Further characterizations of the weak core inverse of matrices and the weak core matrix

1.   Introduction

The weak core inverse was introduced in ^[1] where the authors presented some characterizations and properties. In ^[2], the authors introduced an extension of the weak core inverse. Continuing previous research about the weak core inverse, our purpose is to present new characterizations and representations of the weak core inverse. Additionally, we also give several equivalent conditions for a matrix to be a weak core matrix.

Let $\mathbb{C}^{m\times n}$ be the set of all $m\times n$ complex matrices and $\mathbb{Z^{+}}$ denotes the set of all positive integers. The symbols $\mathcal{R}(A)$, $\mathcal{N}(A)$, $A^{*}$, $r(A)$ and $I_{n}$ will denote the range space, null space, conjugate transpose, rank of $A\in\mathbb{C}^{m\times n}$ and the identity matrix of order $n$. Ind($A$) means the index of $A\in\mathbb{C}^{n\times n}$. Let $\mathbb{C}^{n\times n}_{k}$ be the set consisting of all $n\times n$ complex matrices with index $k$. The symbol dim$(S)$ represents the dimension of a subspace $S\subseteq\mathbb{C}^{n}$. $P_{L}$ stands for the orthogonal projection onto the subspace $L$. $P_{A}$, $P_{A^{*}}$ respectively denote the orthogonal projection onto $\mathcal{R}(A)$ and $\mathcal{R}(A^{*})$, i.e., $P_{A} = AA^{†}$, $P_{A^{*}} = A^{†}A$.

We will now introduce definitions of several generalized inverses that will be used throughout the paper. The Moore-Penrose inverse of $A\in\mathbb{C}^{m\times n}$, denoted by $A^{†}$, is defined as the unique matrix $X\in\mathbb{C}^{n\times m}$ satisfying ^[3]:

In particular, $X$ is an outer inverse of $A$ which is denoted as $A^{(2)}$ if $XAX = X$. For any matrix $A\in\mathbb{C}^{m\times n}$ with $r(A) = r$, let $T\subseteq\mathbb{C}^{n}$, $S\subseteq\mathbb{C}^{m}$ be two subspaces such that dim$(T) = t\leq r$ and dim$(S) = m-t$. Then $A$ has an outer inverse $X$ that satisfies $\mathcal{R}(X) = T$ and $\mathcal{N}(X) = S$ if and only if $AT\oplus S = \mathbb{C}^{m}$. In that case $X$ is unique and denoted by $A^{(2)}_{T, S}$ ^[4].

The Drazin inverse of $A\in\mathbb{C}^{n\times n}_{k}$, denoted by $A^{D}$, is the unique matrix $X\in\mathbb{C}^{n\times n}$ satisfying ^[5]: $XAX = X, \ AX = XA, \ XA^{k+1} = A^{k}.$

For any matrix $A\in\mathbb{C}^{n\times n}_{1}$, a new generalized inverse, which is called core inverse ^[6] was introduced. Two other generalizations of the core inverse for $A\in\mathbb{C}^{n\times n}_{k}$ such as core-EP inverse ^[7], DMP inverse ^[8] were also introduced.

In 2018, Wang and Chen ^[9] defined the weak group inverse of $A\in\mathbb{C}^{n\times n}_{k}$, denoted by $A^{Ⓦ}$, as the unique matrix $X\in\mathbb{C}^{n\times n}$ such that ^[9]: . Moreover, it was verified that .

Recently, Ferreyra et al. introduced a new generalization of core inverse called the weak core inverse of $A\in\mathbb{C}^{n\times n}_{k}$, denoted by $A^{Ⓦ, †}$(or, in short, WC inverse). It is defined as the unique matrix $X\in\mathbb{C}^{n\times n}$ satisfying ^[1]:

where $C = AA^{Ⓦ}A$. Moreover, it is proved that $A^{Ⓦ, †} = A^{D}CA^{†} = A^{Ⓦ}AA^{†}$.

The structure of this paper is as follows: In Section 2, we give some preliminaries which will be made use of later in this paper. In Section 3, we discuss some characterizations of the WC inverse based on its range space, null space and matrix equations. In Section 4, several new representations of the WC inverse are proposed. Section 5 is devoted to deriving some properties of the WC inverse by the core-EP decomposition. Moreover, in Section 6, we present several equivalent conditions for a matrix to be a weak core matrix.

2.   Preliminaries

For convenience, we will use the following notations: $\mathbb{C}^{ \rm{CM}}_{n}$, $\mathbb{C}^{ \rm{EP}}_{n}$, $\mathbb{C}^{ \rm{P}}_{n}$ and $\mathbb{C}^{ \rm{OP}}_{n}$ will denote the subsets of $\mathbb{C}^{n\times n}$ consisting of core matrices, EP matrices, idempotent matrices and Hermitian idempotent matrices, respectively, i.e.,

● $\mathbb{C}^{ \rm{CM}}_{n} = \{A\mid A\in\mathbb{C}^{n\times n}, r(A^{2}) = r(A)\}$;

● $\mathbb{C}^{ \rm{EP}}_{n} = \{A\mid A\in\mathbb{C}^{n\times n}, \mathcal{R}(A) = \mathcal{R}(A^{*})\}$;

● $\mathbb{C}^{ \rm{P}}_{n} = \{A\mid A\in\mathbb{C}^{n\times n}, A^{2} = A\}$;

● $\mathbb{C}^{ \rm{OP}}_{n} = \{A\mid A\in\mathbb{C}^{n\times n}, A^{2} = A = A^{*}\}$.

Before giving characterizations of the WC inverse, we first present the following auxiliary lemmas which will be repeatedly used throughout this paper.

Lemma 2.1. ^[10] Let $A\in\mathbb{C}^{n\times n}_{k}$. Then $A$ can be represented as

where $T\in\mathbb{C}^{t\times t}$ is nonsingular and $t = r(T) = r(A^{k})$, $N$ is nilpotent with index $k$, and $U\in\mathbb{C}^{n\times n}$ is unitary.

