Matrix inverses along the core parts of three matrix decompositions

: New characterizations for generalized inverses along the core parts of three matrix decompositions were investigated in this paper. Let A 1 , ˆ A 1 and ˜ A 1 be the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of A ∈ C n × n , respectively, where EP denotes the EP matrix. A number of characterizations and di ff erent representations of the Drazin inverse, the weak group inverse and the core-EP inverse were given by using the core parts A 1 , ˆ A 1 and ˜ A 1 . One can prove that, the Drazin inverse is the inverse along A 1 , the weak group inverse is the inverse along ˆ A 1 and the core-EP inverse is the inverse along ˜ A 1 . A unified theory presented in this paper covers the Drazin inverse, the weak group inverse and the core-EP inverse based on the core parts of the core-nilpotent decomposition, the core-EP decomposition and EP-nilpotent decomposition of A ∈ C n × n , respectively. In addition, we proved that the Drazin inverse of A is the inverse of A along U and A 1 for any U ∈ { A 1 , ˆ A 1 , ˜ A 1 } ; the weak group inverse of A is the inverse of A along U and ˆ A 1 for any U ∈ { A 1 , ˆ A 1 , ˜ A 1 } ; the core-EP inverse of A is the inverse of A along U and ˜ A 1 for any U ∈ { A 1 , ˆ A 1 , ˜ A 1 } . Let X 1 , X 4 and X 7 be the generalized inverses along A 1 , ˆ A 1 and ˜ A 1 , respectively. In the last section, some useful examples were given, which showed that the generalized inverses X 1 , X 4 and X 7 were di ff erent generalized inverses. For a certain singular complex matrix, the Drazin inverse coincides with the weak group inverse, which is di ff erent from the core-EP inverse. Moreover, we showed that the Drazin inverse, the weak group inverse and the core-EP inverse can be the same for a certain singular complex matrix.


