Further characterizations of the $m$-weak group inverse of a complex matrix

1.   Introduction

Let $\mathbb{C}^{m\times n}$ and $\mathbb{Z}^{+}$ denote the set of all $m\times n$ complex matrices and the set of all positive integers, respectively. The symbols $r(A)$ and ${{\rm{ Ind}}}(A)$ stand for the rank and the index of $A\in\mathbb{C}^{n\times n}$, respectively. For a matrix $A\in\mathbb{C}^{n\times n}$, we assume that $A^{0} = I_{n}$. Let $\mathbb{C}^{n\times n}_{k}$ be the set of all $n\times n$ complex matrices with index $k$. By $\mathbb{C}^{\rm CM}_{n}$ we denote the set of all core matrices (or group invertible matrices), i.e.,

The Drazin inverse ^[1] 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:

Especially, when $A\in \mathbb{C}^{\rm CM}_{n}$, then $X$ that satisfies $(1.1)$ is called the group inverse of $A$ and is denoted by $A^{\#}$. The Drazin inverse has been widely applied in different fields of mathematics and its applications. Here we will mention only some of them. The perturbation theory and additive results for the Drazin inverse were investigated in ^[2,3,4,5]. In ^[6], the algorithms for the computation of the Drazin inverse of a polynomial matrix are presented based on the discrete Fourier transformation. Karampetakis and Stanimirovi$\rm \acute{c}$ ^[7] presented two algorithms for symbolic computation of the Drazin inverse of a given square one-variable polynomial matrix, which was effective with respect to CPU time and the elimination of redundant computations. Some representations of the $W$-weighted Drazin inverse were investigated and the computational complexities of the representations were also estimated in ^[8]. Kyrchei ^[9] generalized the weighted Drazin inverse, the weighted DMP-inverse, and the weighted dual DMP-inverse ^[10,11,12] for the matrices over the quaternion skew field and provided their determinantal representations by using noncommutative column and row determinants. In ^[13], the authors considered the quaternion two-sided restricted matrix equations and gave their unique solutions by the DMP-inverse and dual DMP-inverse. For interesting properties of different kinds of generalized inverses see ^[14].

In 2018, Wang ^[15] introduced the weak group inverse of complex square matrices using the core-EP decomposition ^[16] and gave its certain characterizations.

Definition 1.1. Let $A\in\mathbb{C}^{n\times n}_{k}$. Then the unique solution of the system

is the weak group inverse of $A$ denoted by $A^{{Ⓦ}}.$

Recently, there has been a huge interest in the weak group inverse. For example, Wang et al. ^[17] compared the weak group inverse with the group inverse of a matrix. In ^[18], the weak group inverse was introduced in *-rings and characterized by three equations (see also ^[19,20]). The weak group inverse in the setting of rectangular matrices was considered in ^[21]. In 2021, Zhou and Chen ^[19] introduced the $m$-weak group inverse in the ring and presented its different characterizations.

Definition 1.2. Let $R$ be a unitary ring with involution, $a\in R$ and $m\in \mathbb{Z}^{+}$. If there exist $x\in R$ and $k\in \mathbb{Z}^{+}$ such that

then $x$ is called the $m$-weak group inverse of $a$ and in this case, $a$ is $m$-weak group invertible.

In general, the $m$-weak group inverse of $a$ may not be unique. If the $m$-weak group inverse of $a$ is unique, then it is denoted by $a^{{Ⓦ}_{m}}$.

In ^[22], we can find a relation between the weak core inverse and the $m$-weak group inverse as well as certain necessary and sufficient conditions that the Drazin inverse coincides with the $m$-weak group inverse of a complex matrix. It is interesting to note that $X$ which satisfies $(1.1)$ coincides with the $m$-weak group inverse on complex matrices, in which case $X$ exists for every $A\in\mathbb{C}^{n\times n}$ and is unique.

Now, we consider the system of equations

Motivated by the above discussion, we introduce a new characterization of the $m$-weak group inverse related with $(1.2)$ and proved the existence and uniqueness of a solution of $(1.2)$, for every $A\in\mathbb{C}^{n\times n}$. Some new characterizations of the $m$-weak group inverse are derived in terms of the range space, null space, rank equalities, and projectors. We present some representations of the $m$-weak group inverse involving some known generalized inverses and limit expressions as well as certain relations between the $m$-weak group inverse and other generalized inverses. Finally, we consider a relation between the $m$-weak group inverse and the nonsingular bordered matrix, which is applied to the Cramer's rule for the solution of the restricted matrix equation.

The paper is organized as follows: In Section 2, we present some well-known definitions and lemmas. In Section 3, we provide a new characterization, as well as certain representations and properties of the $m$-weak group inverse of a complex matrix. In Section 4, we provide several expressions of the $m$-weak group inverse which are useful in computation. In Section 5, we present some properties of the $m$-weak group inverse as well as the relationships between the $m$-weak group inverse and other generalized inverses by core-EP decomposition. In Section 6, we show the applications of the $m$-weak group inverse concerned with the bordered matrices and the Cramer's rule for the solution of the restricted matrix equation.

2.   Preliminaries

The symbols $\mathcal{R}(A)$, $\mathcal{N}(A)$ and $A^{*}$ denote the range space, null space and conjugate transpose of $A\in\mathbb{C}^{m\times n}$, respectively. The symbol $I_{n}$ denotes the identity matrix of order $n$. Let $P_{\mathcal{L}, \mathcal{M}}$ be the projector on the space $\mathcal{L}$ along the $\mathcal{M}$, where $\mathcal{L}, \mathcal{M}\leq \mathbb{C}^{n}$ and $\mathcal{L}\oplus \mathcal{M} = \mathbb{C}^{n}.$ For $A\in\mathbb{C}^{m\times n}$, $P_{A}$ represents the orthogonal projection onto $\mathcal{R}(A)$, i.e., $P_{A} = P_{\mathcal{R}(A)} = AA^{†}$. The symbols $\mathbb{C}^{\rm P}_{n}$ and $\mathbb{C}^{\rm H}_{n}$ represent the subsets of $\mathbb{C}^{n\times n}$ consisting of all idempotent and Hermitian matrices, respectively, i.e.,

