:

1. exist;

2. there exist unique idempotent matrices such that , , and .

If any one of the statements holds, then .

###### Proof.

Let . If we define and , then we can easily show that and . Further , , , and . To show the uniqueness of and , assume that there exists two idempotent pairs and which satisfies . Now from , we have and . Using and , we obatin

.

Pre-multiplying , we get . Similarly, we can show the uniqueness of .
Let and be the unique idempotent matrices with , , and . Then , , and for some . Further, and . Now, consider . Then is follows from the following verification:

;

.

###### Theorem 2.14.

Let and be an invetible hermitian matrix. If and , then the following five conditions are true.

1. (;

2. (

3. ;

4. for any ;

5. .

###### Proof.

From Theorem 2.7, we obtain . Now, let . Then

 A\textcircled{\#},MXA\textcircled{\#},M=A#AA(1,3M)XA#AA(1,3M)=A#AA(1,3M)=A\textcircled{\#},M, (2.2)
 XA\textcircled{\#},MX=XA#AA(1,3M)X=A2A(1,3M)A#AA(1,3M)A2A(1,3M)=A2A(1,3M)=X, (2.3)

and

 A\textcircled{\#},MX=A#AA(1,3M)AAA(1,3M)=A2A#A(1,3M)=AAA(1,3M)AA#A(1,3M)=XA\textcircled{\#},M. (2.4)

From equations (2.2)-(2.4), we obtain . Further, we have

 (MA\textcircled{\#},MX)∗ = (MA#AA(1,3M)A2A(1,3M))∗=(MAA(1,3M))∗=MAA(1,3M) (2.5) = MAA#AA(1,3M)=MA#AA(1,3M)A2A(1,3M)=MA\textcircled{\#},MX,

and

 (MXA\textcircled{\#},M)∗ = (MA2A(1,3M)A\textcircled{\#},M)∗=(MA2A(1,3M)A#AA(1,3M))∗=(MAA(1,3M))∗ (2.6) = MAA(1,3M)=MAA#AA(1,3M)