Introduction
Let C be the complex field.The set C m×n denotes the set of all m × n complex matrices over the complex field C. Let A ∈ C m×n .The symbol A * denotes the conjugate transpose of A. Notations R(A) = {y ∈ C m : y = Ax, x ∈ C n } and N(A) = {x ∈ C n : Ax = 0} will be used in the sequel.The smallest positive integer is k, such that rank (A k ) = rank(A k+1 ) is called the index of A ∈ C n×n and denoted by ind(A).
Let A ∈ C m×n .If a matrix X ∈ C n×m satisfies AXA = A, XAX = X, (AX) * = AX, (XA) * = XA, then X is called the Moore-Penrose inverse of A [13,17] and denoted by X = A † .Let A, X ∈ C n×n with ind (A) = k.Then, the algebraic definition of the Drazin inverse is as follows if AXA = A, XA k+1 = A k and AX = XA, then X is called the Drazin inverse of A. If such X exists, then it is unique and denoted by A D [7].Note that for a square complex matrix, the algebraic definition of the Drazin inverse is equivalent to the functional definition of the Drazin inverse.We have the following lemma by the canonical form representation for A and A D in Theorem 7.2.1 [5].
The core inverse and the dual core inverse for a complex matrix was introduced by Baksalary and Trenkler [4].Let A ∈ C n×n .A matrix X ∈ C n×n is called a core inverse of A if it satisfies AX = P A and R(X) ⊆ R(A), where R(A) denotes the column space of A and P A is the orthogonal projector onto R(A).If such a matrix exists, then it is unique (and denoted by A # ).Baksalary and Trenkler gave several characterizations of the core inverse by using the decomposition of Hartwig and Spindelböck [10,11].In [12], Mary introduced a new type of generalized inverse, namely the inverse along an element.This inverse is depended on Green's relations [9].The inverse along an element contains some known generalized inverses, such as group inverse, Drazin inverse and Moore-Penrose inverse.Many existence criterion for the inverse along an element can be found in [12,16].Manjunatha Prasad and Mohana [15] introduced the core-EP inverse of a matrix.Let A ∈ C n×n .If there exists X ∈ C n×n such that XAX = X and R(X) = R(X * ) = R(A k ), then X is called the core-EP inverse of A. If such inverse exists, then it is unique and denoted by A † .The weak group inverse of a complex matrix was introduced by Wang and Chen [22], which is the unique matrix X such that AX 2 = X and AX = A † A and denoted by X = A w .
Let A ∈ C n×n .The core-nilpotent decomposition [14, see Theorem 2.2.21] of A is the sum of two matrices A 1 and A 2 , i.e., A = A 1 + A 2 , such that rank (A 1 ) = rank (A 2 1 ), A 2 is nilpotent and Wang introduced a new matrix decomposition, namely the core-EP decomposition of A ∈ C n×n with ind(A) = k.Given a matrix A ∈ C n×n , then A can be written as the sum of matrices Â1 ∈ C n×n and Â2 ∈ C n×n .That is A = Â1 + Â2 , where Â1 is an index one matrix, Âk 2 = 0 and Â * 1 Â2 = Â2 Â1 = 0.In [21, Theorems 2.3 and 2.4], Wang proved this matrix decomposition is unique and that there exists a unitary matrix U ∈ C n×n such that where T ∈ C r×r is nonsingular, N ∈ C (n−r)×(n−r) is nilpotent and r is number of nonzero eigenvalues of A. In [21, Theorem 2.3], Wang proved that Â1 can be described by using the Moore-Penrose inverse of A k .The explicit expressions of Â1 can be found in the following lemmas.
Let A ∈ C n×n with ind (A) = k.The EP-nilpotent decomposition of A was introduced by Wang and Liu [23].A can be written as A = Ã1 + Ã2 , where Ã1 is an EP matrix, Ãk+1 where T ∈ C r×r is nonsingular, N ∈ C (n−r)×(n−r) is nilpotent and r is the number of nonzero eigenvalues of A.
The core part of the EP-nilpotent decomposition can be expressed by the Moore-Penrose inverse of A k , where ind(A) = k.
If such Y exists, then it is unique (see [1,Definition 4.1] and [19,Definition 1.2]).We also call the (B, C)-inverse of A is the inverse of A along B and C. Note that the (B, C)-inverse was introduced in the setting of semigroups [8].The (B, C)-inverse of A will be denoted by A ∥(B,C) .Note that Bapat et al. [2] investigated an outer inverse in Theorem 5 that is exactly the same as the (y, x)-inverse, where x and y are elements in a semigroup.In [20], Rao and Mitra showed that A ∥(B,C) = B(CAB) − C, where (CAB) − stands for the arbitrary inner inverse of CAB, where CAB is the product of A, B, C ∈ C n×n .
The following lemma shows that the (B, C)-inverse of A is an outer inverse of A, and can be characterized by using the column space of B and the null space of C. The following lemma can be found in [24,Lemma 3.11] for elements in rings, which also shows that the Drazin inverse is the inverse along A k and A k , where k is the index of A.
Lemma 1.7.[8, p1910] Let A ∈ C n×n with ind(A) = k, then the Drazin inverse of A coincides with the (A k , A k )-inverse of A. In particular, the group inverse of A coincides with the (A, A)-inverse of A.
Lemmas 1.8 and 1.9 show that the core-EP inverse of A is a generalization of the core inverse of A. Moreover, the core inverse of A k is the core-EP inverse of A, where k is the index of A.
Lemma 1.8.[8, p1910] Let A ∈ C n×n with ind(A) = 1, then the core inverse of A coincides with the (A, A * )-inverse of A. Lemma 1.9.[19, Theorem 1.10] Let A ∈ C n×n with ind(A) = k, then the core-EP inverse of A coincides with the Based on the core parts of the core-nilpotent decomposition, core-EP decomposition and EPnilpotent decomposition of A ∈ C n×n , respectively, three generalized inverses along two matrices are investigated, namely, the Drazin inverse, the weak group inverse and the core-EP inverse.Let X 1 , X 4 and X 7 be the generalized inverses along A 1 , Â1 and Ã1 , respectively.The major contributions of the article can be highlighted as follows: 1) Three generalized inverses related the core part A 1 of the core-nilpotent decomposition are investigated.
2) Three generalized inverses related the core part Â1 of the core-EP decomposition are investigated.
3) Three generalized inverses related the core part Ã1 of the EP-nilpotent decomposition are investigated.
4) We show that the Drazin inverse, the weak group inverse and the core-EP inverse are different generalized inverses.5) For a singular complex matrix, we can prove that the Drazin inverse coincides with the weak group inverse, which is different from the core-EP inverse.Moreover, we can show that the Drazin inverse, the weak group inverse and the core-EP inverse can be same for a certain singular complex matrix.
The paper is organized as follows.In section two, we prove that X i is the same as X j .Moreover, X j coincides with the Drazin inverse of A, where i, j ∈ {1, 2, 3}.In section three, we can prove that X i is the same as X j and that X j coincides with the weak group inverse of A, where i, j ∈ {4, 5, 6}.In section four, we can prove that X i is the same as X j and X j coincides with the core-EP inverse of A, where i, j ∈ {7, 8, 9}.In section five, relationships between X i and X j for i, j ∈ {1, 2, • • • , 9} are investigated.

