Matrix inverses along the core parts of three matrix decompositions

1.   Introduction

Let $\mathbb{C}$ be the complex field. The set $\mathbb{C}^{m\times n}$ denotes the set of all $m\times n$ complex matrices over the complex field $\mathbb{C}$. Let $A\in \mathbb{C}^{m\times n}$. The symbol $A^{\ast}$ denotes the conjugate transpose of $A$. Notations $\mathcal{R}(A) = \{y\in \mathbb{C}^{m}: y = Ax, x\in \mathbb{C}^{n}\}$ and $\mathcal{N}(A) = \{x\in \mathbb{C}^{n}: Ax = 0\}$ will be used in the sequel. The smallest positive integer is $k$, such that $\text{rank}{(A^{k})} = \text{rank}(A^{k+1})$ is called the index of $A\in \mathbb{C}^{n\times n}$ and denoted by $\text{ind}(A)$.

Let $A\in \mathbb{C}^{m\times n}$. If a matrix $X\in \mathbb{C}^{n\times m}$ satisfies $AXA = A$, $XAX = X$, $(AX)^{\ast} = AX$, $(XA)^{\ast} = XA,$ then $X$ is called the Moore-Penrose inverse of $A$ ^[13,17] and denoted by $X = A^{\dagger}$. Let $A, X\in \mathbb{C}^{n\times n}$ with $\text{ind}{(A)} = k$. Then, the algebraic definition of the Drazin inverse is as follows if

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].

Lemma 1.1. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}{(A)} = k > 0$, then the Drazin inverse exists.

The core inverse and the dual core inverse for a complex matrix was introduced by Baksalary and Trenkler ^[4]. Let $A\in \mathbb{C}^{n\times n}$. A matrix $X\in \mathbb{C}^{n\times n}$ is called a core inverse of $A$ if it satisfies $AX = P_{A}$ and $\mathcal{R}(X)\subseteq \mathcal{R}(A)$, where $\mathcal{R}(A)$ denotes the column space of $A$ and $P_{A}$ is the orthogonal projector onto $\mathcal{R}(A)$. If such a matrix exists, then it is unique (and denoted by ). 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 $A\in \mathbb{C}^{n\times n}$. If there exists $X\in \mathbb{C}^{n\times n}$ such that $XAX = X\; \text{and}\; \mathcal{R}(X) = \mathcal{R}(X^{\ast}) = \mathcal{R}(A^{k})$, then $X$ is called the core-EP inverse of $A$. If such inverse exists, then it is unique and denoted by . 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 and denoted by $X = A^{Ⓦ}$.

Let $A\in \mathbb{C}^{n\times n}$. 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 $\text{rank}{(A_1)} = \text{rank}{(A_1^2)}$, $A_2$ is nilpotent and $A_{1}A_{2} = A_{2}A_{1} = 0$. It is well known that this decomposition is unique. Moreover, $A_1 = AA^DA = A^DA^2 = A^2A^D$ by [5, Definition 7.3.1], if $\text{ind}{(A)}\leqslant 1$, and thus $A$ coincides with $A_{1}$. $A_1$ is called the core part of $A$. Also, $A_2 = A-AA^DA$ is the nilpotent part of $A$. In [21, Theorem 2.1], Wang introduced a new matrix decomposition, namely the core-EP decomposition of $A\in \mathbb{C}^{n\times n}$ with ${\rm ind}(A) = k$. Given a matrix $A\in \mathbb{C}^{n\times n}$, then $A$ can be written as the sum of matrices $\hat{A}_1\in \mathbb{C}^{n\times n}$ and $\hat{A}_2\in \mathbb{C}^{n\times n}$. That is $A = \hat{A}_1+\hat{A}_2$, where $\hat{A}_1$ is an index one matrix, $\hat{A}_2^k = 0$ and $\hat{A}_1^\ast \hat{A}_2 = \hat{A}_2 \hat{A}_1 = 0$. In [21, Theorems 2.3 and 2.4], Wang proved this matrix decomposition is unique and that there exists a unitary matrix $U\in \mathbb{C}^{n\times n}$ such that

where $T\in \mathbb{C}^{r\times r}$ is nonsingular, $N\in \mathbb{C}^{(n-r)\times (n-r)}$ is nilpotent and $r$ is number of nonzero eigenvalues of $A$. In [21, Theorem 2.3], Wang proved that $\hat{A}_{1}$ can be described by using the Moore-Penrose inverse of $A^{k}$. The explicit expressions of $\hat{A}_{1}$ can be found in the following lemmas.

