Two New Matrix Classes Related to the CMP Inverse: CMP and Co-CMP Matrices

Abstract

This paper focuses on two new matrix classes related to the CMP inverse of a square matrix, called the CMP and co-CMP matrices. Some of their characterizations are identified based on the core-EP decomposition and subspace operations. And, we show an equivalence relationship between the co-CMP matrix set and the intersection of the core matrix set and the co-EP matrix set. The nonsingularity of other matrices closely associated with the co-CMP matrices is considered, and their inverses are presented in a decomposition form.

1. Introduction

Let ${\mathbb{C}}^{m\times n}$ be the set of all $m\times n$ complex matrices. For $A\in {\mathbb{C}}^{m\times n}$, the symbols $\mathcal{R}\left(A\right)$, $\mathcal{N}\left(A\right)$, $A^{*}$ and $r\left(A\right)$ denote the range space, null space, conjugate transpose, and rank of A, respectively. The index of $A\in {\mathbb{C}}^{n\times n}$, denoted by $\mathrm{Ind}\left(A\right)$, is the smallest non-negative integer k such that $r\left(A^k\right)=r\left(A^{k+1}\right)$. In addition, ${\mathbb{C}}_k^{n\times n}$ is the set consisting of all $n\times n$ complex matrices with index k, and $I_n$ is the identity matrix of order n. For two subspaces $\mathcal{T},\mathcal{S}\subseteq {\mathbb{C}}^{n\times 1}$, $P_{\mathcal{T},\mathcal{S}}$ stands for the projector on $\mathcal{T}$ along $\mathcal{S}$.

The Moore–Penrose inverse [1] of $A\in {\mathbb{C}}^{m\times n}$, denoted by $A^{†}$, is the unique matrix $X\in {\mathbb{C}}^{n\times m}$ satisfying needed.

$$\left(1\right)AXA=A,\left(2\right)XAX=X,\left(3\right){\left(AX\right)}^{*}=AX,\left(4\right){\left(XA\right)}^{*}=XA.$$

For $A\in {\mathbb{C}}_k^{n\times n}$, if $X\in {\mathbb{C}}^{n\times n}$ satisfies $XAX=X, AX=XA, X{A}^{k+1}=A^k$, then X is called the Drazin inverse [2] of A and is denoted by $A^D$. Using the Moore–Penrose inverse and Drazin inverse, Malik and Thome [3] defined the DMP inverse of $A\in {\mathbb{C}}_k^{n\times n}$, $A^{D,†}=A^{D}A{A}^{†}$, and the dual DMP inverse of A, $A^{†,D}=A^{†}A{A}^D$. The CMP inverse of $A\in {\mathbb{C}}^{n\times n}$, introduced by Mehdipour and Salemi [4], is the unique matrix $X\in {\mathbb{C}}^{n\times n}$ such that

$$XAX=X,AXA=A_1,AX=A_1{A}^{†},XA=A^{†}{A}_1,$$

where $A_1=A{A}^{D}A$ is the core part of the core-nilpotent decomposition of A and is denoted by $A^{c,†}$. The CMP inverse of A is a reflexive g-inverse of $A_1$, that is, $A^{c,†}{A}_1{A}^{c,†}=A^{c,†},A_1{A}^{c,†}{A}_1=A_1$, but it is not a reflexive g-inverse of A, which means that the CMP inverse is related to the Moore–Penrose inverse and Drazin inverse. And we note that $A^{c,†}$ satisfies $A{A}^{c,†}=A_1{A}^{c,†}$ and $A^{c,†}A=A^{c,†}{A}_1$ (see [4] (Proposition 2.2)), which inspired us to study $A{A}^{c,†}$ and $A^{c,†}A$. Mosić et al. [5] and Pablos [6] extended the concept of the CMP inverse from a square matrix to a rectangular matrix and finite potent endomorphisms, respectively. For more details about the CMP inverse, readers are referred to [7,8,9,10].