Three generalized inverses related the core part A 1 of the core-nilpotent decomposition
In this section, three generalized inverses along the core parts of matrix decompositions are introduced.In Table 1, one can see that we denoted the generalized inverse along the core parts of the core-nilpotent decomposition as X 1 by using the symbol of the generalized inverse along two matrices.In a similar way, X 2 denotes the generalized inverse along the core part of the core-EP decomposition and the core part of the core-nilpotent decomposition.X 3 denotes the generalized inverse along the core part of the EP-nilpotent decomposition and the core part of the core-nilpotent decomposition.In addition, we prove that X i is the same as X j and that X j coincides with the Drazin inverse of A, where i, j ∈ {1, 2, 3}.
Table 1.Three generalized inverses related A 1 of the core-nilpotent decomposition.
Three generalized inverses Core part The generalized inverses along the core part type I Theorem 2.1.Let A ∈ C n×n with ind(A) = k.The X 1 coincides with the Drazin inverse of A. That is the Drazin inverse of A is the inverse along A 1 , where A 1 is the core part of the core-nilpotent decomposition.
Proof.Let A 1 be the core part of the core-nilpotent decomposition, then A 1 = A D A 2 and we have Thus, we have For any x ∈ N(A 1 ), then For any y ∈ N(A D ), then by the Eqs (2.3) and (2.4).For any u ∈ N(A D ), then For any v ∈ N(A k ), then by the Eqs (2.6) and (2.7).Thus, we have by the Eqs (2.5) and (2.8).Therefore, X 1 coincides with the Drazin inverse by Eqs (2.2) and (2.9) and Lemmas 1.7 and 1.10.□ Theorem 2.2.Let A ∈ C n×n with ind(A) = k.The X 2 coincides with the Drazin inverse of A. That is the Drazin inverse of A is the inverse along Â1 and A 1 , where Â1 is the core part of the core-EP decomposition and A 1 is the core part of the core-nilpotent decomposition.Proof.Since