Moreover, the representation of $A$ given by $(2.1)$ is unique [10,Theorem 2.4]. In that case, we have that

where $\widetilde{T} = \sum\limits_{j = 0}^{k-1}T^{j}SN^{k-1-j}.$

Lemma 2.2. ^{[1,9,10,11,12]} Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$. Then :

.

where $\widetilde{T} = \sum\limits_{j = 0}^{k-1}T^{j}SN^{k-1-j}$ and $\triangle = [TT^{*}+S(I_{n-t}-N^{†}N)S^{*}]^{-1}$.

$\widetilde{T}$ and $\triangle$ will be often used throughout this paper.

Lemma 2.3. ^[1] Let $A\in\mathbb{C}^{n\times n}_{k}$. Then

(a) $A^{Ⓦ, †} = A^{(2)}_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2}A^{†})};$

(b) $AA^{Ⓦ, †} = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2}A^{†})}$;

(c) $A^{Ⓦ, †}A = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2})}$.

Lemma 2.4. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $C = AA^{Ⓦ}A$. The following conditions hold:

(a) ^[1]

(b) ^[1] $r(A^{Ⓦ, †}) = r(A^{k})$;

(c) ^[1] $CA^{†}C = C$;

(d) ^[9] $A^{Ⓦ}A^{k+1} = A^{k}$;

(e) $C^{k} = A^{k}A^{Ⓦ}A$.

Proof. Item $(e)$ can be directly verified by $(2.1)$, $(2.2)$ and $(2.8)$.

3.   Some characterizations of the WC inverse

Applying existing results for the WC inverse with respect to $\mathcal{R}(X) = \mathcal{R}(A^{k})$ and $\mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{2}A^{†})$, some new results can be obtained for the WC inverse in the next result.

Theorem 3.1. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $C = AA^{Ⓦ}A$. The following statements are equivalent:

(a) $X = A^{Ⓦ, †}$;

(b) $\mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{2}A^{†})$ and $XAA^{*} = A^{Ⓦ}AA^{*}$;

(c) $\mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{2}A^{†})$ and $XA = A^{Ⓦ}A$;

(d) $\mathcal{R}(X) = \mathcal{R}(A^{k})$ and $A^{*}AX = A^{*}CA^{†}$;

(e) $\mathcal{R}(X) = \mathcal{R}(A^{k})$ and $AX = CA^{†}$;

(f) $\mathcal{R}(X) = \mathcal{R}(A^{k})$ and $A^{k}X = C^{k}A^{†}$;

(g) $\mathcal{R}(X) = \mathcal{R}(A^{k})$, $\mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{2}A^{†})$ and $XAA^{Ⓦ} = A^{Ⓦ}$;

(h) $\mathcal{R}(X) = \mathcal{R}(A^{k})$, $\mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{2}A^{†})$ and $XA^{k+1} = A^{k}$.

Proof. $(a)\Rightarrow (b)$. By the definition of $A^{Ⓦ, †}$, we have that $XAA^{*} = A^{D}CA^{*} = A^{D}AA^{Ⓦ}AA^{*} = A^{Ⓦ}AA^{*}$. Hence, by $(a)$ of Lemma 2.3, we now obtain that $(b)$ holds.

$(b)\Rightarrow (c)$. Postmultiplying $XAA^{*} = A^{Ⓦ}AA^{*}$ by $(A^{†})^{*}$, we obtain that $XA = A^{Ⓦ}A$.

$(c)\Rightarrow (d)$. From $\mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{2}A^{†})$, we have that $\mathcal{N}(AA^{†})\subseteq\mathcal{N}((A^{k})^{*}A^{2}A^{†}) = \mathcal{N}(X)$, which leads to $X = XAA^{†}$. Thus we get that $X = XAA^{†} = A^{Ⓦ}AA^{†} = A^{Ⓦ, †}$ by $XA = A^{Ⓦ}A$. Hence, by the definition of $A^{Ⓦ, †}$ and $(a)$ of Lemma 2.3, we have that $(d)$ holds.

$(d)\Rightarrow (e)$. Evidently.

$(e)\Rightarrow (f)$. Since $C = AA^{Ⓦ}A$ and $C^{k} = A^{k}A^{Ⓦ}A$, premultiplying $AX = CA^{†}$ by $A^{k-1}$, we have that $A^{k}X = C^{k}A^{†}$.

$(f)\Rightarrow (g)$. From $(2.2)$ and $\mathcal{R}(X) = \mathcal{R}(A^{k})$, we can set ${X = U \left[\begin{array}{cc} X_{1} & X_{2} \\ 0 & 0 \\ \end{array} \right] U^{*}},$ where $X_{1}\in\mathbb{C}^{t\times t}$, $X_{2}\in\mathbb{C}^{t\times (n-t)}$ and $t = r(A^{k})$. Furthermore, it follows from $A^{k}X = C^{k}A^{†}$ and $(2.9)$ that $X = A^{Ⓦ, †}$. Therefore, by the definition of $A^{Ⓦ, †}$ and $(a)$ of Lemma 2.3, we obtain that $(g)$ holds.

$(g)\Rightarrow (h)$. It follows from $A^{Ⓦ}A^{k+1} = A^{k}$ and $XAA^{Ⓦ} = A^{Ⓦ}$ that $XA^{k+1} = XAA^{Ⓦ}A^{k+1} = A^{Ⓦ}A^{k+1} = A^{k}$.

$(h)\Rightarrow (a)$. By $\mathcal{R}(X) = \mathcal{R}(A^{k})$ and $XA^{k+1} = A^{k}$, we get that $XAX = X$. Hence, by $(a)$ of Lemma 2.3, we get that $X = A^{Ⓦ, †}$.

Now we will consider other characterizations of the WC inverse by the fact that $A^{Ⓦ, †}AA^{Ⓦ, †} = A^{Ⓦ, †}$.

Theorem 3.2. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $C = AA^{Ⓦ}A$. The following statements are equivalent:

(a) $X = A^{Ⓦ, †}$;

(b) $XAX = X$, $\mathcal{R}(X) = \mathcal{R}(A^{k})$ and $\mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{2}A^{†})$;

(c) $XAX = X$, $\mathcal{R}(X) = \mathcal{R}(A^{k})$ and $AX = CA^{†}$;

(d) $XAX = X$, $AX = CA^{†}$ and $XA^{k} = A^{Ⓦ}A^{k}$;

(e) $XAX = X$, $XA = A^{Ⓦ}A$ and $A^{k}X = C^{k}A^{†}$;

(f) $XAX = X$, $XA = A^{Ⓦ}A$ and $\mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{2}A^{†})$.