Let $A\in\mathbb{C}^{m\times n}$. The MP-inverse $A^{†}$ of $A$ is the unique matrix $X\in\mathbb{C}^{n\times m}$ satisfying the following four Penrose equations (see ^[14,23,24]):

A matrix $X\in\mathbb{C}^{n\times m}$ that satisfies condition (1) above is called an inner inverse of $A$ and the set of all inner inverses of $A$ is denoted by $A\{1\}$, while a matrix $X\in\mathbb{C}^{n\times m}$ that satisfies condition (2) above is called an outer inverse of $A$. A matrix $X\in\mathbb{C}^{n\times m}$ that satisfies both conditions (1) and (2) is called a reflexive g-inverse of $A$. If a matrix $X\in\mathbb{C}^{n\times m}$ satisfies

where $\mathcal{T}$ and $\mathcal{S}$ are the subspaces of $\mathbb{C}^{n}$ and $\mathbb{C}^{m}$ respectively, then $X$ is an outer inverse of $A$ with prescribed range and null space and it is denoted by $A^{(2)}_{\mathcal{T}, \mathcal{S}}$. If $A^{(2)}_{\mathcal{T}, \mathcal{S}}$ exists, then it is unique. The notion of the core inverse on the $\mathbb{C}^{\rm CM}_{n}$ was proposed and was denoted by ^[25,26,27]. The core inverse of $A\in\mathbb{C}^{n\times n}_{k}$ is the unique matrix $X\in\mathbb{C}^{n\times n}$ satisfying

In addition, it was proved that

The core-EP inverse of $A\in\mathbb{C}^{n\times n}_{k}$, denoted by is given in ^[28,29,30]. The core-EP inverse of $A\in\mathbb{C}^{n\times n}_{k}$ is the unique matrix $X\in\mathbb{C}^{n\times n}$ satisfying

Moreover, it was proved that

The DMP-inverse of $A\in\mathbb{C}^{n\times n}_{k}$, denoted by $A^{D, †}$ was introduced in ^[10,11]. The DMP-inverse of $A\in\mathbb{C}^{n\times n}_{k}$ is the unique matrix $X\in\mathbb{C}^{n\times n}$ satisfying

Moreover, it was shown that

Also, the dual DMP-inverse of $A$ was introduced in ^[10], as $A^{†, D} = A^{†}AA^{D}$.

The $(B, C)$-inverse of $A\in\mathbb{C}^{m\times n}$, denoted by $A^{(B, C)}$ ^[31,32], is the unique matrix $X\in\mathbb{C}^{n\times m}$ satisfying

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

To discuss further properties of the $m$-weak group inverse, several auxiliary lemmas will be given. The first lemma gives the core-EP decomposition of a matrix $A\in\mathbb{C}^{n\times n}_{k}$ which will be a very useful tool throughout this paper.

Lemma 2.1. ^[16] Let $A\in\mathbb{C}^{n\times n}_{k}$. Then there exists a unitary matrix $U\in\mathbb{C}^{n\times n}$ such that

where $T\in\mathbb{C}^{t\times t}$ is nonsingular with $t = r(T) = r(A^{k})$ and $N$ is nilpotent of index $k$. The representation $(2.1)$ is called the core-EP decomposition of $A$, while $A_{1}$ and $A_{2}$ are the core part and nilpotent part of $A$, respectively.

Following the representation $(2.1)$ of a matrix $A\in\mathbb{C}^{n\times n}_{k}$, we have the following representations of certain generalized inverses (see ^[15,16,33]):

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

By direct computations, we get that $A\in \mathbb{C}^{\rm CM}_{n}$ is equivalent with $N = 0$, in which case

and

Let $A\in\mathbb{C}^{n\times n}_{k}$ be of the form $(2.1)$ and let $m\in\mathbb{Z^{+}}$. The notations below will be frequently used in this paper:

Lemma 2.2. [34,Lemma 6] Let $A\in\mathbb{C}^{n\times n}_{k}$ be of the form $(2.1)$. Then

From $(2.8)$ and [16,Theorem 2.2], we get that

where $t = r(A^{k}).$

Lemma 2.3. ^[29,35,36] Let $A\in\mathbb{C}^{n\times n}_{k}$ and let $m\in\mathbb{Z^{+}}$. Then

Lemma 2.4. Let $A\in\mathbb{C}^{n\times n}_{k}$ and let $m\in\mathbb{Z^{+}}$. Then

Proof. Assume that $A\in\mathbb{C}^{n\times n}_{k}$ is of the form $(2.1)$. By $(2.3)$, $(2.12)$ and $(2.13)$, it follows that

3.   $m$-weak group inverse on complex matrices

In this section, using the core-EP decomposition of a matrix $A\in\mathbb{C}^{n\times n}_{k}$ we will give another definition of the $m$-weak group inverse. Furthermore, some properties of the $m$-weak group inverse will be derived.

Theorem 3.1. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$ and let $X\in\mathbb{C}^{n\times n}$ and $m\in\mathbb{Z^{+}}$. The system of equations

is consistent and has a unique solution $X$ given by

Proof. If $m = 1$, then $X$ coincides with $A^{{Ⓦ}}$. Clearly, $X$ is the unique solution of $(3.1)$ according to the definition of the weak group inverse. If $m\neq1$, by $(3.1)$, Lemmas 2.3 $(d)$ and 2.4, it follows that

Thus, by $(2.3)$ and $(2.12)$, we have that

Definition 3.2. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $m\in\mathbb{Z^{+}}$. The $m$-weak group inverse of $A$, denoted by $A^{{Ⓦ}_{m}}$, is the unique solution of the system $(3.1)$.

Remark 3.3. The $m$-weak group inverse is in some sense a generalization of the weak group inverse and Drazin inverse. We have the following:

$(a)$ If $m = 1$, then $1$-weak group inverse of $A\in\mathbb{C}^{n\times n}_{k}$ coincides with the weak group inverse of $A$;

$(b)$ If $m\geq k$, then $m$-weak group inverse of $A\in\mathbb{C}^{n\times n}_{k}$ coincides with the Drazin inverse of $A$.