Proof. By the equalities Â1
we have R(A k+1 ) = R( Ã1 ), which implies (2.12) Thus, X 2 coincides the inverse along Thus, the condition (3.6) can be replaced by R( Â1 ) = R(A k ) and we have the following theorem.Thus, X 3 coincides with the Drazin inverse of A by Lemma 1.10, and the proof of Theorem 2.1.□ Theorem 2.4.Let A ∈ C n×n with ind(A) = k, then, X i is the same as X j .Moreover, X j coincides with the Drazin inverse of A, where i, j ∈ {1, 2, 3}.
Proof.It is trivial by Theorems 2.1-2.3.□ 3. Three generalized inverses related the core part Â1 of the core-EP decomposition In this section, three generalized inverses along the core parts of matrix decompositions are introduced.In Table 2, one can see that we denoted the generalized inverse along the core parts of the core-EP decomposition as X 4 by using the symbol of the generalized inverse along two matrices.In a similar way, X 5 denotes the generalized inverse along the core part of the core-nilpotent decomposition and the core part of the core-EP decomposition.X 6 denotes the generalized inverse along the core part of the EP-nilpotent decomposition and the core part of the core-EP decomposition decomposition.In addition, we prove that X i is the same as X j and that X j coincides with the weak group inverse of A, where i, j ∈ {4, 5, 6}.
Table 2. Three generalized inverses related Â1 of the core-EP decomposition.
Three generalized inverses Core parts The generalized inverses along the core part type IV Â1 X 4 = A ∥( Â1 , Â1 ) type V A 1 and Â1 X 5 = A ∥(A 1 , Â1 ) type VI Ã1 and Â1 X 6 = A ∥( Ã1 , Â1 ) Theorem 3.1.Let A ∈ C n×n with ind(A) = k, then the generalized inverse X 4 coincides with the Proof.Let Â1 be the core part of the core-EP decomposition as (1.1), the Â1 For any y ∈ N( Â1 ), we have Thus, we have by Eqs (3.1) and (3.2).Also, we have Equations (3.3) and (3.4) imply Thus, X 4 coincides the inverse along ).Thus, condition (3.6) can be replaced by R( Â1 ) = R(A k ) and we have the following theorem.Theorem 3.2.Let A ∈ C n×n with ind(A) = k, then X 4 coincides with the (A k , (A k ) * A)-inverse.
For the square matrix A 1 , an inner inverse of A 1 with columns belonging to the linear manifold generated by the columns of A 1 and rows belonging to the linear manifold generated by the rows of A 1 will be called a generalized constrained inverse of A and denoted by and RS(X) ⊆ RS(A 1 ), then X = A − gRC .In the following lemmas, one can see that the generalized constrained inverse of A coincides with the weak group inverse by Lemma 3.3.Moreover, the weak group inverse of A coincides with the group inverse of Â1 by Lemma 3.5, thus the generalized constrained inverse of A coincides with the group inverse of Â1 .By Lemma 3.4 and Theorem 3.2, we have that X 4 coincides with the generalized constrained inverse of A. Proof.It is trivial by Theorems 2.3 and 3.1.□ Theorem 3.10.Let A ∈ C n×n with ind(A) = k, then X i is the same as X j .Moreover, X j coincides with the weak group inverse of A, where i, j ∈ {4, 5, 6}.

Three generalized inverses related the core part Ã1 of the EP-nilpotent decomposition
In this section, three generalized inverses along the core parts of matrix decompositions are introduced.In Table 3, one can see that we denoted the generalized inverse along the core parts of the EP-nilpotent decomposition as X 7 by using the symbol of the generalized inverse along two matrices.In a similar way, X 8 denotes the generalized inverse along the core part of the core-nilpotent decomposition and the core part of the EP-nilpotent decomposition.X 9 denotes the generalized inverse along the core part of the core-EP decomposition and the core part of the EP-nilpotent decomposition decomposition.In addition, we prove that X i is the same as X j and that X j coincides with the core-EP inverse of A, where i, j ∈ {7, 8, 9}.
Table 3.Three generalized inverses related Ã1 of the EP-nilpotent decomposition.
Three generalized inverses Core parts The generalized inverses along the core part type VII Ã1 we have R(A k+1 ) = R( Ã1 ), which implies For any u ∈ N( Ã1 ), and we have inverse coincides with the weak group inverse, which is different from the core-EP inverse.Moreover, we show that the Drazin inverse, the weak group inverse and the core-EP inverse can be the same for a certain singular complex matrix.
Then, it is easy to check that ind(A) = 2 and It is trivial that the generalized inverses X 1 , X 4 and X 7 are different generalized inverses by Example 5.1.Thus, we have the following theorem.
Theorem 5.2.Let A ∈ C n×n with ind A = k, then generalized inverses X 1 , X 4 and X 7 are different generalized inverses.
Then, it is easy to check that ind(A) = 2, and For a singular complex matrix, Example 5.3 shows that the Drazin inverse coincides with the weak group inverse, which is different from the core-EP inverse.
Then, it is easy to check that ind(A) = 1, and Example 5.4 shows that the Drazin inverse, the weak group inverse and the core-EP inverse can be the same for a certain singular complex matrix.
Theorem 5.2 and Example 5.1 show that the generalized inverses X 1 , X 4 and X 7 are different generalized inverses.Thus, we have the following Tables 5 and 6.
Table 5. Counterexamples related the inverse X 1 to X 9 .Table 6.Examples related the inverse X 1 to X 9 .