Proof. $(a)\Rightarrow (b)$. The proof can be demonstrated by $(a)$ of Lemma 2.3.

$(b)\Rightarrow (c)$. By the definition of $A^{Ⓦ, †}$ and $(b)$ of Lemma 2.3, we get that $AX\in\mathbb{C}^{\mathrm{P}}_{n}$, $\mathcal{R}(AX) = A\mathcal{R}(X) = \mathcal{R}(A^{k+1}) = \mathcal{R}(A^{k}) = \mathcal{R}(AA^{Ⓦ, †}) = \mathcal{R}(CA^{†})$ and $\mathcal{N}(AX) = \mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{2}A^{†}) = \mathcal{N}(AA^{Ⓦ, †}) = \mathcal{N}(CA^{†})$. On the other hand, Lemma 2.4 $(c)$ implies $CA^{†}\in\mathbb{C}^{\mathrm{P}}_{n}$, hence $AX = CA^{†}$.

$(c)\Rightarrow (d)$. By item $(c)$ of Lemma 2.3, we obtain that $\mathcal{R}(X) = \mathcal{R}(A^{k}) = \mathcal{R}(A^{Ⓦ, †}A)$. So we get that $A^{Ⓦ, †}AX = X$, which implies that $XA^{k} = A^{Ⓦ, †}AXA^{k} = A^{Ⓦ, †}CA^{†}A^{k} = A^{Ⓦ, †}AA^{Ⓦ}AA^{†}A^{k} = A^{Ⓦ}A^{k}$.

$(d)\Rightarrow (e)$. By conditions and $AA^{Ⓦ} = A^{k}(A^{Ⓦ})^{k}$, we can infer that $X = XCA^{†} = XAA^{Ⓦ}AA^{†} = XA^{k}(A^{Ⓦ})^{k}AA^{†} = A^{Ⓦ}A^{k}(A^{Ⓦ})^{k}AA^{†} = A^{Ⓦ, †}$. Hence, by $A^{Ⓦ, †} = A^{Ⓦ}AA^{†}$ and $A^{k}A^{Ⓦ}A = C^{k}$, we obtain that $(e)$ holds.

$(e)\Rightarrow (f)$. Since $XAX = X$, $XA = A^{Ⓦ}A$, we have that $\mathcal{R}(X) = \mathcal{R}(XA) = \mathcal{R}(A^{Ⓦ}A) = \mathcal{R}(A^{k})$. We now obtain that $X = A^{Ⓦ, †}$ by $(f)$ of Theorem 3.1. Hence $(f)$ holds by $(a)$ of Lemma 2.3.

$(f)\Rightarrow (a)$. It follows from $XAX = X$ that $\mathcal{N}(AX) = \mathcal{N}(X)$, by conditions and $(a)$ of Lemma 2.3. We now obtain that $X = XAX = A^{Ⓦ}AX = A^{Ⓦ}AA^{†}AX = A^{Ⓦ, †} P_{\mathcal{R}(AX), \ \mathcal{N}(AX)} = A^{Ⓦ, †}$.

Notice the fact that $XA^{k+1} = A^{k}$ if $X = A^{Ⓦ, †}$. Therefore, we will characterize the WC inverse in terms of $A^{Ⓦ, †}A^{k+1} = A^{k}$.

Theorem 3.3. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $C = AA^{Ⓦ}A$. The following statements are equivalent:

(a) $X = A^{Ⓦ, †}$;

(b) $XA^{k+1} = A^{k}$, $A^{*}AX = A^{*}CA^{†}$ and $r(X) = r(A^{k})$;

(c) $XA^{k+1} = A^{k}$, $AX = CA^{†}$ and $r(X) = r(A^{k})$;

(d) $XA^{k+1} = A^{k}$, $A^{k}X = C^{k}A^{†}$ and $r(X) = r(A^{k})$.

Proof. $(a)\Rightarrow (b)$. Since $A^{Ⓦ, †} = A^{Ⓦ}AA^{†}$, we can show that $XA^{k+1} = A^{k}$, $A^{*}AX = A^{*}CA^{†}$. Then, by $(b)$ of Lemma 2.4, we get that $(b)$ holds.

$(b)\Rightarrow (c)$. Obviously.

$(c)\Rightarrow (d)$. Premultiplying $AX = CA^{†}$ by $A^{k-1}$, we have that $A^{k}X = C^{k}A^{†}$ from $A^{k}A^{Ⓦ}A = C^{k}$.

$(d)\Rightarrow (a)$. It follows from $XA^{k+1} = A^{k}$ and $r(X) = r(A^{k})$ that $\mathcal{R}(X) = \mathcal{R}(A^{k})$. Hence, we obtain that $X = A^{Ⓦ, †}$ from $(f)$ of Theorem 3.1.

In the following example, we show that the condition $r(X) = r(A^{k})$ in Theorem 3.3 is necessary.

Example 3.4. Let

Then ${\rm Ind}(A) = 2$,

It can be directly verified that $XA^{3} = A^{2}$, $A^{*}AX = A^{*}CA^{†}$ and $r(X)\neq r(A^{2})$, but $X \neq A^{Ⓦ, †}$. The other cases follow similarly.

By Lemma 2.3, it is clear that $AX = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2}A^{†})}$ and $XA = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2})}$ if $X = A^{Ⓦ, †}$. However, the converse is invalid as shown in the next example:

Example 3.5. Let $A$, $X$ be the same as in Example 3.4. Then

It can be directly verified that $AX = P_{\mathcal{R}(A^{2}), \ \mathcal{N}((A^{2})^{*}A^{2}A^{†})}$ and $XA = P_{\mathcal{R}(A^{2}), \ \mathcal{N}((A^{2})^{*}A^{2})}$, but $X \neq A^{Ⓦ, †}$.