In the following example, we will show that the $m$-weak group inverse is different from some known generalized inverses.

Example 3.4. Let $A = \left[\begin{array}{cc} I_{3} & I_{3} \\ 0 & N \\ \end{array} \right],$ where $N = \left[\begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \\ \end{array} \right].$ It can be verified that ${{\rm{ Ind}}}(A) = 3.$ By computations, we can check the following:

where $H_{1} = \left[\begin{array}{ccc} \frac{1}{2} & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array} \right],$ $H_{2} = \left[\begin{array}{ccc} 1 & 1 & 1\\ 0 & 1 & 1 \\ 0 & 0 & 1 \\ \end{array} \right],$ $H_{3} = \left[\begin{array}{ccc} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 0\\ \end{array} \right],$ $H_{4} = \left[\begin{array}{ccc} \frac{1}{2} & \frac{1}{2} & \frac{1}{2}\\ 0 & 1 & 1\\ 0 & 0 & 0\\ \end{array} \right]$ and $N^{†} = \left[\begin{array}{ccc} 0 & 0 & 0\\ 1 & 0 & 0\\ 0 & 1 & 0\\ \end{array} \right].$

It is clear that

Theorem 3.5. Let $A\in\mathbb{C}^{n\times n}_{k}$ be decomposed by $A = A_{1}+A_{2}$ as in $(2.1)$ and let $m\in\mathbb{Z^{+}}$. Then

$(a)$ $A^{{Ⓦ}_{m}}$ is an outer inverse of $A;$

$(b)$ $A^{{Ⓦ}_{m}}$ is a reflexive g-inverse of $A_{1}.$

Proof. $(a)$ By Lemmas 2.3 $(d)$, 2.4 and the definition of $A^{{Ⓦ}_{m}}$, it follows that

$(b)$ By $(2.2)$ and $(3.2)$, we get that

From [16,Theorem 3.4], we get . By the fact that and the statement $(a)$ above, it follows that

Hence $A^{{Ⓦ}_{m}}$ is a reflexive g-inverse of $A_{1}.$

Theorem 3.6. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $m\in\mathbb{Z^{+}}$. Then

$(a)$ $r(A^{{Ⓦ}_{m}}) = r(A^{k}).$

$(b)$ $\mathcal{R}(A^{{Ⓦ}_{m}}) = \mathcal{R}(A^{k}),$ $\mathcal{N}(A^{{Ⓦ}_{m}}) = \mathcal{N}((A^{k})^{*}A^{m}).$

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

Proof. $(a)$ Assume that $A$ is given by $(2.1)$. From $(2.11)$ and $(3.2)$, it is clear that $r(A^{{Ⓦ}_{m}}) = t = r(A^{k}).$

$(b)$ Since implies that and since $r(A^{{Ⓦ}_{m}}) = r(A^{k})$, we get $\mathcal{R}(A^{{Ⓦ}_{m}}) = \mathcal{R}(A^{k}).$ From and we get If we get that Then , and by $r(A^{{Ⓦ}_{m}}) = r((A^{k})^{*}A^{m})$, it follows that

$(c)$ It is a direct consequence from Theorems 3.5 $(a)$ and 3.6 $(b)$.

Theorem 3.7. Let $A\in\mathbb{C}^{n\times n}_{k}$ and $m\in\mathbb{Z^{+}}$. Then

$(a)$ $AA^{{Ⓦ}_{m}} = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m})};$

$(b)$ $A^{{Ⓦ}_{m}}A = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}.$

Proof. $(a)$ From Theorem 3.5 $(a)$, it follows that $AA^{{Ⓦ}_{m}}\in \mathbb{C}^{\rm P}_{n}.$ By the definition of $A^{{Ⓦ}_{m}}$ and $(3.2)$, it can be proved that and $r(AA^{{Ⓦ}_{m}}) = r(A^{{Ⓦ}_{m}}) = r(A^{k}) = t.$ Hence $\mathcal{R}(AA^{{Ⓦ}_{m}}) = \mathcal{R}(A^{k}).$ Similarly, we get that $\mathcal{N}(AA^{{Ⓦ}_{m}}) = \mathcal{N}(A^{{Ⓦ}_{m}}) = \mathcal{N}((A^{k})^{*}A^{m}).$ Therefore, $AA^{{Ⓦ}_{m}} = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m})}.$

$(b)$ The proof follows similarly as for the part $(a)$.

4.   Some characterizations of the $m$-weak group inverse

In this part, we represent some characterizations of the $m$-weak group inverse in terms of the range space, null space, rank equalities, and projectors.

The next theorem gives several characterizations of $A^{{Ⓦ}_{m}}$.

Theorem 4.1. Let $A\in\mathbb{C}^{n\times n}_{k}$, $X\in\mathbb{C}^{n\times n}$ and let $m\in\mathbb{Z^{+}}$. Then the following hold:

$(a)$ $X = A^{{Ⓦ}_{m}}.$

$(c)$ $\mathcal{R}(X) = \mathcal{R}(A^{k}),$ $A^{m+1}X = P_{A^{k}}A^{m}.$

$(d)$ $\mathcal{R}(X) = \mathcal{R}(A^{k}),$ $(A^{k})^{*}A^{m+1}X = (A^{k})^{*}A^{m}.$

Proof. $(a)\Rightarrow (b):$ This follows directly by Theorem 3.6 $(b)$ and the definition of $A^{{Ⓦ}_{m}}.$

$(b)\Rightarrow (c):$ Premultiplying by $A^m$, and by Lemma 2.4, it follows that

$(c)\Rightarrow (d):$ Premultiplying $A^{m+1}X = P_{A^{k}}A^{m}$ by $(A^{k})^{*}$, it follows that

$(d)\Rightarrow (a):$ Let $A$ be of the form $(2.1)$. By $(2.11)$ and $\mathcal{R}(X) = \mathcal{R}(A^{k}),$ we obtain that

where $X_{1}\in\mathbb{C}^{t\times t}$ and $X_{2}\in\mathbb{C}^{t\times (n-t)}$. Thus $(A^{k})^{*}A^{m+1}X = (A^{k})^{*}A^{m}$ implies that

i.e., $X_{1} = T^{-1}\ and\ X_{2} = (T^{m+1})^{-1}T_{m}$, which imply $X = U \left[\begin{array}{cc} T^{-1} & (T^{m+1})^{-1}T_{m}\\ 0 & 0\\ \end{array} \right] U^{*} = A^{{Ⓦ}_{m}}$.