Related generalized inverses Examples
Example 5.4

Conclusions
New characterizations for generalized inverses along the core parts of three matrix decompositions were investigated in this paper.A number of characterizations and different representations of the Drazin inverse, the weak group inverse and the core-EP inverse were given by using the core parts A 1 , Â1 and Ã1 .Some useful examples were given, which showed that the generalized inverses X 1 , X 4 and X 7 are different generalized inverses.We believe that investigation related to the generalized inverses along the core parts of related matrix decompositions will attract attention, and we describe perspectives for further research: 1) Considering the matrix partial orders based on the generalized inverses can relate the core parts of matrix decompositions.2) Extending the generalized inverses can relate the core parts of matrix decompositions to an element in rings.
3) The column space and the null space of a complex matrix can be described by the core parts of matrix decompositions.

Use of AI tools declaration
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

2 =
0 and Ã2 Ã1 = 0.By the proof of [23, Theorem 2.2], one can get the following lemma.Lemma 1.3.[23, Theorem 2.1] Let A ∈ C n×n with ind (A) = k and A = Ã1 + Ã2 be the EP-nilpotent decomposition of A. Then, there exists a unitary matrix U ∈ C n×n such that

Theorem 2 . 3 .
Thus, the condition (3.6) can be replaced by R( Â1 ) = R(A k ) and we have the following theorems.Thus, X 2 coincides with the Drazin inverse of A by Lemma 1.10, and the proof of Theorem 2.1.□Let A ∈ C n×n with ind(A) = k.The X 3 coincides with the Drazin inverse of A, that is the Drazin inverse of A is the inverse along Ã1 and A 1 , where Ã1 is the core part of the EP-nilpotent decomposition and A 1 is the core part of the core-nilpotent decomposition.

Lemma 3 . 3 . 1 . 3 . 6 .Theorem 3 . 7 .Theorem 3 . 9 .
[6, Theorem 3.4] Let A ∈ C n×n .If X ∈ C n×n is a generalized constrained inverse of A, then this generalized constrained inverse of A is unique.Moreover, the generalized constrained inverse of A coincides with the weak group inverse; that is, A − gRC = A w .Lemma 3.4.[6, Theorem 4.4] Let A ∈ C n×n with ind (A) = k.The generalized constrained inverse of A coincides with the (A k , (A k ) * A)-inverse of A. Lemma 3.5.[22, Theorem 3.7] Let A ∈ C n×n with ind(A) = k and A = Â1 + Â2 be the core-EP decomposition of A as given in (1.1).The weak group inverse of A coincides with the group inverse of Â1 ; that is, A w = Â# Lemma Let A ∈ C n×n with ind(A) = k and A = Â1 + Â2 be the core-EP decomposition of A as given in (1.1).The weak group inverse of A coincides with the ( Â1 , Â1 )-inverse of A. Proof.It is trivial by Lemmas 3.5 and 1.7.□ Let A ∈ C n×n with ind(A) = k and A = Â1 + Â2 be the core-EP decomposition of A as (1.1).Then, the inverse X 4 coincides with the weak group inverse of A. Proof.It is trivial by Lemma 3.6 and the definition of the inverse of X 2 .□ Theorem 3.8.Let A ∈ C n×n with ind(A) = k, then the generalized inverse X 5 coincides with the (A k , (A k ) * A)-inverse of A. Proof.It is trivial by Theorems 2.1 and 3.1.□ Let A ∈ C n×n with ind(A) = k, then the generalized inverse X 6 coincides with the (A k , (A k ) * A)-inverse of A.   