Lemma 1.2. Let $A\in \mathbb{C}^{n \times n}$ with $\text{ind}(A) = k$. If $A = \hat{A}_1+\hat{A}_2$ is the core-EP decomposition of $A$, then $\hat{A}_{1} = A^{k}(A^{k})^{\dagger}A$ and $\hat{A}_{2} = A-A^{k}(A^{k})^{\dagger}A$.

Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}{(A)} = k$. The EP-nilpotent decomposition of $A$ was introduced by Wang and Liu ^[23]. $A$ can be written as $A = \tilde{A}_{1}+\tilde{A}_{2}$, where $\tilde{A}_{1}$ is an EP matrix, $\tilde{A}_{2}^{k+1} = 0$ and $\tilde{A}_{2}\tilde{A}_{1} = 0$. By the proof of [23, Theorem 2.2], one can get the following lemma.

Lemma 1.3. [23, Theorem 2.1] Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}{(A)} = k$ and $A = \tilde{A}_{1}+\tilde{A}_{2}$ be the EP-nilpotent decomposition of $A$. Then, there exists a unitary matrix $U\in \mathbb{C}^{n\times n}$ such that

where $T\in \mathbb{C}^{r\times r}$ is nonsingular, $N\in \mathbb{C}^{(n-r)\times (n-r)}$ 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 $\text{ind}(A) = k$.

Lemma 1.4. [23, Theorem 2.2] Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$ and $A = \tilde{A}_{1}+\tilde{A}_{2}$ be the EP-nilpotent decomposition of $A$ as $(1.2)$, then $\tilde{A}_{1} = AA^{k}(A^{k})^{\dagger}$.

Let $A, B, C\in \mathbb{C}^{n\times n}$. We say that $Y\in \mathbb{C}^{n\times n}$ is a $(B, C)$-inverse of $A$ if we have

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 $A, B, C\in \mathbb{C}^{n\times n}$.

Lemma 1.5. [18, Lemma 2.2.6(g)] Let $A, B, C\in \mathbb{C}^{n\times n}$. If $\text{rank}{(CAB)} = \text{rank}{(B)} = \text{rank}{(C)}$, then $B(CAB)^{-}C$ is invariant for any choice of $(CAB)^{-}$.

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 $C$.

Theorem 1.6. [8, Theorem 2.1 (ii) and Proposition 6.1] Let $A, B, C\in \mathbb{C}^{n\times n}$. Then, $Y\in \mathbb{C}^{n\times n}$ is the $(B, C)$-inverse of $A$ if, and only if, $YAY = Y$, $\mathcal{R}(Y) = \mathcal{R}(B)$ and $\mathcal{N}(Y) = \mathcal{N}(C)$.

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 $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, 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 $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = 1$, then the core inverse of $A$ coincides with the $(A, A^{\ast})$-inverse of $A$.

Lemma 1.9. [19, Theorem 1.10] Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, then the core-EP inverse of $A$ coincides with the $(A^{k}, (A^{k})^{\ast})$-inverse of $A$.

Lemma 1.10. [3, Remark 2.2 (i)] Let $A, B, C, U, V\in \mathbb{C}^{n\times n}$. If $\mathcal{R}(B) = \mathcal{R}(U)$ and $\mathcal{N}(C) = \mathcal{N}(V)$, then $A$ is $(B, C)$-invertible if and only if $A$ is $(U, V)$-invertible. In this case, we have $A^{\|(B, C)} = A^{\|(U, V)}$.

Based on the core parts of the core-nilpotent decomposition, core-EP decomposition and EP-nilpotent decomposition of $A\in \mathbb{C}^{n\times n}$, 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 $A_{1}$, $\hat{A}_{1}$ and $\tilde{A}_{1}$, 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 $\hat{A}_{1}$ of the core-EP decomposition are investigated.

3) Three generalized inverses related the core part $\tilde{A}_{1}$ 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 $i, j\in\{1, 2, 3\}$. 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 $i, j\in\{4, 5, 6\}$. 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 $i, j\in\{7, 8, 9\}$. In section five, relationships between $X_{i}$ and $X_{j}$ for $i, j\in\{1, 2, \cdots, 9\}$ are investigated.

2.   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 $i, j\in\{1, 2, 3\}$.