By Theorem 3.5, it is known that $A^{{Ⓦ}_{m}}$ is an outer inverse of $A\in\mathbb{C}^{n\times n}_{k}$, i.e., $A^{{Ⓦ}_{m}}AA^{{Ⓦ}_{m}} = A^{{Ⓦ}_{m}}$. Using this result, we obtain some characterizations of $A^{{Ⓦ}_{m}}$.

Theorem 4.2. Let $A\in\mathbb{C}^{n\times n}_{k}$, $X\in\mathbb{C}^{n\times n}$ and let $m\in\mathbb{N^{+}}$. Then the following conditions are equivalent:

$(a)$ $X = A^{{Ⓦ}_{m}}.$

$(b)$ $XAX = X$, $\mathcal{R}(X) = \mathcal{R}(A^{k}),$ $\mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{m}).$

$(d)$ $XAX = X$, $\mathcal{R}(X) = \mathcal{R}(A^{k}),$ $(A^{m})^{*}A^{m+1}X\in\mathbb{C}^{\rm H}_{n}.$

Proof. $(a)\Rightarrow (b):$ It is a direct consequence of Theorem 3.6 $(c)$.

$(b)\Rightarrow (c):$ By $XAX = X$ and $\mathcal{R}(X) = \mathcal{R}(A^{k})$, it follows that

and

Since $AX$, , we have and $XAX = X$, we obtain that $XA^{k+1} = A^{k}.$

$(c)\Rightarrow (d):$ We have that

and by $\mathcal{R}(A^{k}) = \mathcal{R}(XA^{k+1})\subseteq\mathcal{R}(X)$, we get $\mathcal{R}(XA) = \mathcal{R}(A^{k}).$ Since , it follows that

$(d)\Rightarrow (a):$ Assume that $A$ is of the form $(2.1)$. From $XAX = X$ and $\mathcal{R}(X) = \mathcal{R}(A^{k})$, we get that $XA^{k+1} = A^{k}$. Then it is easy to conclude that

where $X_{2}\in\mathbb{C}^{t\times (n-t)}$.

Since

we obtain that $X_{2} = T^{-(m+1)}T_{m}$. Hence $X = U\left[\begin{array}{cc}T^{-1} & T^{-(m+1)}T_{m}\\0 & 0 \\\end{array}\right]U^{*} = A^{{Ⓦ}_{m}}.$

Motivated by the first two matrix equations $XA^{k+1} = A^{k}$ and $XAX = X$, we provide several characterizations of $A^{{Ⓦ}_{m}}.$

Theorem 4.3. Let $A\in\mathbb{C}^{n\times n}_{k}$, $X\in\mathbb{C}^{n\times n}$ and let $m\in\mathbb{Z^{+}}$. Then the following conditions are equivalent:

$(a)$ $X = A^{{Ⓦ}_{m}}.$

$(b)$ $XA^{k+1} = A^{k}$, $AX^{2} = X,$ $(A^{m})^{*}A^{m+1}X\in\mathbb{C}^{\rm H}_{n}.$

$(c)$ $XA^{k+1} = A^{k}$, $AX^{2} = X,$ $A^{m+1}X = P_{A^{k}}A^{m}.$

Proof. $(a)\Leftrightarrow(b):$ This follows by Proposition 4.2 in ^[19].

$(a)\Rightarrow (c):$ It is a direct consequence of Theorems 4.1 $(c)$ and 4.2 $(c)$.

$(c)\Rightarrow (d):$ Assume that $A$ is given by $(2.1)$. By $XA^{k+1} = A^{k}$, we get that

where $X_{2}\in\mathbb{C}^{t\times (n-t)}$ and $X_{4}\in\mathbb{C}^{(n-t)\times (n-t)}$.

By $AX^{2} = X$, we have that $X_{4} = N{X_{4}}^{2}$, which implies that

Using $(2.12)$ and $(2.13)$ and that $A^{m+1}X = P_{A^{k}}A^{m}$, we get

Now, the proof follows directly.

$(d)\Rightarrow (a):$ Since $XA^{k+1} = A^{k}$, it follows that $\mathcal{R}(A^{k}) = \mathcal{R}(XA^{k+1})\subseteq \mathcal{R}(X)$ and by $r(X) = r(A^{k}),$ we get $\mathcal{R}(A^{k}) = \mathcal{R}(X).$ Hence, according to Theorem 4.1 $(b)$, we get $X = A^{{Ⓦ}_{m}}.$

According to Theorem 3.7, it follows that $AX = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m})}$ and $XA = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}$ when $X = A^{{Ⓦ}_{m}}$. Conversely, the implication does not hold. Here's an example below.

Example 4.4. Let $A = \left[\begin{array}{cc} I_{3} & L\\ 0 & N \\ \end{array} \right],$ $X = \left[\begin{array}{cc} I_{3} & L\\ 0 & L \\ \end{array} \right],$ where $N = \left[\begin{array}{ccc} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0\\ \end{array} \right],$ $L = \left[\begin{array}{ccc} 0 & 0 & 1\\ 0 & 0 & 0\\ 0 & 0 & 0\\ \end{array} \right].$ Then it is clear that $k = {\rm{ Ind}}$$(A) = 3$ and $A^{{Ⓦ}_{2}} = \left[\begin{array}{cccccc} I_{3} & L\\ 0 & 0 \\ \end{array} \right].$ It can be directly verified that $AX = P_{\mathcal{R}(A^{3}), \mathcal{N}((A^{3})^{*}A^{2})}, XA = P_{\mathcal{R}(A^{3}), \mathcal{N}((A^{3})^{*}A^{3})}$. However, $X\neq A^{{Ⓦ}_{2}}.$

Based on the example above, the next theorem, we consider other characterizations of $A^{{Ⓦ}_{m}}$ by using $AX = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m})}$ and $XA = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}$.