In the next result, we will present some new equivalent conditions for the converse implication:

Theorem 3.6. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $X\in\mathbb{C}^{n\times n}$. The following statements are equivalent:

(a) $X = A^{Ⓦ, †}$;

(b) $AX = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2}A^{†})}, \; XA = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2})}$ and $r(X) = r(A^{k})$;

(c) $AX = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2}A^{†})}, \; XA = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2})}$ and $XAX = X$;

(d) $AX = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2}A^{†})}, \; XA = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2})}$ and $AX^{2} = X$.

Proof. $(a)\Rightarrow (b)$. The proof can be demonstrated by $(b)$ and $(c)$ of Lemma 2.3 and $(b)$ of Lemma 2.4.

$(b)\Rightarrow (c)$. By $\mathcal{R}(XA) = \mathcal{R}(A^{k})$ and $r(X) = r(A^{k})$, we obtain that $\mathcal{R}(X) = \mathcal{R}(XA) = \mathcal{R}(A^{k})$, hence we further derive that $XAX = X$.

$(c)\Rightarrow (d)$. By conditions and $(a)$ of Lemma 2.3, we have that $X = A^{Ⓦ, †}$. Therefore, by $(2.9)$, it can be directly verified that $AX^{2} = X$.

$(d)\Rightarrow (a)$. From $AX^{2} = X$, we have that $X = AX^{2} = A^{2}X^{3} = \dots = A^{k}X^{k+1}$, which implies $\mathcal{R}(X)\subseteq\mathcal{R}(A^{k})$. Combining with the condition $\mathcal{R}(A^{k}) = \mathcal{R}(XA)\subseteq\mathcal{R}(X)$, we get that $\mathcal{R}(X) = \mathcal{R}(A^{k})$. From $(2.2)$, we now set ${X = U \left[\begin{array}{cc} X_{1} & X_{2} \\ 0 & 0 \\ \end{array} \right] U^{*}},$ where $X_{1}\in\mathbb{C}^{t\times t}$, $X_{2}\in\mathbb{C}^{t\times (n-t)}$ and $t = r(A^{k})$. On the other hand, it follows from $\mathcal{N}(AX) = \mathcal{N}((A^{k})^{*}A^{2}A^{†})$ that $(A^{k})^{*}A^{2}A^{†} = (A^{k})^{*}A^{2}X$, which yields $X_{1} = T^{-1}$ and $X_{2} = T^{-2}SNN^{†}$. Therefore, by $(2.9)$, we obtain that $X = A^{Ⓦ, †}$.

In ^[1], the authors introduced the definition of $A^{Ⓦ, †}$ with an algebraic approach. In the next result, we will consider characterization of $A^{Ⓦ, †}$ with a geometrical point of view.

Theorem 3.7. Let $A\in\mathbb{C}^{n\times n}_{k}$. Then:

(a) $A^{Ⓦ, †}$ is the unique matrix $X$ that satisfies:

(b) $A^{Ⓦ, †}$ is the unique matrix $X$ that satisfies:

Proof. $(a)$. Since $\mathcal{R}(A^{D}) = \mathcal{R}(A^{k})$, it is a consequence of [2,Corollary 3.2] by properities of Drazin and MP inverse.

$(b)$. Since items $(c)$ of Lemma 2.3, $A^{Ⓦ, †}$ satisfies $XA = P_{\mathcal{R}(A^{k}), \ \mathcal{N}((A^{k})^{*}A^{2})}.$ Additionally, we derive that $\mathcal{N}(A^{*}) = \mathcal{N}(A^{†})\subseteq\mathcal{N}(A^{Ⓦ}AA^{†}) = \mathcal{N}(X).$ Now it remains to prove that $X$ is unique.

Assume that $X_{1}$, $X_{2}$ satisfy $(3.2)$, then $X_{1}A = X_{2}A$, \ $\mathcal{N}(A^{*})\subseteq\mathcal{N}(X_{1})$ and $\mathcal{N}(A^{*})\subseteq\mathcal{N}(X_{2})$. Furthermore, we get that $(X_{1}-X_{2})A = 0$ and $\mathcal{R}(X_{i}^{*})\subseteq\mathcal{R}(A)$ for $i = 1, 2$, which further imply that $A^{*}(X_{1}^{*}-X_{2}^{*}) = 0$ and $\mathcal{R}(X_{1}^{*}-X_{2}^{*})\subseteq\mathcal{R}(A)$. Therefore we have that $\mathcal{R}(X_{1}^{*}-X_{2}^{*})\subseteq\mathcal{R}({A})\cap\mathcal{N}(A^{*}) = \mathcal{R}({A})\cap\mathcal{R}(A)^{\perp} = \{0\}$. Thus, $X_{1}^{*} = X_{2}^{*}$, i.e., $X_{1} = X_{2}$.

Remark 3.8. In Theorem $3.7$, $\mathcal{R}(X)\subseteq\mathcal{R}(A^{k})$ in $(3.1)$ can be replaced by $\mathcal{R}(X) = \mathcal{R}(A^{k})$. However, if we replace $\mathcal{N}(A^{*})\subseteq\mathcal{N}(X)$ with $\mathcal{N}(A^{*}) = \mathcal{N}(X)$ in $(3.2)$, item $(b)$ of Theorem $3.7$ does not hold.

Characterizations of some generalized inverses by using its block matrices have been investigated in ^{[13,14,15,16,17]}. In [18,Theorem 3.2], the authors presented a characterization for the WC inverse using its block matrices. Next we will give another proof of it by using characterization of projection operator.

Theorem 3.9. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $r(A^{k}) = t$. Then there exist a unique matrix $P$ such that

a unique matrix $Q$ such that

and a unique matrix $X$ such that

Furthermore, $X$ is the WC inverse $A^{Ⓦ, †}$ of $A$ and

Proof. It is not difficult to prove that

Similarly, we can show that $(3.4)$ have the unique solution $Q = P_{\mathcal{N}((A^{k})^{*}A^{2}A^{†}), \ \mathcal{R}(A^{k})}$.

Furthermore, comparing $(3.6)$ and items $(b)$ and $(c)$ of Lemma 2.3 immediately leads to the conclusion that

By $(3.5)$, we obtain that $X = A^{Ⓦ, †}$.