Theorem 2.1. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$. 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 $x\in \mathcal{N}(A_{1})$, then

For any $y\in \mathcal{N}(A^{D})$, then

So,

by the Eqs (2.3) and (2.4). For any $u\in \mathcal{N}(A^{D})$, then

For any $v\in \mathcal{N}(A^{k})$, then

So,

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 $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$. The $X_{2}$ coincides with the Drazin inverse of $A$. That is the Drazin inverse of $A$ is the inverse along $\hat{A}_{1}$ and $A_{1}$, where $\hat{A}_{1}$ is the core part of the core-EP decomposition and $A_{1}$ is the core part of the core-nilpotent decomposition.

Proof. By the equalities $\hat{A}_{1} = A^{k}(A^{k})^{\dagger}A = A^{k}[(A^{k})^{\dagger}A^{k}]^{\ast}(A^{k})^{\dagger}A = A^{k}(A^{k})^{\ast}[(A^{k})^{\dagger}]^{\ast}(A^{k})^{\dagger}A$ and $A^{k}(A^{k})^{\ast} = A^{k}(A^{k})^{\dagger}A^{k}(A^{k})^{\ast} = A^{k}(A^{k})^{\dagger}AA^{k-1}(A^{k})^{\ast} = \hat{A}_{1}A^{k-1}(A^{k})^{\ast}$, we have

Thus, $X_{2}$ coincides the inverse along $A^{k}(A^{k})^{\ast}$ and $(A^{k})^{\ast}A$. That is $X_{2} = A^{\parallel(A^{k}(A^{k})^{\ast}, (A^{k})^{\ast}A)}$, which is equivalent to $X_{2}$ as the $(A^{k}(A^{k})^{\ast}, (A^{k})^{\ast}A)$-inverse. By $\hat{A}_{1} = A^{k}(A^{k})^{\dagger}A$ and $A^{k} = A^{k}(A^{k})^{\dagger}A^{k} = A^{k}(A^{k})^{\dagger}AA^{k-1} = A_{1}A^{k-1}$, we have $\mathcal{R}(\hat{A}_{1}) = \mathcal{R}(A^{k})$. Thus, the condition (2.10) can be replaced by $\mathcal{R}(\hat{A}_{1}) = \mathcal{R}(A^{k})$ 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. □

Theorem 2.3. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$. The $X_{3}$ coincides with the Drazin inverse of $A$, that is the Drazin inverse of $A$ is the inverse along $\tilde{A}_{1}$ and $A_{1}$, where $\tilde{A}_{1}$ is the core part of the EP-nilpotent decomposition and $A_{1}$ is the core part of the core-nilpotent decomposition.

Proof. Since

we have $\mathcal{R}(A^{k+1}) = \mathcal{R}(\tilde{A}_{1})$, which implies

Thus, $X_{2}$ coincides the inverse along $A^{k}(A^{k})^{\ast}$ and $(A^{k})^{\ast}A$. That is $X_{2} = A^{\parallel(A^{k}(A^{k})^{\ast}, (A^{k})^{\ast}A)}$, which is equivalent to $X_{2}$ as the $(A^{k}(A^{k})^{\ast}, (A^{k})^{\ast}A)$-inverse. By $\hat{A}_{1} = A^{k}(A^{k})^{\dagger}A$ and $A^{k} = A^{k}(A^{k})^{\dagger}A^{k} = A^{k}(A^{k})^{\dagger}AA^{k-1} = A_{1}A^{k-1}$, we have $\mathcal{R}(\hat{A}_{1}) = \mathcal{R}(A^{k})$. Thus, the condition (2.10) can be replaced by $\mathcal{R}(\hat{A}_{1}) = \mathcal{R}(A^{k})$ 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 $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, then, $X_{i}$ is the same as $X_{j}$. Moreover, $X_{j}$ coincides with the Drazin inverse of $A$, where $i, j\in\{1, 2, 3\}$.

Proof. It is trivial by Theorems 2.1–2.3. □

3.   Three generalized inverses related the core part $\hat{A}_{1}$ 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 $i, j\in\{4, 5, 6\}$.

Theorem 3.1. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, then the generalized inverse $X_{4}$ coincides with the $(A^{k}(A^{k})^{\ast}, (A^{k})^{\ast}A)$-inverse of $A$.