Theorem 4.5. Let $A\in\mathbb{C}^{n\times n}_{k}$ be of the form $(2.1)$, $X\in\mathbb{C}^{n\times n}$ and $m\in\mathbb{Z^{+}}$. Then the following statements are equivalent:

$(a)$ $X = A^{{Ⓦ}_{m}};$

$(a)$ $AX = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m})}, XA = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}\ {\rm and}\ r(X) = r(A^{k});$

$(a)$ $AX = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m})}, XA = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}\ {\rm and}\ XAX = X;$

$(a)$ $AX = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m})}, XA = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}\ {\rm and}\ AX^{2} = X.$

Proof. $(a)\Rightarrow (b):$ It is a direct consequence of Theorems 3.6 $(a)$ and 3.7.

$(b)\Rightarrow (c):$ Since $XA = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}\ {\rm and}\ r(X) = r(A^{k})$, we get that $\mathcal{R}(X) = \mathcal{R}(A^{k})$ and by $XA = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}$, we obtain $XAX = X$.

$(c)\Rightarrow (d):$ From $XA = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}\ {\rm and}\ r(X) = r(A^{k})$, we have that $\mathcal{R}(X) = \mathcal{R}(A^{k})$ and by $AX = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m})}$, it follows that $AX^{2} = X$.

$(d)\Rightarrow (a):$ By $XA = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}\ {\rm and}\ AX^{2} = X,$ it follows that

which implies $\mathcal{R}(X) = \mathcal{R}(A^{k}).$ By $AX = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m})},$ we get that

According to Theorem 4.1 $(d)$, we have that $X = A^{{Ⓦ}_{m}}.$

Analogously, we characterize $A^{{Ⓦ}_{m}}$ using that $AX = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m})}$ or $XA = P_{\mathcal{R}(A^{k}), \mathcal{N}((A^{k})^{*}A^{m+1})}$ as follows:

Theorem 4.6. Let $A\in\mathbb{C}^{n\times n}_{k}$, $X\in\mathbb{C}^{n\times n}$ and let $m\in\mathbb{Z^{+}}$. Then

$(a)$ $X = A^{{Ⓦ}_{m}}$ is the unique solution of the system of equations:

$(b)$ $X = A^{{Ⓦ}_{m}}$ is the unique solution of the system of equations:

Proof. $(a)$ By Theorems 3.6 $(b)$ and 3.7 $(a)$, it follows that $X = A^{{Ⓦ}_{m}}$ is a solution of the system of Eq $(4.1)$. Conversely, if the system $(4.1)$ is consistent, it follows that $(A^{k})^{*}A^{m}AX = (A^{k})^{*}A^{m}$. Hence by Theorem 4.1, $X = A^{{Ⓦ}_{m}}$ $(d)$.

$(b)$ By Theorems 3.6 $(b)$ and 3.7 $(b)$, it is evident that $X = A^{{Ⓦ}_{m}}$ is a solution of $(4.2)$. Next, we prove the uniqueness of the solution.

Assume that $X_{1}, X_{2}$ satisfy the system of Eq $(4.2)$. Then $X_{1}A = X_{2}A$ and $\mathcal{N}(X_{1}) = \mathcal{N}(X_{2}) = \mathcal{N}((A^{k})^{*}A^{m}).$ Thus, we get that $\mathcal{R}(X_{1}^{*}-X_{2}^{*})\subseteq \mathcal{N}(A^{*})\subseteq \mathcal{N}((A^{k})^{*})$ and $\mathcal{R}(X_{1}^{*}-X_{2}^{*})\subseteq\mathcal{R}((A^{m})^{*}A^{k}).$ For any $\eta\in \mathcal{N}((A^{k})^{*})\cap\mathcal{R}((A^{m})^{*}A^{k}),$ we obtain that $(A^{k})^{*}\eta = 0$, $\eta = (A^{m})^{*}A^{k}\xi$ for some $\xi\in\mathbb{C}^{n}.$ Since Ind$(A) = k$, we derive that $\mathcal{R}(A^{k}) = \mathcal{R}(A^{k+m})$, and it follows that $A^{k}\xi = A^{k+m}\xi_{0}$ for some $\xi_{0}\in\mathbb{C}^{n\times n}.$ Then we have that

Premultiplying the equation above by $\xi_{0}^{*}$, we derive that $(A^{k+m}\xi_{0})^{*}A^{k+m}\xi_{0} = 0$, which implies $A^{k+m}\xi_{0} = 0$. Hence $\eta = 0$, i.e., $\mathcal{R}(X_{1}^{*}-X_{2}^{*}) = \{0\}$, which implies $X_{1} = X_{2}$.

Remark 4.7. Notice that the condition $\mathcal{R}(X) = \mathcal{R}(A^{k})$ in Theorem 4.6 $(a)$ can be replaced by $\mathcal{R}(X)\subseteq\mathcal{R}(A^{k})$. Also the condition $\mathcal{N}(X) = \mathcal{N}((A^{k})^{*}A^{m})$ in Theorem 4.6 $(b)$ can be replaced by $\mathcal{N}(X)\supseteq\mathcal{N}((A^{k})^{*}A^{m})$.

5.   Representations of the $m$-weak group inverse

From Theorem 3.1, we get an expression of $A^{{Ⓦ}_{m}}$ in terms of . In the next results, we present several expressions of $A^{{Ⓦ}_{m}}$ in terms of certain generalized inverses.

Theorem 5.1. Let $A\in\mathbb{C}^{n\times n}_{k}$ and let $m\in\mathbb{Z^{+}}$. Then the following statements hold:

$(a)$ $A^{{Ⓦ}_{m}} = (A^{D})^{m+1}P_{A^{k}}A^{m}.$

$(c)$ $A^{{Ⓦ}_{m}} = (A^{k})^{\#}A^{k-m-1}P_{A^{k}}A^{m} \ (k\geq m+1).$

$(d)$ $A^{{Ⓦ}_{m}} = (A^{m+1}P_{A^{k}})^{†}A^{m}.$

$(e)$ $A^{{Ⓦ}_{m}} = A^{m-1}P_{A^{k}}(A^{m})^{{Ⓦ}}.$

Proof. Assume that $A$ is given by $(2.1)$. By $(2.3)$–$(2.7)$ and $(2.11)$–$(2.13)$, we get that

$(a)$ By $(2.12)$, $(2.13)$ and $(5.3)$, it follows that

Hence $A^{{Ⓦ}_{m}} = (A^{D})^{m+1}P_{A^{k}}A^{m}$.