4.   Some representations of the WC inverse

In ^[19], Drazin introduced the $(b, c)$-inverse in semigroup. In ^[20], Benítez et al. investigated the $(B, C)$-inverse of $A\in\mathbb{C}^{m\times n}$, as the unique matrix $X\in\mathbb{C}^{n\times m}$ satisfying ^[20]:

where $B, C\in\mathbb{C}^{n\times m}$. In the next result, we will show that the WC inverse is a special $(B, C)$-inverse.

Theorem 4.1. Let $A\in\mathbb{C}^{n\times n}_{k}$. Then

Proof. According to Lemma 2.3, we get that

Observe that $A^{Ⓦ, †}AA^{k} = A^{Ⓦ}A^{k+1} = A^{k}$ and $(A^{k})^{*}A^{2}A^{†}AA^{Ⓦ, †} = (A^{k})^{*}A^{2}A^{Ⓦ}AA^{†} = (A^{k})^{*}A^{2}A^{†}.$ Thus, we obtain $A^{Ⓦ, †} = A^{(A^{k}, (A^{k})^{*}A^{2}A^{†})}.$

In ^[21], the authors introduced the Bott-Duffin inverse of $A\in\mathbb{C}^{n\times n}$ when $AP_{L}+P_{L^{\perp}}$ is nonsingular, i.e., $A^{(-1)}_{L} = P_{L}(AP_{L}+P_{L^{\perp}})^{-1} = P_{L}(AP_{L}+I-P_{L})^{-1}$. In ^[22], the authors showed the weak group inverse by a special Bott-Duffin inverse. Next we will show that the WC inverse of $A$ is indeed the Bott-Duffin inverse of $A^{2}$ with respect to $\mathcal{R}(A^{k})$.

Theorem 4.2. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$. Then

Proof. It follows from $(2.3)$ and $(2.2)$ that

We now obtain that

Similarly, by a direct calculation, we can derive that $A^{Ⓦ, †} = (P_{A^{k}}A^{2}P_{A^{k}})^{†}AP_{A}.$

Working with the fact that $P = P_{\mathcal{N}((A^{k})^{*}A^{2}), \ \mathcal{R}(A^{k})}$ and $Q = P_{\mathcal{N}((A^{k})^{*}A^{2}A^{†}), \ \mathcal{R}(A^{k})}$ in Theorem 3.9, we will consider other representations of $A^{Ⓦ, †}$ in the next theorem.

Theorem 4.3. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $P = P_{\mathcal{N}((A^{k})^{*}A^{2}), \ \mathcal{R}(A^{k})}, \ \ Q = P_{\mathcal{N}((A^{k})^{*}A^{2}A^{†}), \ \mathcal{R}(A^{k})}.$ Then for any $a, b\neq 0$, we have

Proof. From items $(b)$ and $(c)$ of Lemma 2.3, it is not difficult to conclude that

Now we only need to show the invertibility of $A+aP$. Assume that $\alpha = U\left[\begin{array}{cc|c} \alpha_{1} \\ \alpha_{2}\\ \end{array} \right]\in\mathbb{C}^{n}$ such that $(A+aP)\alpha = 0$, i.e., $A\alpha = -aP\alpha$, where $\alpha_{1}\in\mathbb{C}^{p}$, $\alpha_{2}\in\mathbb{C}^{n-p}$. Now it follows from condition $(c)$ of Lemma 2.3 and $(3.6)$ that

implying $\alpha_{1} = 0$ and $\alpha_{2} = 0$ since $a\neq 0$, $N$ is nilpotent and $T$ is non singular. Thus $A+aP$ is nonsingular.

Analogously, we can prove that $A+bQ$ is invertible and $A^{Ⓦ, †} = (I-P)(A+bQ)^{-1}.$

The limit expressions for some generalized inverses of matrices have been given in ^{[14,15,16,17,23,24]}. Similarly, the WC inverse can also be characterized as limit value as shown in the next result:

Theorem 4.4. Let $A\in\mathbb{C}^{n\times n}_{k}$. Then:

(a) $A^{Ⓦ, †} = \begin{matrix}\lim_{\lambda \to 0}A^{k}(\lambda I_{n}+(A^{k})^{*}A^{k+2})^{-1}(A^{k})^{*}A^{2}A^{*}(\lambda I_{n}+AA^{*})^{-1} \end{matrix}$;

(b) $A^{Ⓦ, †} = \begin{matrix}\lim_{\lambda \to 0}A^{k}(A^{k})^{*}A(\lambda I_{n}+A^{k+1}(A^{k})^{*}A)^{-1}AA^{*}(\lambda I_{n}+AA^{*})^{-1} \end{matrix}$;

(c) $A^{Ⓦ, †} = \begin{matrix}\lim_{\lambda \to 0}(\lambda I_{n}+A^{k}(A^{k})^{*}A^{2})^{-1}A^{k}(A^{k})^{*}A^{2}A^{*}(\lambda I_{n}+AA^{*})^{-1} \end{matrix}$;

(d) $A^{Ⓦ, †} = \begin{matrix}\lim_{\lambda \to 0}A^{k}(A^{k})^{*}A^{2}A^{*}(\lambda I_{n}+AA^{*})^{-1}(\lambda I_{n}+A^{k+1}(A^{k})^{*}A^{2}A^{*}(\lambda I_{n}+AA^{*})^{-1})^{-1}. \end{matrix}$

Proof. According to condition $(a)$ of Lemma 2.3, it is not hard to show that

Thus, by [25,Theorem 2.1], we have the following results:

$(a)$ Let $X = A^{k}$, $Y = (A^{k})^{*}A^{2}A^{†}$ and by $A^{†} = \begin{matrix}\lim_{\lambda \to 0}A^{*}(\lambda I_{n}+AA^{*})^{-1} \end{matrix}$. We have

$(b)$ Let $X = A^{k}(A^{k})^{*}A$, $Y = AA^{†}$ and by $A^{†} = \begin{matrix}\lim_{\lambda \to 0}A^{*}(\lambda I_{n}+AA^{*})^{-1} \end{matrix}$. We have

$(c)$ Let $X = I_{n}$, $Y = A^{k}(A^{k})^{*}A^{2}A^{†}$ and by $A^{†} = \begin{matrix}\lim_{\lambda \to 0}A^{*}(\lambda I_{n}+AA^{*})^{-1}. \end{matrix}$ We have

$(d)$ Let $X = A^{k}(A^{k})^{*}A^{2}A^{†}$, $Y = I_{n}$ and by $A^{†} = \begin{matrix}\lim_{\lambda \to 0}A^{*}(\lambda I_{n}+AA^{*})^{-1} \end{matrix}$. We have

We end up this section with three examples of computing the WC inverse of a matrix using three different expressions in Theorems 4.2–4.4.

