Weak group inverse

In this paper, we introduce a weak group inverse (called the WG inverse in the present paper) for square matrices of an arbitrary index, and give some of its characterizations and properties. Furthermore, we introduce two orders: one is a pre-order and the other is a partial order, and derive several characterizations of the two orders. At last, one characterization of the core-EP order is derived by using the WG inverses.


Introduction
In this paper, we use the following notations. The symbol C m,n is the set of m × n matrices with complex entries; A * , R(A) and rk (A) represent the conjugate transpose, range space (or column space) and rank of A ∈ C m,n . Let A ∈ C n,n , the smallest positive integer k, which satisfies rk A k+1 = rk A k , is called the index of A and is denoted as Ind(A). The symbol C CM n stands for the set of n × n matrices of index equal to one. The Moore-Penrose inverse of A ∈ C m,n is defined as the unique matrix X ∈ C n,m satisfying the equations: (1) AXA = A, (2) XAX = X, (3) (AX) * = AX, (4) (XA) * = XA, In this paper, our main tools are two decompositions: one is the core decomposition, the other is the core-EP decomposition. The aim of the paper is to introduce a generalized group inverse, consider its applications and derive some of its characterizations and properties.

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In this section, we present some preliminary results.
[3] Let A ∈ C n,n be with Ind(A) = k. Then LEMMA 2.2. [1,7,16] Let A ∈ C n,n be with Ind(A) = k. Then there exists a unitary matrix U such that

Furthermore, A is core invertible if and only if ΣK is non-singular. When
A ∈ C CM n , (2.2) is called the core decomposition of A and

4)
where T = ΣK and S = ΣL. 30 It is well known that the core-nilpotent decomposition has been widely used Similarly, Wang introduced the notion of the core-EP decomposition in [16]: then A can be written as the sum of matrices A 1 and Here one or both of A 1 and A 2 can be null. 40 LEMMA 2.5. [16] Let the core-EP decomposition of A ∈ C n,n be as in Lemma

5)
where T is non-singular, and N is nilpotent. Furthermore, the core-EP inverse of A is

WG inverse
In this section, we apply the core-EP decomposition to introduce a generalized group inverse (i.e. the WG inverse) and consider some characterizations of the generalized inverse.

Definition and properties of the WG inverse 45
Let A ∈ C n,n be with Ind(A) = k, and consider the system of equations Let the core-EP decomposition of A be as in (2.5). Then the core-EP inverse A † of A can be formed as: Suppose that Substituting (3.3) for X in (3.1) and applying (3.2 ), we derive Therefore, (3.3) is the solution of the system to equations (3.1).
Furthermore, suppose that both X and X satisfy (3.1), then that is, the solution to the system of equations (3.1) is unique. We have the following: In the following example, we explain that the WG inverse is different from the Drazin, DMP, core-EP and B-T inverses.
. It is easy to check that Ind(A) = 2, the Moore-Penrose inverse A † and the Drazin inverse A D are , and the core-EP inverse A † and the WG inverse A W are

Characterizations of the WG inverse
Let A = A 1 + A 2 be the core-nilpotent decomposition of A ∈ C n,n . Then Applying Lemma 2.4, (2.5) and (3.3), we have the following theorem.
THEOREM 3.2. Let the core-EP decomposition of A ∈ C n,n be as in (2.5). Then Since we have the following theorem: Let the core-EP decomposition of A be as in (2.5). Then Therefore, we have the following theorem.
THEOREM 3.4. Let A ∈ C n,n be with Ind(A) = k. Then It is known that the Drazin inverse is one generalization of the group inverse.
We will see the similarities and differences between the Drazin inverse and the WG inverse from the following corollaries.
It is well known that A 2 D = A D 2 , but the same is not true for the WG inverse. Applying the core-EP decomposition (2.5) of A, we have and (3.9) Therefore, invertible, we derive the following Corollary 3.6.

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COROLLARY 3.6. Let the core-EP decomposition of A ∈ C n,n be as in (2.5).
The commutativity is one of the main characteristics of the group inverse.
The Drazin inverse has the characteristic, too. It is of interest to inquire whether the same is true or not for the WG inverse. Applying the core-EP decomposition (3.10b) Therefore, we have the following Corollary 3.7.
COROLLARY 3.7. Let the core-EP decomposition of A ∈ C n,n be as in (2.5).
Let SN = 0, then by applying Corollary 3.6 and Corollary 3.7, we derive Let t be a positive integer. It follows from applying (2.1), (2.4) and (2.6) that Therefore, we have the following Corollary 3.8..
COROLLARY 3.8. Let A ∈ C n,n be with Ind(A) = k, the core-EP decomposition of A be as in (2.5) and SN = 0. Then where t is a positive integer.

Two Orders
A binary operation on a set S is said to be a pre-order on S if it is reflexive and transitive. If the pre-order is also anti-symmetric, we call it a partial order [13,Chap 1]. Let S 1 and S 2 be sets, and S 2 ⊆ S 1 , then a partial order is said to be implied by a partial order in which A = A 1 + A 2 and B = B 1 + B 2 are the core-nilpotent decompositions 80 of A and B, respectively. Similarly, in this section, we apply the core-EP decomposition to introduce two orders: one is the WG order and the other is the C-E order.

WG order
Consider the binary operation:   In the following two examples, we see some differences between the WG order and the Drazin order.

It is easy to check that
. Therefore, the WG order does not 100 imply the Drazin order.
Let A WG ≤ B, A = A 1 + A 2 and B = B 1 + B 2 are the core-EP decompositions of A and B, A 1 and A 2 be as given in (2.5), and partition By applying (4.4) and (4.5), we have

It follows from
Therefore, in which B 22 is an arbitrary matrix of an appropriate size. From (4.5) and (4.6), we obtain Since B 1 is core invertible, B 22 is core invertible. Let the core decomposition of B 22 be as It is easy to see that U is a unitary matrix. Let SU 1 be partitioned as follows: Then and

C-E partial order 110
Consider the binary operation:  Proof. Reflexivity is trivial.
Let A and B be of the forms as given in (4.13a) and (4.13b), then A = A 1 +A 2 and B = B 1 + B 2 are the core-EP decompositions of A and B, respectively, and It is easy to check that

Characterizations of the core-EP order
As is noted in [16], the core-EP order is given: A † ≤ B : A, B ∈ C n,n , A † A = A † B and AA † = BA † . (5.1) Some characterizations of the core-EP order are given in [16].