The proofs of $(b)$–$(e)$ are analogous to that of $(a)$.

Next, we consider the accuracy of the expression in Theorem 5.1 $(a)$ for computing the $m$-weak group inverse.

Example 5.2. Let

Assume that $A$ is given by $(2.1)$. Then

It is clear that $k$ = ${\rm{ Ind}}$$(A)$ = 3. According to $(2.12)$, $(2.13)$, $(3.2)$ and $(5.3)$, a straightforward computation shows that

Let $K = (A^{D})^{3}P_{A^{3}}A^{2}$. Then

and

where $\parallel \cdot \parallel$ is the Frobenius norm.

Hence, Theorem 5.1 $(a)$ gives a good result in terms of computational accuracy.

In the following theorem, we present a connection between the $(B, C)$-inverse and the $m$-weak group inverse showing that the $m$-weak group inverse of $A\in\mathbb{C}^{n\times n}_{k}$ is its $(A^{k}, (A^{k})^{*}A^{m})$-inverse.

Theorem 5.3. Let $A\in\mathbb{C}^{n\times n}_{k}$ and let $m\in\mathbb{Z^{+}}$. Then $A^{{Ⓦ}_{m}} = A^{(A^{k}, (A^{k})^{*}A^{m})}.$

Proof. By Theorem 3.7 we have that $A^{{Ⓦ}_{m}}AA^{k} = A^{k}$ and $((A^{k})^{*}A^{m})AA^{{Ⓦ}_{m}} = (A^{k})^{*}A^{m}.$ From Theorem 3.6 $(b)$, we derive that $\mathcal{R}(A^{{Ⓦ}_{m}}) = \mathcal{R}(A^{k})$ and $\mathcal{N}(A^{{Ⓦ}_{m}}) = \mathcal{N}((A^{k})^{*}A^{m})$. Evidently, $A^{{Ⓦ}_{m}} = A^{(A^{k}, (A^{k})^{*}A^{m})}.$

Now we will give some limit expressions of $A^{{Ⓦ}_{m}}$, but before we need the next auxiliary lemma:

Lemma 5.4. ^[37] Let $A\in\mathbb{C}^{m\times n}, X\in\mathbb{C}^{n\times p}$ and $Y\in\mathbb{C}^{p\times m}.$ Then the following hold:

$(a)$ $\lim_{\lambda \to 0}X(\lambda I_{p}+YAX)^{-1}Y$ exists;

$(b)$ $r(XYAXY) = r(XY);$

$(c)$ $A^{(2)}_{\mathcal{R}(XY), \mathcal{N}(XY)}$ exists,

in which case,

Theorem 5.5. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$ and let $m\in\mathbb{N^{+}}$. Then the following statements hold:

$(a)$ $A^{{Ⓦ}_{m}} = \lim_{\lambda \to 0}{A^{k}(\lambda I_{n}+(A^{k})^{*}A^{k+m+1})^{-1}}(A^{k})^{*}A^{m};$

$(b)$ $A^{{Ⓦ}_{m}} = \lim_{\lambda \to 0}{A^{k}(A^{k})^{*}(\lambda I_{n}+A^{k+m+1}(A^{k})^{*})^{-1}}A^{m};$

$(c)$ $A^{{Ⓦ}_{m}} = \lim_{\lambda \to 0}{A^{k}(A^{k})^{*}A^{m}(\lambda I_{n}+A^{k+1}(A^{k})^{*}A^{m})^{-1}};$

$(d)$ $A^{{Ⓦ}_{m}} = \lim_{\lambda \to 0}{(\lambda I_{n}+A^{k}(A^{k})^{*}A^{m+1})^{-1}}A^{k}(A^{k})^{*}A^{m}.$

Proof. $(a)$ It is easy to check that $r(A^{k}(A^{k})^{*}A^{m}) = r((A^{k})^{*}A^{m}) = r(A^{k}) = t$. By Theorem 3.6, we get that $\mathcal{R}(A^{k}) = \mathcal{R}(A^{k}(A^{k})^{*}A^{m})$, $\mathcal{N}((A^{k})^{*}A^{m}) = \mathcal{N}(A^{k}(A^{k})^{*}A^{m})$. From Theorem 3.6, we get

Let $X = A^{k}, Y = (A^{k})^{*}A^{m}$. By Lemma 5.4, we get that

The statements $(b)$–$(d)$ can be similarly proved.

The following example will test the accuracy of expression in Theorem 5.5 $(a)$ for computing the $m$-weak group inverse.

Example 5.6. Let

with $k$ = ${\rm{ Ind}}$$(A)$ = 3. By , we get

Together with $(3.2)$, it follows that

Let $L = \lim_{\lambda \to 0}{A^{3}(\lambda I_{n}+(A^{3})^{*}A^{6})^{-1}}(A^{3})^{*}A^{2}$. Then

and

where $\parallel \cdot \parallel$ is the Frobenius norm. Hence, the representation in Theorem 5.5 (a) is efficient for computing the $m$-weak group inverse.

6.   Relationships between the $m$-weak group inverse and other generalized inverses

In this section, we consider some relations between the $m$-weak group inverse and other generalized inverses as well as certain matrix classes. The symbols $\mathbb{C}^{\rm OP}_{n}$, $\mathbb{C}^{\rm EP}_{n}$, $\mathbb{C}^{i-{\rm EP}}_{n}$ and stand for the subsets of $\mathbb{C}^{n\times n}$ consisting of orthogonal projectors (Hermitian idempotent matrices), EP (Range-Hermitian) matrices, $i$-EP matrices and $k$-core-EP matrices, respectively, i.e.,

First, we will state the following lemma auxiliary lemma:

Lemma 6.1. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$. Then $T_{m} = 0$ if and only if $S = 0.$

Proof. Notice that $T_{m} = 0$ can be equivalently expressed by the equation below:

Multiplying the equation above from the right side by $N^{k-1}$, we get $SN^{k-1} = 0.$ Then multiplying from the right by $N^{k-2}$, we get $SN^{k-1} = 0,$. Similarly, we get $SN^{k-3} = 0, \cdots, SN = 0.$ Now by $(6.1)$, it follows that $T^{m-1}S = 0$, i.e, $S = 0.$

The next theorem provides some necessary and sufficient conditions for $A^{{Ⓦ}_{m}}$ to be equal to various transformations of $A\in\mathbb{C}^{n\times n}_{k}$.