A great deal of mathematical effort has been devoted to the study of the nonsingularity of the difference and sum of two idempotent elements (see [11,12,13] and the references therein). In 2010, Benítez and Rakoćević [14] used the nonsingularity of the difference of a pair of special idempotent matrices to define the co-EP matrices, i.e., $A\in {\mathbb{C}}^{n\times n}$ is a co-EP matrix if $A{A}^{†}-A^{†}A$ is nonsingular. Significantly, the definition of the co-EP matrices is complementary with that of the EP matrices, i.e., $A\in {\mathbb{C}}^{n\times n}$ is an EP matrix if $A{A}^{†}=A^{†}A$. More information about the co-EP matrices can been seen in [14,15,16,17]. Recently, we [18] defined a co-GBD matrix $A\in {\mathbb{C}}^{n\times n}$ if $A{A}_{\left(\mathcal{L}\right)}^{(†)}-A_{\left(\mathcal{L}\right)}^{(†)}A$ is nonsingular, where $A_{\left(\mathcal{L}\right)}^{(†)}$ is the generalized Bott–Duffin inverse of A with respect to $\mathcal{L}$, and we [19] discussed a co-BD matrix A such that $A{A}_{\left(\mathcal{L}\right)}^{(-1)}-A_{\left(\mathcal{L}\right)}^{(-1)}A$ is nonsingular. Inspired by the above work, we studied the two new matrix classes related to the CMP inverse: $\{A\in {\mathbb{C}}^{n\times n}|A{A}^{c,†}=A^{c,†}A\}$ and $\{A\in {\mathbb{C}}^{n\times n}|A{A}^{c,†}-A^{c,†}A$ is nonsingular}, and we explored the relationships between EP matrices and $\{A\in {\mathbb{C}}^{n\times n}|A{A}^{c,†}=A^{c,†}A$ }, co-EP matrices and $\{A\in {\mathbb{C}}^{n\times n}|A{A}^{c,†}-A^{c,†}A$ is nonsingular}. For the convenience of narrative, we give the following definition:

Definition 1.

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

(1)

If $A{A}^{c,†}=A^{c,†}A$, then A is called a CMP matrix.

(2)

If $A{A}^{c,†}-A^{c,†}A$ is nonsingular, then A is called a co-CMP matrix.

This paper is organized as follows. In Section 2, we recall some necessary notations and lemmas. A few equivalent conditions for a matrix to be a CMP matrix are established by the core-EP decomposition in Section 3. Section 4 gives various characterizations of a co-CMP matrix in terms of the core-EP decomposition and subspace operations and shows an interesting relationship among co-CMP, core, and co-EP matrices. An interesting result is that the matrix A is a co-CMP matrix if and only if it is both a core matrix and a co-EP matrix. In Section 5, we investigate the nonsingularity of $A{A}^{c,†}+A^{c,†}A$ and $I-A{\left(A^{c,†}\right)}^{2}A$, which is closely related to the co-CMP matrices. The conclusion is stated in Section 6.

2. Premliminaries

For convenience, we first introduce some notations: ${\mathbb{C}}_n^{\mathrm{P}}$, ${\mathbb{C}}_n^{\mathrm{CM}}$, ${\mathbb{C}}_n^{\mathrm{EP}}$, ${\mathbb{C}}_n^{co-\mathrm{EP}}$, ${\mathbb{C}}_n^{\mathrm{CMP}}$ and ${\mathbb{C}}_n^{co-\mathrm{CMP}}$ denote the sets consisting of all idempotent, core, EP, co-EP, CMP, and co-CMP matrices, respectively, i.e.,

$$\begin{array}{ccc}& & {\mathbb{C}}_n^{\mathrm{P}}=\{A\mid A\in {\mathbb{C}}^{n\times n},A^2=A\},\hfill \\ & & {\mathbb{C}}_n^{\mathrm{CM}}=\{A\mid A\in {\mathbb{C}}^{n\times n},r\left(A\right)=r\left(A^2\right)\},\hfill \\ & & {\mathbb{C}}_n^{co-\mathrm{EP}}=\{A\mid A\in {\mathbb{C}}^{n\times n},A{A}^{†}-A^{†}A\mathrm{is}\mathrm{nonsingular}\},\hfill \\ & & {\mathbb{C}}_n^{\mathrm{CMP}}=\{A\mid A\in {\mathbb{C}}^{n\times n},A{A}^{c,†}=A^{c,†}A\},\hfill \\ & & {\mathbb{C}}_n^{co-\mathrm{CMP}}=\{A\mid A\in {\mathbb{C}}^{n\times n},A{A}^{c,†}-A^{c,†}A\mathrm{is}\mathrm{nonsingular}\}.\hfill \end{array}$$

It is well known that the core-EP decomposition plays an irreplaceable role in studying the generalized inverses. Next, we review this decomposition and some properties of the CMP inverse.

Lemma 1

(core-EP decomposition [20]). Let $A\in {\mathbb{C}}_k^{n\times n}$ and $t=r\left(A^k\right)$. Then, there exists a unitary matrix U such that

$$A=U\left[\begin{array}{cc}T& S\\ 0& N\end{array}\right]U^{*},$$

(1)

where $T\in {\mathbb{C}}^{t\times t}$ is nonsingular, $N\in {\mathbb{C}}^{(n-t)\times (n-t)}$ is nilpotent with $N^k=0$, and $S\in {\mathbb{C}}^{t\times (n-t)}$.