Proof. Let $\hat{A}_{1}$ be the core part of the core-EP decomposition as (1.1), the $\hat{A}_{1} = A^{k}(A^{k})^{\dagger}A$ by Lemma 1.2. For any $x\in \mathcal{N}((A^{k})^{\dagger}A)$, we have

For any $y\in \mathcal{N}(\hat{A}_{1})$, we have

Thus, we have

by Eqs (3.1) and (3.2). Also, we have

by

and

Equations (3.3) and (3.4) imply

By $\hat{A}_{1} = A^{k}(A^{k})^{\dagger}A = A^{k}(A^{k})^{\dagger}A^{k}(A^{k})^{\dagger}A = A^{k}[(A^{k})^{\dagger}A^{k}]^{\ast}(A^{k})^{\dagger}A = A^{k}(A^{k})^{\ast}[(A^{k})^{\dagger}]^{\ast}(A^{k})^{\dagger}A$ and $A^{k}(A^{k})^{\ast} = A^{k}(A^{k})^{\dagger}A^{k}(A^{k})^{\ast} = A^{k}(A^{k})^{\dagger}AA^{k-1}(A^{k})^{\ast} = \hat{A}_{1}A^{k-1}(A^{k})^{\ast}$, we have

Thus, $X_{4}$ coincides the inverse along $A^{k}(A^{k})^{\ast}$ and $(A^{k})^{\ast}A$. That is, $X_{4} = A^{\parallel(A^{k}(A^{k})^{\ast}, (A^{k})^{\ast}A)}$, which is equivalent to $X_{4}$ as the $(A^{k}(A^{k})^{\ast}, (A^{k})^{\ast}A)$-inverse. □

We have $\hat{A}_{1} = A^{k}(A^{k})^{\dagger}A$ and $A^{k} = A^{k}(A^{k})^{\dagger}A^{k} = A^{k}(A^{k})^{\dagger}AA^{k-1} = A_{1}A^{k-1}$ by Lemma 1.2, so $\mathcal{R}(\hat{A}_{1}) = \mathcal{R}(A^{k})$. Thus, condition (2.10) can be replaced by $\mathcal{R}(\hat{A}_{1}) = \mathcal{R}(A^{k})$ and we have the following theorem.

Theorem 3.2. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, then $X_{4}$ coincides with the $(A^{k}, (A^{k})^{\ast}A)$-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 $A_{gRC}^{-}$[6, Definition 3.1]. That is, if $X\in \mathbb{C}^{n \times n}$ satisfies $A_{1}XA_{1} = A_{1}$, $\mathcal{R}(X)\subseteq \mathcal{R}(A_{1})$ and $\mathcal{RS}(X)\subseteq \mathcal{RS}(A_{1})$, 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 $\hat{A}_1$ by Lemma 3.5, thus the generalized constrained inverse of $A$ coincides with the group inverse of $\hat{A}_1$. By Lemma 3.4 and Theorem 3.2, we have that $X_{4}$ coincides with the generalized constrained inverse of $A$.

Lemma 3.3. [6, Theorem 3.4] Let $A\in \mathbb{C}^{n\times n}$. If $X\in \mathbb{C}^{n \times n}$ 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, $A_{gRC}^{-} = A^{Ⓦ}$.

Lemma 3.4. [6, Theorem 4.4] Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}{(A)} = k$. The generalized constrained inverse of $A$ coincides with the $(A^{k}, (A^{k})^{\ast}A)$-inverse of $A$.

Lemma 3.5. [22, Theorem 3.7] Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$ and $A = \hat{A}_1+\hat{A}_2$ be the core-EP decomposition of $A$ as given in $(1.1)$. The weak group inverse of $A$ coincides with the group inverse of $\hat{A}_1$; that is, $A^{Ⓦ} = \hat{A}_1^{\#}$.

Lemma 3.6. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$ and $A = \hat{A}_1+\hat{A}_2$ be the core-EP decomposition of $A$ as given in $(1.1)$. The weak group inverse of $A$ coincides with the $(\hat{A}_1, \hat{A}_1)$-inverse of $A$.

Proof. It is trivial by Lemmas 3.5 and 1.7. □

Theorem 3.7. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$ and $A = \hat{A}_1+\hat{A}_2$ 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 $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, then the generalized inverse $X_{5}$ coincides with the $(A^{k}, (A^{k})^{\ast}A)$-inverse of $A$.

Proof. It is trivial by Theorems 2.1 and 3.1. □

Theorem 3.9. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, then the generalized inverse $X_{6}$ coincides with the $(A^{k}, (A^{k})^{\ast}A)$-inverse of $A$.