Example 4.5. Let

Then ${\rm Ind}(A) = 2$ and the weak core inverse of $A$ is

Firstly, using the expression $(4.1)$ to compute the WC inverse of $A$. Then

After simplification, it follows that $(A^{2})^{(-1)}_{(\mathcal{R}(A^{2}))}AP_{A} = A^{Ⓦ, †}$ and $(P_{A^{2}}A^{2}P_{A^{2}})^{†}AP_{A} = A^{Ⓦ, †}.$

Secondly, using the expression $(4.4)$, we obtain

Therefore, it can be directly verified $(A-6P)^{-1}(I-Q) = A^{Ⓦ, †}$, $(I-P)(A+\frac{1}{5}Q)^{-1} = A^{Ⓦ, †}$.

Finally, using the limit expressions of item $(a)$ in Theorem 4.4.

Let $B = A^{2}(\lambda I_{n}+(A^{2})^{*}A^{4})^{-1}(A^{2})^{*}A^{2}A^{*}(\lambda I_{n}+AA^{*})^{-1}$, then

where $\lambda_{1} = \lambda^{4}+38734\lambda^{3}+1470569\lambda^{2}+197888\lambda+580$, $\lambda_{2} = \lambda^{3}+38697\lambda^{2}+38812\lambda+116$.

After simplification, it follows that

The other cases in Theorem 4.4 can be similarly verified.

5.   Some properties of the WC inverse

In this section, we discuss some properties of the WC inverse and consider the connection between the WC inverse and other known classes of matrices.

Lemma 5.1. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$. Then:

(a) $A\in\mathbb{C}^{\mathrm{EP}}_{n}\Leftrightarrow S = 0$ and $N = 0$;

(b) $A\in\mathbb{C}^{\mathrm{P}}_{n}\Leftrightarrow T = I_{t}$ and $N = 0$;

(c) $A\in\mathbb{C}^{\mathrm{OP}}_{n}\Leftrightarrow T = I_{t}, \ \ S = 0$ and $N = 0$.

Proof. $(a)$ The proof can be easily verified from $(2.1)$ and $(2.3)$.

(b) By $(2.1)$, we obtain that $A\in\mathbb{C}^{\mathrm{P}}_{n}$ is equivalent with

which is further equivalent with $T^{2} = T$, $TS+SN = S$ and $N^{2} = N$. Hence, by non singularity of $T$ and $N^{k} = 0$, we can conclude that $A\in\mathbb{C}^{\mathrm{P}}_{n}$ if and only if $T = I_{t}$ and $N = 0$.

$(c)$ Since $\mathbb{C}^{\rm OP}_{n}\subseteq\mathbb{C}^{\rm P}_{n}$, it is a direct consequence of item $(b)$ and $(2.1)$.

Theorem 5.2. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$. The following statements hold:

(a) $A^{Ⓦ, †} = 0\Leftrightarrow A$ is nilpotent $;

(b) $A^{Ⓦ,†} = A\Leftrightarrow A\in\mathbb{C}^{\mathrm{EP}}_{n}$ and $A^{3} = A$;

(c) $A^{Ⓦ,†} = A^{*}\Leftrightarrow A\in\mathbb{C}^{\mathrm{EP}}_{n}$ and $AA^{*} = P_{A^{k}}$;

(d) $A^{Ⓦ,†} = P_{A}\Leftrightarrow A\in\mathbb{C}^{\mathrm{P}}_{n}$;

(e) $A^{Ⓦ,†} = P_{A^{*}}\Leftrightarrow A\in\mathbb{C}^{\mathrm{OP}}_{n}$.

Proof. $(a)$ By $(2.1)$ and $(2.9)$, we directly get that

$(b)$ It follows $(2.1)$, $(2.9)$ and $(a)$ of lemma 5.1 that

$(c)$ By $(2.1)$, $(2.9)$ and $(a)$ of lemma 5.1, we have that

$(d)$ From $(2.9)$, $(4.2)$ and $(b)$ of lemma 5.1, we obtain that

$(e)$ It follows from $(2.1)$ and $(2.3)$ that

By $(2.9)$ and $(5.1)$, we now get that $A^{Ⓦ, †} = P_{A^{*}}$ is equivalent with

which is further equivalent with $T^{-1} = T^{*}\triangle T$, $(I_{n-t}-N^{†}N)S^{*}\triangle T = 0$ and $N^{†}N+(I_{n-t}-N^{†}N)S^{*}\triangle S(I_{n-t}-N^{†}N) = 0$. Hence, by nonsingularity of $\triangle T$ and $(c)$ of lemma 5.1, we can conclude that $A^{Ⓦ, †} = P_{A^{*}}$ if and only if $A\in\mathbb{C}^{\mathrm{OP}}_{n}$.

From Lemma 2.3, we know that both $AA^{Ⓦ, †}$ and $A^{Ⓦ, †}A$ are oblique projectors. The next theorem will further discuss other characteriations for $AA^{Ⓦ, †}$ and $A^{Ⓦ, †}A$.

Theorem 5.3. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$. The following statements hold:

(a) $AA^{Ⓦ, †} = P_{A}\Leftrightarrow A\in\mathbb{C}^{\mathrm{CM}}_{n}$;       (b) $AA^{Ⓦ, †} = P_{A^{*}}\Leftrightarrow A\in\mathbb{C}^{\mathrm{EP}}_{n}$;

(c) $A^{Ⓦ, †}A = P_{A}\Leftrightarrow A\in\mathbb{C}^{\mathrm{EP}}_{n}$;       (d) $A^{Ⓦ, †}A = P_{A^{*}}\Leftrightarrow A\in\mathbb{C}^{\mathrm{EP}}_{n}$.