For $A\in {\mathbb{C}}_k^{n\times n}$ given by (1), we have

$$\begin{array}{c}\hfill ▵={[T{T}^{*}+S(I_{n-t}-N^{†}N)S^{*}]}^{-1}\mathrm{and}\tilde{T}=\sum _{j=0}^{k-1}{T}^{j}S{N}^{k-1-j},\end{array}$$

which is used throughout this paper.

Lemma 2 

([3] (Proposition 2.2), [10] (Theorem 4.6) and [8] (Corollary 3.12)). Let $A\in {\mathbb{C}}_k^{n\times n}$.

$\left(1\right)$

$A^{c,†}=A^{†}A{A}^{D}A{A}^{†}$;

$\left(2\right)$

$A{A}^{c,†}=P_{\mathcal{R}\left(A^k\right),\mathcal{N}\left(A^k{A}^{†}\right)}$ and $A^{c,†}A=P_{\mathcal{R}\left(A^{†}{A}^k\right),\mathcal{N}\left(A^k\right)}$.

$\left(3\right)$

If A is given by (1), then

$$A^{c,†}=U\left[\begin{array}{cc}{T}^{*}▵& T^{*}▵T^{-k}\tilde{T}N{N}^{†}\\ (I_{n-t}-N^{†}N)S^{*}▵& (I_{n-t}-N^{†}N)S^{*}▵T^{-k}\tilde{T}N{N}^{†}\end{array}\right]U^{*}.$$

(2)

Several results of characterizing the nonsingularity of the difference and the sum of two idempotent matrices are recalled in the following lemmas.

Lemma 3 

([13] (Corollary 1.5) and [11] (Theorem 2.3)). Let $P,Q\in {\mathbb{C}}_n^{\mathrm{P}}$. The following statements are equivalent:

$\left(1\right)$

$P-Q$ is nonsingular;

$\left(2\right)$

$P+Q$ and $I_n-PQ$ are nonsingular;

$\left(3\right)$

$\mathcal{R}\left(P\right)\cap \mathcal{R}\left(Q\right)=\left\{0\right\}$ and $\mathcal{N}\left(P\right)\cap \mathcal{N}\left(Q\right)=\left\{0\right\}$;

$\left(4\right)$

$\mathcal{R}\left(P(I_n-Q)\right)\oplus \mathcal{R}\left((I_n-P)Q\right)={\mathbb{C}}^{n\times 1}$;

$\left(5\right)$

$\mathcal{N}\left(P(I_n-Q)\right)\oplus \mathcal{N}\left((I_n-P)Q\right)={\mathbb{C}}^{n\times 1}$.

Lemma 4 

([12] (Theorem 3.2)). Let $P,Q\in {\mathbb{C}}_n^{\mathrm{P}}$. The following statements are equivalent:

$\left(1\right)$

$P+Q$ is nonsingular;

$\left(2\right)$

$\mathcal{R}\left(Q(I_n-P)\right)\cap \mathcal{R}\left(P\right)=\left\{0\right\}$ and $\mathcal{N}\left((I_n-P)Q\right)\cap \mathcal{N}\left(P\right)=\left\{0\right\}$;

$\left(3\right)$

$\mathcal{R}\left(Q(I_n-P)\right)+\mathcal{R}\left(P\right)={\mathbb{C}}^{n\times 1}$ and $\mathcal{N}\left((I_n-P)Q\right)+\mathcal{N}\left(P\right)={\mathbb{C}}^{n\times 1}$.

3. Characterizations of the CMP Matrices

In this section, we characterize the CMP matrices using the core-EP decomposition.

Theorem 1.

Let $A\in {\mathbb{C}}_k^{n\times n}$ be given by (1). Then, the following statements are equivalent:

$\left(1\right)$

$A\in {\mathbb{C}}_n^{\mathrm{CMP}}$;

$\left(2\right)$

$A_1{A}^{†}=A^{†}{A}_1$, where $A_1=A{A}^{D}A$;

$\left(3\right)$

$S{N}^{†}N=S$ and $\tilde{T}N{N}^{†}=\tilde{T}$;

$\left(4\right)$

$A{A}^{c,†}-A^{c,†}A\in {\mathbb{C}}_n^{\mathrm{P}}$.

Proof. 

