Further results on weighted core-EP inverse of matrices

05/06/2020 ∙ by Jajati Keshari Sahoo, et al. ∙ BITS Pilani University of Central Florida 0

In this paper, we introduce the notation of E-weighted core-EP and F-weighted dual core-EP inverse of matrices. We then obtain a few explicit expressions for the weighted core-EP inverse of matrices through other generalized inverses. Further, we discuss the existence of generalized weighted Moore-Penrose inverse and additive properties of the weighted core-EP inverse of matrices. In addition to these, we propose the star weighted core-EP and weighted core-EP star class of matrices for solving the system of matrix equations. We further elaborate on this theory by producing a few representation and characterization of star weighted core-EP and weighted core-EP star classes of matrices.

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1 Introduction

The core and core-EP inverses of matrices have been intensively studied in recent years to solve a certain type of matrix equations baks ; BakTr14 . Hence, a significant number of papers explored the characterizations of the core inverse and its applications in baskett1969 ; Kurata_2018 ; LiChen18 ; PreMo20 . A few properties of the core inverse and interconnections with different generalized inverses were discussed in baks ; Kurata_2018 ; rakic ; Wang_2014 . The core-EP inverse of matrices, introduced by Prasad and Mohana PrasadMo14 , have significantly impacted for square matrices. Then several characterizations of the core-EP inverse and its extension to rectangular matrices were discussed in ferreyra2018revisiting . In this connection, the authors of gao2018representations have discussed the weighted core-EP inverse and several representations in terms of matrix decomposition. Further, a few characterizations and properties of the core-EP inverse with other inverses are discussed in Gao_2019 ; PrasadMo14 ; PredragKM17 . The vast literature on core-EP, weighted core-EP inverses of matrices along with its multifarious extensions MaH19 ; HaiTing19 ; zhou2019core ; mosic2019 , motivate us to study introduce E-weighted core-EP and F-weighted dual core-EP inverse of matrices.

We mention below a summary of the main points of the discussion.

  1. The notation of -weighted core-EP and -weighted dual core-EP inverses are proposed. Through these definitions, the existence of generalized weighted Moore-Penrose inverse is discussed.

  2. Introduce several explicit expression for the weighted core-EP inverse of matrices through other generalized inverses, like, Drazin inverses, weighted core inverse, and generalized Moore-Penrose inverses.

  3. We have discussed additive properties of the -weighted core-EP and -weighted dual core-EP inverse of matrices.

  4. Introduce star weighted core-EP and weighted core-EP star matrices to solve the system of matrix equations.

  5. A few characterization and representation of star weighted core-EP and weighted core-EP star classes of matrices are discussed.

The main objective of this paper to investigate a new fruitful way for developing the relation of the weighted core-EP inverse of matrices with different generalized inverses, like, Drazin inverses, weighted core inverse, and generalized Moore-Penrose inverses. The results of these approaches will help the necessary freedom to deal with different types of inverses and flexibility to choose generalized inverses depending on applications. It is worth to mention the work of Mosić in Mosic20 in which they have introduced the Drazin-star and star-Drazin Matrices for solving some kind of system of matrix equations, very recently. The author also discussed the maximal classes of matrices for generating the most general form of this class of matrices. It also motivates us to introduce the star weighted core-EP and weighted core-EP star class of matrices and then provide characterizations of -weighted core-EP inverse and -weighted dual core inverse of matrices in the form of the outer inverse of the Moore-Penrose inverse.

The outline of the paper is as follows. We present some necessary definitions and notation in Section 2. Definition, existence, and several explicit expression for the weighted core-EP inverse of matrices are considered in Section 3. In Section 4, we discuss the new class of matrices (i.e., star weighted core-EP and weighted core-EP star) to solve the system of the matrix equation. In addition to these, we discuss a few characterizations of the new class of matrices. The work is concluded along with a few future perspective problems in Section 5.