Proof. It follows from $(2.1)$ and $(2.9)$ that

$(a)$ By $(4.2)$ and $(5.2)$, the result can be directly verified.

$(b)$ By $(5.1)$ and $(5.2)$, we can show that $AA^{Ⓦ, †} = P_{A^{*}}$ if and only if $(I_{n-t}-N^{†}N)S^{*}\triangle T = 0$ and $N^{†}N+(I_{n-t}-N^{†}N)S^{*}\triangle S(I_{n-t}-N^{†}N) = 0$, which is further equivalent with $S = 0$ and $N = 0$, i.e., $A\in\mathbb{C}^{\mathrm{EP}}_{n}$.

$(c)$ It follows from $(4.2)$ and $(5.3)$ that $A^{Ⓦ, †}A = P_{A}$ is equivalent with $A\in\mathbb{C}^{\mathrm{EP}}_{n}$.

$(d)$ From $(5.1)$ and $(5.3)$, it is similar to the proof of $(b)$.

Recall from ^[6] that the core inverse is necessarily EP. The next Theorem shows that this is not the case with the WC inverse.

Theorem 5.4. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$ and $t\in\mathbb{Z^{+}}$. The following statements are equivalent:

(a) $A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{EP}}_{n}$;       (b) $SN = 0$;

(c) ;       (d) ;

(e) $A^{Ⓦ, †}A^{t} = A^{t}A^{Ⓦ}$.

Proof. $(a)\Leftrightarrow(b)$. Since $A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{EP}}_{n}$ is equivalent with $\mathcal{R}(A^{Ⓦ, †}) = \mathcal{R}((A^{Ⓦ, †})^{*})$. Using $(2.9)$, we have that $A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{EP}}_{n}$ if and only if $SN = 0$.

$(c)\Leftrightarrow(b)$. By $(2.5)$ and $(5.3)$, it can be directly verified that if and only if $SN = 0$.

$(d)\Leftrightarrow(b)$. By $(2.5)$ and $(2.9)$, it follows that

$(e)\Leftrightarrow(b)$. From $(2.8)$ and $(2.9)$, it follows that

where $T_{t} = \sum\limits_{j = 0}^{t-1}T^{j}SN^{t-1-j}.$

In ^[26], the authors introduced that a matrix $A$ to be a weak group matrix if $A\in\mathbb{C}^{\mathrm{WG}}_{n}$, which is equivalent with $SN = 0$. Therefore, we have that following remark:

Remark 5.5. It is worth noting that conditions $(a)$, $(c)$–$(e)$ in Theorem 5.4 are equivalent with $A\in\mathbb{C}^{\mathrm{WG}}_{n}$.

The next theorems provide some equivalent conditions for $A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{P}}_{n}$ and $A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{OP}}_{n}$.

Theorem 5.6. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$. The following statements are equivalent:

(a) $A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{P}}_{n}$;            (b) $T = I_{t}$;

(c) $AA^{Ⓦ, †} = A^{Ⓦ, †}$;           (d) $A^{Ⓦ, †}A^{k} = A^{k}$;

(e) $A^{k}(A^{Ⓦ, †})^{k} = A^{Ⓦ, †}$;         (f) $A(A^{Ⓦ, †})^{k} = (A^{Ⓦ, †})^{k}$.

Proof. $(a)\Leftrightarrow(b)$. From $(2.9)$, it is not hard to prove that $A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{P}}_{n}$ is equivalent with $T = I_{t}$.

$(c)\Leftrightarrow(b)$. From $(2.9)$ and $(5.3)$, it follows that $AA^{Ⓦ, †} = A^{Ⓦ, †}$ if and only if $T = I_{t}$.

$(d)\Leftrightarrow(b)$. By $(2.2)$ and $(2.9)$, it is easy to verify that $A^{Ⓦ, †}A^{k} = A^{k}$ if and only if $T = I_{t}$.

The proofs $(e)\Leftrightarrow(b)$ and $(f)\Leftrightarrow(b)$ are similar to the proof $(d)\Leftrightarrow(b)$.

Theorem 5.7. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$. The following statements are equivalent:

(a) $A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{OP}}_{n}$;            (b) $T = I_{t}$ and $SN = 0$;

(c) $AA^{Ⓦ, †} = (A^{Ⓦ, †})^{*}$;         (d) $A^{Ⓦ, †}A = A^{Ⓦ}$;

(e) $(A^{Ⓦ, †})^{k}A^{k} = A^{Ⓦ}$;         (f) $(A^{Ⓦ, †})^{k}A = (A^{Ⓦ})^{k}$.

Proof. $(a)\Leftrightarrow(b)$. From $(2.9)$ and Theorem 5.6, we can show that $A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{OP}}_{n}$ is equivalent with $T = I_{t}$ and $SN = 0$.

$(c)\Leftrightarrow(b)$. By $(2.9)$ and $(5.3)$, it follows from $AA^{Ⓦ, †} = (A^{Ⓦ, †})^{*}$ that

Hence, we get that $AA^{Ⓦ, †} = (A^{Ⓦ, †})^{*}$ is equivalent with $T = I_{t}$ and $SN = 0$.

The proofs of $(d)\Leftrightarrow(b)$, $(e)\Leftrightarrow(b)$ and $(f)\Leftrightarrow(b)$ are similar to the proof of $(c)\Leftrightarrow(b)$.

Corollary 5.8. Let $A\in\mathbb{C}^{n\times n}_{k}$. Then $A\in\mathbb{C}^{\mathrm{OP}}_{n}$ if and only if $A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{P}}_{n}\cap\mathbb{C}^{\mathrm{EP}}_{n}$.

Proof. It is a direct consequence from Theorem 5.4 and Theorem 5.6.

Working with Theorem 5.6 and Theorem 5.7, we have the following corollary.