Proof. It is trivial by Theorems 2.3 and 3.1. □

Theorem 3.10. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, then $X_{i}$ is the same as $X_{j}$. Moreover, $X_{j}$ coincides with the weak group inverse of $A$, where $i, j\in\{4, 5, 6\}$.

Proof. It is obvious by Theorems 3.1, 3.8 and 3.9. □

4.   Three generalized inverses related the core part $\tilde{A}_{1}$ 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 $i, j\in\{7, 8, 9\}$.

Theorem 4.1. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$. The $X_{7}$ coincides with the inverse of $A$ along $A^{k+1}(A^{k})^{\ast}$ and $(A^{k})^{\ast}$, that is $X_{7}$ is the $(A^{k}, (A^{k})^{\ast})$-inverse. Moreover, the generalized inverse $X_{7}$ is the core-EP inverse of $A$.

Proof. Let $X_{3}$ be the $(\tilde{A}_{1}, \tilde{A}_{1})$-inverse of $A$. That is the $(A^{k+1}(A^{k})^{\dagger}, A^{k+1}(A^{k})^{\dagger})$-inverse of $A$ by Lemma 1.4. Since

we have $\mathcal{R}(A^{k+1}) = \mathcal{R}(\tilde{A}_{1})$, which implies

For any $u\in \mathcal{N}(\tilde{A}_{1})$,

by Lemma 1.1. For any $v\in \mathcal{N}((A^{k})^{\ast})$,

and we have

Thus, $X_{7}$ coincides with the inverse of $A$ along $A^{k+1}(A^{k})^{\ast}$ and $(A^{k})^{\ast}$ by (2.12), (4.5) and Lemma 1.10. Therefore, the generalized inverse $X_{7}$ is the core-EP inverse of $A$ by Lemma 1.8 and the condition $X_{7}$ is the $(A^{k}, (A^{k})^{\ast})$-inverse. □

Theorem 4.2. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, then $X_{8}$ coincides with the inverse of $A$ along $A^{k+1}(A^{k})^{\ast}$ and $(A^{k})^{\ast}$, that is $X_{8}$ is the $(A^{k}, (A^{k})^{\ast})$-inverse. Moreover, the generalized inverse $X_{8}$ is the core-EP inverse of $A$.

Proof. It is trivial by Theorems 2.1 and 4.1. □

Theorem 4.3. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, then $X_{9}$ coincides with the inverse of $A$ along $A^{k+1}(A^{k})^{\ast}$ and $(A^{k})^{\ast}$, that is $X_{9}$ is the $(A^{k}, (A^{k})^{\ast})$-inverse. Moreover, the generalized inverse $X_{9}$ is the core-EP inverse of $A$.

Proof. It is trivial by Theorems 3.1 and 4.1. □

Theorem 4.4. Let $A\in \mathbb{C}^{n\times n}$ with $\text{ind}(A) = k$, then $X_{i}$ is the same as $X_{j}$. Moreover, $X_{j}$ coincides with the core-EP inverse of $A$, where $i, j\in\{7, 8, 9\}$.

Proof. It is obvious by Theorems 4.1–4.3. □

Let $X_{1}$, $X_{4}$ and $X_{7}$ be the generalized inverses along $A_{1}$, $\hat{A}_{1}$ and $\tilde{A}_{1}$, respectively. Note that $X_{1}$ denotes the inverse along $A_{1}$ and $A_{1}$; $X_{2}$ denotes the inverse along $\hat{A}_{1}$ and $A_{1}$; $X_{3}$ denotes the inverse along $\tilde{A}_{1}$ and $A_{1}$; $X_{4}$ denotes the inverse along $A_{1}$ and $\hat{A}_{1}$; $X_{5}$ denotes the inverse along $\hat{A}_{1}$ and $\hat{A}_{1}$; $X_{6}$ denotes the inverse along $\tilde{A}_{1}$ and $\hat{A}_{1}$; $X_{7}$ denotes the inverse along $A_{1}$ and $\tilde{A}_{1}$; $X_{8}$ denotes the inverse along $\hat{A}_{1}$ and $\tilde{A}_{1}$ and $X_{9}$ denotes the inverse along $\tilde{A}_{1}$ and $\tilde{A}_{1}$. Table 4 shows that $X_{1}$, $X_{2}$ and $X_{3}$ have the same column and nilpotent parts and $\mathcal{R}(X_{i}) = \mathcal{R}(A^{k})$ and $\mathcal{N}(X_{i}) = \mathcal{N}(A^{k})$ for $i = 1, 2, 3$; $X_{4}$, $X_{5}$ and $X_{6}$ have the same column and nilpotent parts and that $\mathcal{R}(X_{j}) = \mathcal{R}(A^{k})$ and $\mathcal{N}(X_{j}) = \mathcal{N}((A^{k})^{\ast}A)$ for $j = 4, 5, 6$ and $X_{7}$, $X_{8}$ and $X_{9}$ have the same column and nilpotent parts and $\mathcal{R}(X_{k}) = \mathcal{R}(A^{k})$ and that $\mathcal{N}(X_{k}) = \mathcal{N}((A^{k})^{\ast})$ for $k = 7, 8, 9$.