2 Preliminaries

For convenience, throughout this paper, stands for the set of matrices over complex numbers. In addition, we assume the matrices and to be invertible, and hermitian. Further, we use the notation , and for the range space, null space and conjugate transpose of respectively. The index of is the smallest non-negative integer , such that , which is denoted by . The Drazin inverse discussed in cline1980 for a rectangular matrix, however, it was introduced Drazin58 earlier in the context of associative rings and semigroups. Let , the Drazin inverse of is the unique matrix which satisfies the following equations

Let us recall the generalized weighted Moore-Penrose inverse sheng2007generalized ; ben of a matrix, as follow.

Definition 2.1.

Let . A matrix satisfying

 ,  ,  ,  ,

is called the generalized weighted inverse of and denoted by .

Note that generalized weighted Moore-Penrose inverse of a matrix does not exist alwayssheng2007generalized . But the positive definite of and leads to the existence of . The uniqueness of can be verified easily. Recall the definition of weighted core and dual inverse of a matrix as follows.

Definition 2.2.

Let . If a matrix satisfies

(6) (7) , ,

is called the -weighted core inverse of .

This inverse denoted by and the uniqueness of it can found in RJR . At the same time, the -weighted dual core inverse of denoted by and defined as follows.

Definition 2.3 (Rjr ).

Let . A matrix is called -weighted dual core inverse of if satisfies

(8) (9) , and .

Let us recall a few useful results from RJR and ben .

Theorem 2.4.

(RJR ) Let and . If , then the following five conditions are true.

  1. (;

  2. (

  3. ;

  4. for any ;

  5. .

Lemma 2.5 (ben ).

Let with . Then has index 1 and for all .

Lemma 2.6 (Proposition 3.2, Rjr ).

For , if , then and .

Lemma 2.7.

Let . If , then .

Proof.

Let . Then , and . Using these, we obtain

Similarly, the following result follows for weighted dual core inverse.

Lemma 2.8.

Let . If , then .

Lemma 2.9 (Rjr ).

Let and with . Then .

Theorem 2.10.

Let and . If and , then .

Proof.

Let . Then there exists a such that . Now the hermitian of follows from the below expression.

(1)

Using equation (1), we obtain

.

3 One-sided weighted core-EP inverse

In this section, we introduce -weighted core-EP inverse and -weighted dual core-EP inverses of square matrices. In addition, a few results and characterizations are established. Now we present the definition of the -weighted core-EP inverse.

Definition 3.1.

Let and . A matrix is called the -weighted core-EP inverse of if it satisfies

, and .

The -weighted core-EP inverse of is denoted by . Next, we define the -weighted dual core-EP inverse.

Definition 3.2.

Let and with . A matrix satisfies

, ,

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

In support of the Definition 3.1 and 3.2, we have worked out the following example.

Example 3.3.

Let , and . It is easy to verify ,

and

Now, we discuss a few useful results which will be frequently used in the subsequent sections.

Proposition 3.4.

Let and . Then the following are holds.

  1. If a matrix then for any .

  2. If , then and .

  3. If a matrix then for any .

Proof.

Let . Then .
Let . Then by part , we have

.

Further, from and , we obtain

The characterization of -weighted core-EP inverse through the range condition is presented below.

Theorem 3.5.

Let and . Then the following statements are equivalent.

  1. ,

  2. and .

Proof.

By Proposition 3.4, it enough to show . Let . Then

Thus . From

,

we get . Hence .
From , we have , for some . Now

.

Let and . Then there exists such that and . Using these results, we obtain

(2)
(3)

In view of Eqns. (2), (3), and invertibility of , we have . Thus

. Hence is the -weighted core-EP inverse of .

Similarly, the following result holds for -weighted dual core-EP inverse.

Theorem 3.6.

Let and . Then the following statements are equivalent.

  1. ,

  2. and .

The representation of the Drazin inverse is constructed through inverse as follow.