Corollary 5.9. Let $A\in\mathbb{C}^{n\times n}_{k}$ and for any $l\in\mathbb{N}$, $l\geq k$. The following statements statements hold:

$(a) \; A\in\mathbb{C}^{\mathrm{P}}_{n}\Leftrightarrow A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{P}}_{n}$ and $A^{l} = A$;

$(b) \; A\in\mathbb{C}^{\mathrm{OP}}_{n}\Leftrightarrow A^{Ⓦ, †}\in\mathbb{C}^{\mathrm{P}}_{n}$ and $A^{l} = A^{*}$.

Proof. $(a)$ The result can be easily derived by lemma 5.1 and Theorem 5.6,

$(b)$ From Lemma 5.1 and Theorem 5.7, we can show that $(b)$ holds.

6.   Different characterizations of weak core matrix

Ferreyra et al.^[1] introduced the weak core matrix. The set of all $n\times n$ weak core matrices is denoted by $\mathbb{C}^{ \rm{WC}}_{n}$, that is:

In this section, we discuss some equivalent conditions satisfied by a matrix $A$ such that $A\in\mathbb{C}^{WC}_{n}$ using the core-EP decomposition. For convenience, we introduce a necessary lemma.

Lemma 6.1. ^[1] Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$. Then the following statements are equivalent:

(a) $A\in\mathbb{C}^{\mathrm{WC}}_{n}$;

(b) $SN^{2} = 0$;

(c) $A^{Ⓦ} = A^{D}$.

Theorem 6.2. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$ and $t\in\mathbb{Z^{+}}$. The following statements are equivalent:

(a) $A\in\mathbb{C}^{\mathrm{WC}}_{n}$;

(b) $A^{Ⓦ}A = A^{D}A$;

(c) $A^{t}A^{Ⓦ}A = A^{t}A^{D}A$;

(d) $A^{t}A^{Ⓦ, †} = A^{t}A^{D, †}$;

(e) $A^{k}A^{Ⓦ, †} = A^{k}A^{†}$;

(f) $A^{k}A^{Ⓦ}A = A^{k}$.

Proof. $(a)\Rightarrow (b)$. It is a direct consequence from condition $(c)$ of Lemma $6.1$.

$(b)\Rightarrow (c)$. Evident.

$(c)\Rightarrow (a)$. By condition, it follows from $(2.4)$ and $(2.8)$ that

which implies $T^{t-k-1}(T^{k-2}SN^{2}+\dots+TSN^{k-1}) = 0$. We now obtain that $SN^{2} = 0$ since $T$ is invertible. By Lemma $6.1$, we obtain that $A\in\mathbb{C}^{\mathrm{WC}}_{n}$.

$(a)\Leftrightarrow(d)$. It follows form $(2.6)$, $(2.9)$ and Lemma $6.1$ that

$(a)\Rightarrow (e)$. From the definition of the weak core matrix, we have that $A^{k}A^{Ⓦ, †} = A^{k}A^{D, †} = A^{k}A^{†}$.

$(e)\Rightarrow (f)$. Evident.

$(f)\Rightarrow (a)$. If $A^{k}A^{Ⓦ}A = A^{k}$, by $(2.2)$ and $(2.8)$, we can conclude that $SN^{2}$. Hence item $(a)$ holds.

Corollary 6.3. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$. Then $A\in\mathbb{C}^{\mathrm{WC}}_{n}$ if and only if $A^{k} = C^{k}$, where $C = AA^{Ⓦ}A$.

Proof. Since $A^{k}A^{Ⓦ}A = C^{k}$, the result is a direct consequence of item $(f)$ of Theorem $6.2$.

Theorem 6.4. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$ and for some $t\in\mathbb{Z^{+}}$. The following statements are equivalent:

(a) $A\in\mathbb{C}^{\mathrm{WC}}_{n}$;

(b) $A(A^{Ⓦ})^{t}A = (A^{Ⓦ})^{t}A^{2}$;

(c) $(A^{Ⓦ})^{t}A = (A^{Ⓦ})^{t+1}A^{2}$;

(d) $A(A^{Ⓦ})^{t}A$ commutes with $(A^{Ⓦ})^{t}A^{2}$;

(e) $(A^{Ⓦ})^{t}A$ commutes with $(A^{Ⓦ})^{t+1}A^{2}$.

Proof. By $(2.1)$ and $(2.8)$, we get that

$(a)\Leftrightarrow(b)$. By $(6.1)$, $(6.2)$ and Lemma $6.1$, we get that $A(A^{Ⓦ})^{t}A = (A^{Ⓦ})^{t}A^{2}$ if and only if $A\in\mathbb{C}^{\mathrm{WC}}_{n}$.

$(a)\Leftrightarrow(c)$. Similar to the part $(a)\Leftrightarrow(b)$.

$(a)\Leftrightarrow(d)$. It follows from $(6.1)$, $(6.2)$ and Lemma $6.1$ that

which implies that $A(A^{Ⓦ})^{t}A$ commutes with $(A^{Ⓦ})^{t}A^{2}$ if and only if $A\in\mathbb{C}^{\mathrm{WC}}_{n}$.

$(a)\Leftrightarrow(e)$. It is analogous to that of the part $(a)\Leftrightarrow(d)$.

Corollary 6.5. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $t\in\mathbb{Z^{+}}$. The following statements are equivalent:

(a) $A\in\mathbb{C}^{\mathrm{WC}}_{n}$;

Proof. Since , Corollary $6.5$ can be directly verified.

7.   Conclusions

In this paper, new characterizations and properties of the WC inverse are derived by using range, null space, matrix equations, respectively. Several expressions of the WC inverse are also given. Finally, we show various characterizations of the weak core matrix.

According to the current research background, more characterizations and applications for the WC inverse are worthy of further discussion which as follows:

1) Characterizing the WC inverse by maximal classes of matrices, full rank decomposition, integral expressions and so on;

2) New iterative algorithms and splitting methods for computing the WC inverse;

3) Using the WC inverse to solve appropriately constrained systems of linear equations;

4) Investigating the WC inverse of tensors.

Acknowledgements

This work was supported by the Natural Science Foundation of China under Grants 11961076. The authors are thankful to two anonymous referees for their careful reading, detailed corrections and pertinent suggestions on the first version of the paper, which enhanced the presentation of the results distinctly.

Conflict of interest

All authors read and approved the final manuscript. The authors declare no conflict of interest.