5.   Relationships between $X_{i}$ and $X_{j}$ for $i, j\in\{1, 2, \cdots, 9\}$

Let $A\in\mathbb{C}^{n\times n}$ with $\text{ind}{A} = k$. In this section, we will show that the generalized inverses $X_{1}$, $X_{4}$ and $X_{7}$ are different generalized inverses. 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 show that the Drazin inverse, the weak group inverse and the core-EP inverse can be the same for a certain singular complex matrix.

Example 5.1. Let $A = \left[\begin{array}{ccc}1 & 1 & 0 \\1 & 1 & 1 \\ 1 & 1 & \frac{1}{2} \\\end{array}\right] \in \mathbb{C}^{3\times 3}$. Then, it is easy to check that $\text{ind}(A) = 2$ and

However, $X_{1}\neq X_{4}$, $X_{1}\neq X_{7}$, $X_{4}\neq X_{7}$.

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 $A\in\mathbb{C}^{n\times n}$ with $\text{ind}{A} = k$, then generalized inverses $X_{1}$, $X_{4}$ and $X_{7}$ are different generalized inverses.

Example 5.3. Let $A = \left[\begin{array}{cccc}1 & 0 & 1 & 2 \\0 & 1 & 1 & 2 \\0 & 0 & 2 & 4 \\0 & 0 & -1 & -2 \\ \end{array}\right] \in \mathbb{C}^{4\times 4}$. Then, it is easy to check that $\text{ind}(A) = 2$, and

However, $X_{1}\neq X_{7}$, $X_{4}\neq X_{7}$.

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.

Example 5.4. Let $A = \left[\begin{array}{cccc}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & \frac{1}{5} & \frac{2}{5} \\0 & 0 & \frac{2}{5} & \frac{4}{5} \\ \end{array} \right] \in \mathbb{C}^{4\times 4}$. Then, it is easy to check that $\text{ind}(A) = 1$, and

However, $X_{1} = X_{2} = X_{3} = X_{4} = X_{5} = X_{6} = X_{7} = X_{8} = X_{9} = \left[\begin{array}{cccc}1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & \frac{1}{5} & \frac{2}{5} \\0 & 0 & \frac{2}{5} & \frac{4}{5} \\\end{array}\right]$.

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.

Example 5.5. Let $A = \left[\begin{array}{cccc} 1 & 0 & 1 & -1 \\ 0 & 1 & 1 & -1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{array} \right] \in \mathbb{C}^{4\times 4}$, $B = \left[\begin{array}{cccc}1 & -1 & 1 & 0 \\0 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\ \end{array}\right] \in \mathbb{C}^{4\times 4}$. Then, it is easy to check that $\text{ind}(A) = 2$ and $\text{ind}(B) = 3$, but

with $X_{1}\neq X_{4}$, $X_{1}\neq X_{7}$, $X_{4}\neq X_{7}$, and

with $Y_{1}\neq Y_{4}$, $Y_{1}\neq Y_{7}$, $Y_{4}\neq Y_{7}$.

Example 5.5 shows that the difference index of the complex matrices does not affect the relationships between the Drazin inverse, the weak group inverse and the core-EP inverse.

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.

6.   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 $A_{1}$, $\hat{A}_{1}$ and $\tilde{A}_{1}$. 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.

Acknowledgments

The research article is supported by the National Natural Science Foundation of China (No. 12001223), the Qing Lan Project of Jiangsu Province, the Natural Science Foundation of Jiangsu Province of China (No. BK20220702), the Natural Science Foundation of Jiangsu Education Committee (No. 22KJB110010), "Five-Three-Three" Talents of Huaian City and College Students Innovation and Entrepreneurship Training Program (No. 202311049024Z).

Conflict of interest

The authors declare no conflict of interest.