Theorem 6.2. Let $A\in\mathbb{C}^{n\times n}_{k}$ and let $m\in\mathbb{Z^{+}}$. Then the following statements hold:

$(a)$ $A^{{Ⓦ}_{m}}\in A\{1\}$ if and only if $A\in\mathbb{C}^{\rm CM}_{n}.$

$(b)$ $A^{{Ⓦ}_{m}}\in\mathbb{C}^{\rm CM}_{n}.$

$(c)$ $A^{{Ⓦ}_{m}} = A$ if and only if $A = A^{3}.$

$(d)$ $A^{{Ⓦ}_{m}} = A^{*}$ if and only if $AA^{*}\in\mathbb{C}^{\rm OP}_{n}$ and $A\in\mathbb{C}^{\rm EP}_{n}.$

$(e)$ $A^{{Ⓦ}_{m}} = P_{A}$ if and only if $A\in\mathbb{C}^{\rm OP}_{n}.$

Proof. Let $A$ be given by $(2.1)$.

$(a)$ By $(3.2)$, it follows that

$(b)$ By $(3.2)$, it is clear that $r(A^{{Ⓦ}_{m}}) = r((A^{{Ⓦ}_{m}})^{2}) = t$, which implies $A^{{Ⓦ}_{m}}\in\mathbb{C}^{\rm CM}_{n}.$

$(c)$ From $(3.2)$, we get that

$(d)$ According to $(3.2)$, we obtain that

$(e)$ By $(2.9)$ and $(3.2)$, it follows that

Hence $A^{{Ⓦ}_{m}} = P_{A}$ if and only if $A\in\mathbb{C}^{\rm OP}_{n}$.

Using the core-EP decompositio, we proved that $A\in \mathbb{C}^{i-{\rm EP}}_{n}$ if and only if $A^{{Ⓦ}_{m}}\in \mathbb{C}^{\rm EP}_{n}$. Therefore, we will consider certain equivalent conditions for $A^{{Ⓦ}_{m}}\in \mathbb{C}^{\rm EP}_{n}$.

Lemma 6.3. ^[17] Let $A\in\mathbb{C}^{n\times n}_{k}$ be of the form $(2.1)$. Then $A\in \mathbb{C}^{i-{\rm EP}}_{n}$ if and only if $S = 0.$

Moreover, $S = 0$ if and only if

Theorem 6.4. Let $A\in\mathbb{C}^{n\times n}_{k}$ and let $m\in\mathbb{Z^{+}}$. The following statements are equivalent:

$(a)$ $A^{{Ⓦ}_{m}}\in \mathbb{C}^{\rm EP}_{n};$

$(b)$ $A\in \mathbb{C}^{i-{\rm EP}}_{n};$

$(c)$ $A^{{Ⓦ}}\in \mathbb{C}^{\rm EP}_{n};$

Proof. Let $A\in\mathbb{C}^{n\times n}_{k}$ be of the form $(2.1)$. According to Lemma 6.3, we will prove that each of the statements $(a)$, $(c)$, $(d)$ and $(e)$ is equivalent to $S = 0$.

$(a)$ According to $(3.2)$ and Lemma 6.1, it follows that

$(c)$ By $(2.4)$, we get that

$(d)$ By $(2.3)$, $(3.2)$ and Lemma 6.1, it follows that

$(e)$ From $(2.3)$, $(3.2)$ and Lemma 6.1, we get that

In ^[22], the authors proved that $A^{{Ⓦ}_{m}} = A^{D}$ if and only if $SN^{m} = 0.$ In the following results, we investigate the relation between the $m$-weak group inverse and other generalized inverses such as the MP-inverse, group inverse, core inverse, DMP-inverse, dual DMP-inverse, weak group inverse by core-EP decomposition.

Theorem 6.5. Let $A\in\mathbb{C}^{n\times n}_{k}$ be given by $(2.1)$ and let $m\in\mathbb{Z^{+}}$. Then the following statements hold:

$(a)$ $A^{{Ⓦ}_{m}} = A^{†}\Longleftrightarrow A\in\mathbb{C}^{\rm EP}_{n};$

$(b)$ $A^{{Ⓦ}_{m}} = A^{\#}\Longleftrightarrow A\in\mathbb{C}^{\rm CM}_{n};$

$(d)$ $A^{{Ⓦ}_{m}} = A^{D, †}\Longleftrightarrow T^{k-m}T_{m} = T_{k}NN^{†};$

$(e)$ $A^{{Ⓦ}_{m}} = A^{†, D}\Longleftrightarrow SN^{m} = 0$ and $S = SN^{†}N;$

$(f)$ $A^{{Ⓦ}_{m}} = A^{{Ⓦ}}\Longleftrightarrow SN = 0$ $(m > 1).$

Proof. $(a)$ It follows from $(2.8)$ and $(3.2)$ that

$(b)$ Since $A^{\#}$ exits if and only if $A\in\mathbb{C}^{\rm CM}_{n}$, which is equivalent to $N = 0$, we get by $(2.6)$ and $(3.2)$ the following:

$(c)$ The proof follows similarly as in $(b)$.