Lemma 3.7.

Let and . For , if , then the following are holds.

  1. .

  2. .

Proof.

Let . Then by Proposition 3.4 , we have

, and

.

Thus is a inverse of .
Let . Then applying Proposition 3.4, we have

, and

Hence

The following result can be proved in similar manner for -weighed dual core-EP inverse.

Lemma 3.8.

Let and . For , if , then the following are holds.

  1. .

  2. .

We now have the following characterization of the class of -inverse.

Lemma 3.9.

Let with . For if , then .

Proof.

Let . Then

, and .

Using these, we obtain

Similarly, we obtain the following result for -weighted dual core-EP inverse.

Lemma 3.10.

Let and . For if , then

The uniqueness of -weighted core-EP inverse is discussed below.

Theorem 3.11.

Let and . Then is unique.

Proof.

Suppose there exist two -weighted core-EP inverses of . Then using Proposition 3.4 and Lemma 3.7, we obtain

and .

By Lemma 3.9, we have . ∎

Similarly, the uniqueness of -weighted dual core-EP inverse can be verified.

Theorem 3.12.

Let and . Then is unique.

Now we discuss one of our important results, which gives a method of construction of the weighted core-EP inverse using a -inverse of matrix.

Theorem 3.13.

Let and . Then , where be any positive integer satisfying and is a inverse of .

Proof.

Let . Then

,

and . ∎

Similarly, construction of the -weighted dual core-EP inverse through -inverse of a matrix is presented below.

Theorem 3.14.

Let and . Then , where be any positive integer satisfying and is a inverse of .

In view of Theorem 3.13 and 3.14, we state the following as a corollary for construction of weighted core-EP inverse.

Corollary 3.15.

Let and . For , if exists, then

  1. .

  2. , .

  3. .

  4. , .

In conjunction with Lemma 2.9 and Corollary 3.15, we obtain the following representations for the weighted core-EP inverse through weighted core and weighted Moore-Penrose inverse.

Proposition 3.16.

Let be positive definite matrices and with . Then

  1. .

  2. ,

where is any positive integer satisfying .

Proof.

From Lemma 2.5 and 2.9, we get

(4)

Using equation (4) and Corollary 3.15 , we obtain

.

In similar way, we can show the result. ∎

We next discuss a necessary and sufficient condition in connection to the characterization of and inverses.

Proposition 3.17.

Let . Then is non-empty if and only if for some .

Proof.

Let . Then

, where .

Conversely, let and . Then . Further, and

.

Thus is an inverse of . ∎

Proposition 3.18.

Let . Then is non-empty if and only if for some .

In view of the above Proposition 3.17, we obtain the following necessary and sufficient condition for inverse.

Lemma 3.19.

Let . Then for some if and only if for some .

Proof.

Let . Then by taking , we have . Applying Proposition 3.17, we have . Thus

, where .

Conversely, let for some . Then , where . ∎

At the same time, we get the following result for inverse through the Proposition 3.18.

Lemma 3.20.

Let . Then for some if and only if for some .

In view of Theorem 3.13, Proposition 3.17, and Lemma 3.19, we have the following sufficient condition for -weighted core-EP inverse.

Theorem 3.21.

Let and . If for some , then .

Proof.

Let for some .Then by Lemma 3.19, we obtain for some . Using Proposition 3.17, we have . Thus by applying Theorem 3.13, we get . ∎

In view of -weighted dual core-EP inverse can be written as follow.

Theorem 3.22.

Let and . If for some , then .

We, next discuss the existence of the power of an -weighted core-EP inverse.

Theorem 3.23.

Let and Then exists if and only if exists. Moreover, and .

Proof.

Let and . Then by Proposition 3.4 and Lemma 3.7 ,

,

, and

.

Thus .
Conversely, let and . Then by Lemma 2.9, we obtain

.

Applying Lemma 2.6, we have