$(d)$ Using $(2.5)$ and $(2.9)$ to $A^{D, †} = A^{D}AA^{†}$, we derive

and by $(3.2)$, it follows that

$(e)$ Using $(2.5)$ and $(2.10)$ and the faact that $A^{†, D} = A^{†}AA^{D}$, we obtain that

which together with $(3.2)$, gives

$(f)$ If $m > 1$, from $(2.4)$ and $(3.2)$, we get

Clearly, $(T^{m+1})^{-1}T_{m} = T^{-2}S$ is equivalent to $T^{-3}SN+\cdots +(T^{m+1})^{-1}SN^{m-1} = 0$, which is further equivalent to $SN = 0.$ Hence $A^{{Ⓦ}_{m}} = A^{{Ⓦ}}$ if and only if $SN = 0$.

7.   Applications of the $m$-weak group inverse

In this section, we consider a relation between the $m$-weak group inverse and the nonsingular bordered matrix, which will be applied to the Cramer's rule for the solution of the restricted matrix equation.

Theorem 7.1. Let $A\in\mathbb{C}^{n\times n}_{k}$ be such that $r(A^{k}) = t$ and let $m\in\mathbb{Z^{+}}$. Let $B\in\mathbb{C}^{n\times (n-t)}$ and $C^{*}\in\mathbb{C}^{n\times (n-t)}$ be of full column rank such that $\mathcal{N}((A^{k})^{*}A^{m}) = \mathcal{R}(B)$ and $\mathcal{R}(A^{k}) = \mathcal{N}(C)$. Then the bordered matrix

is invertible and its inverse is given by

Proof. Let $X = \left[\begin{array}{cc} A^{{Ⓦ}_{m}} & \ \ (I_{n}-A^{{Ⓦ}_{m}}A)C^{†}\\ B^{†}(I_{n}-AA^{{Ⓦ}_{m}}) & \ \ B^{†}(AA^{{Ⓦ}_{m}}A-A)C^{†}\\ \end{array}\right].$ Since $\mathcal{R}(A^{{Ⓦ}_{m}}) = \mathcal{R}(A^{k}) = \mathcal{N}(C)$, we have that $CA^{{Ⓦ}_{m}} = 0$. Since $C$ is a full row rank matrix, it is right invertible and $CC^{†} = I_{n-t}$. From

we get $BB^{†}(I_{n}-AA^{{Ⓦ}_{m}}) = I_{n}-AA^{{Ⓦ}_{m}}.$ Hence,

Thus, $X = K^{-1}$.

In the next result, we will discuss the solution of the restricted matrix equation

using the $m$-weak group inverse.

Theorem 7.2. Let $A\in\mathbb{C}^{n\times n}_{k}$, $X\in\mathbb{C}^{n\times p}$ and $D\in\mathbb{C}^{n\times p}$. If $\mathcal{R}(D)\subseteq\mathcal{R}(A^{k})$, then the restricted matrix equation

has a unique solution $X = A^{{Ⓦ}_{m}}D$.

Proof. Since $\mathcal{R}(A^{k}) = \mathcal{R}(AA^{k})$ and $\mathcal{R}(D)\subseteq \mathcal{R}(A^{k})$, we get that $\mathcal{R}(D)\subseteq A\mathcal{R}(A^{k})$, which implies solvability of the matrix Eq $(7.1)$. Obviously, $X = A^{{Ⓦ}_{m}}D$ is a solution of $(7.1)$. Then we prove the uniqueness of $X$. If $X_{1}$ also satisfies $(7.1)$, then

Based on the nonsingularity of the bordered matrix given in Theorem 7.1, we will show in the next theorem how the Cramer's rule can be used for solving the restricted matrix Eq $(7.1)$.

Theorem 7.3. Let $A\in\mathbb{C}^{n\times n}_{k}$ be such that $r(A^{k}) = t$ and let $X\in\mathbb{C}^{n\times p}$ and $D\in\mathbb{C}^{n\times p}$. Let $B\in\mathbb{C}^{n\times (n-t)}$ and $C^{*}\in\mathbb{C}^{n\times (n-t)}$ be full column rank matrices such that $\mathcal{N}((A^{k})^{*}A^{m}) = \mathcal{R}(B)$ and $\mathcal{R}(A^{k}) = \mathcal{N}(C)$. Then the unique solution of the restricted matrix Eq $(7.1)$ is given by $X = [x_{ij}]$, where

where $d_{j}$ denotes the j-th column of $D$.

Proof. Since $X$ is the solution of the restricted matrix Eq $(7.1)$, we get that $\mathcal{R}(X)\subseteq\mathcal{R}(A^{k}) = \mathcal{N}(C)$, which implies $CX = 0$. Then the restricted matrix Eq $(7.1)$ can be rewritten as

By Theorem 7.1, we have that $\left[\begin{array}{cc} A & B\\ C & 0\\ \end{array}\right]$ is invertible. Consequently, $(7.2)$ follows from the Cramer's rule for the above equation.

Example 7.4. Let

It can be verified that ${{{\rm{ Ind}}}}(A) = 3.$ Then we get that

It is easy to check that

satisfies the restricted matrix equation $AX = D$ and $\mathcal{R}(X)\subseteq\mathcal{R}(A^{3})$. By simple calculations, we can also get that the components of $X$ can be expressed by $(7.2)$.

8.   Conclusions

This paper gives a new definition of the $m$-weak group inverse for the complex matrices, which extends the Drazin inverse and the weak group inverse. Some characterizations of the $m$-weak group inverse in terms of the range space, null space, rank, and projectors are presented. Several representations of the $m$-weak group inverse involving some known generalized inverses as well as limitations are also derived. The representation in Theorem 5.1 gives a better result in terms of the computational accuracy (see Examples 5.2 and 5.6). The $m$-weak group inverses are concerned with the solution of a restricted matrix Eq $(7.1)$. The solution of $(7.1)$ can also be expressed by the Cramer's rule (see Theorem 7.3). In ^[38,39,40], there are some iterative methods and algorithms to compute the outer inverses. Motivated by these, further investigations deserve more attention as follows:

(1) The applications of the $m$-weak group inverse in linear equations and matrix equations;

(2) Perturbation formulae as well as perturbation bounds for the $m$-weak group inverse;

(3) Iterative algorithm, a splitting method for computing the $m$-weak group inverse;

(4) Other representations of the $m$-weak group inverse.

Acknowledgement

This research is supported by the National Natural Science Foundation of China (No. 11961076).

Conflict of interest

The authors declare no conflict of interest.