$\left(1\right)\iff \left(2\right)$. This is obvious from Lemma 2(1).

$\left(1\right)\iff \left(3\right)$. Since $N^k=0$, we have

$$\begin{array}{cc}\hfill S+T^{-k}\tilde{T}N& =S+T^{1-k}\sum _{j=0}^{k-1}{T}^{j-1}S{N}^{k-j}\hfill \\ \hfill & =T^{1-k}\left[T^{k-1}S+\sum _{j=1}^{k-1}{T}^{j-1}S{N}^{k-j}+T^{-1}S{N}^k\right]=T^{1-k}\tilde{T}.\hfill \end{array}$$

(3)

Then, using (1)–(3), we easily have that

$$A{A}^{c,†}=U\left[\begin{array}{cc}{I}_t& T^{-k}\tilde{T}N{N}^{†}\\ 0& 0\end{array}\right]U^{*},$$

(4)

$$A^{c,†}A=U\left[\begin{array}{cc}{T}^{*}▵T& T^{*}▵T^{1-k}\tilde{T}\\ (I_{n-t}-N^{†}N)S^{*}▵T& (I_{n-t}-N^{†}N)S^{*}▵T^{1-k}\tilde{T}\end{array}\right]U^{*}.$$

(5)

Hence, $A\in {\mathbb{C}}_n^{\mathrm{CMP}}$, i.e., $A{A}^{c,†}=A^{c,†}A$, if and only if

$$\begin{array}{ccc}{T}^{*}▵T=I_t,& T^{-k}\tilde{T}N{N}^{†}=T^{*}▵T^{1-k}\tilde{T},& (I_{n-t}-N^{†}N)S^{*}▵T=0.\end{array}$$

(6)

Since T and $▵$ are nonsingular, it is easy to verify that

$$(I_{n-t}-N^{†}N)S^{*}▵T=0\iff S=S{N}^{†}N,$$

implying $▵={[T{T}^{*}+S(I_{n-t}-N^{†}N)S^{*}]}^{-1}={\left(T^{*}\right)}^{-1}{T}^{-1}$. Thus, if $S=S{N}^{†}N$, then $T^{*}▵T=I_t$ and

$$T^{-k}\tilde{T}N{N}^{†}=T^{*}▵T^{1-k}\tilde{T}\iff \tilde{T}N{N}^{†}=\tilde{T}.$$

Therefore, (6) is equivalent to $\tilde{T}N{N}^{†}=\tilde{T}$ and $S=S{N}^{†}N$.

$\left(4\right)\iff \left(3\right)$. Similarly, using (4) and (5), we have

$$\begin{array}{cc}\hfill A{A}^{c,†}-A^{c,†}A\in {\mathbb{C}}_n^{\mathrm{P}}& \iff {(A{A}^{c,†}-A^{c,†}A)}^2=A{A}^{c,†}-A^{c,†}A\hfill \\ & \iff 2A^{c,†}A=A{A}^{c,†}{A}^{c,†}A+A^{c,†}AA{A}^{c,†}\hfill \\ \hfill & \iff \left\{\begin{array}{c}{T}^{1-k}\tilde{T}N{N}^{†}(I_{n-t}-N^{†}N)S^{*}▵T=0,\hfill \\ T^{*}▵T^{1-k}\tilde{T}(N{N}^{†}-I_{n-t})+T^{-k}\tilde{T}N{N}^{†}(I_{n-t}-N^{†}N)S^{*}▵T^{1-k}\tilde{T}=0,\hfill \\ (I_{n-t}-N^{†}N)S^{*}▵T=0,\hfill \\ 2(I_{n-t}-N^{†}N)S^{*}▵T^{1-k}\tilde{T}=(I_{n-t}-N^{†}N)S^{*}▵T^{1-k}\tilde{T}{T}^{-k}\tilde{T}N{N}^{†}\hfill \end{array}\right.\hfill \\ & \iff \tilde{T}N{N}^{†}=\tilde{T},S=S{N}^{†}N.\hfill \end{array}$$

This completes the proof. □

Remark 1.

It is not difficult to obtain some relationships between the matrix classes from the CMP matrices. Let ${\mathbb{C}}_n^{k-\mathrm{EP}}=\{A\mid A\in {\mathbb{C}}_k^{n\times n},A^k{A}^{†}=A^{†}{A}^k\}$ and ${\mathbb{C}}_n^{k-\mathrm{CMP}}=\{A\mid A\in {\mathbb{C}}_k^{n\times n},A^k{A}^{c,†}=A^{c,†}{A}^k\}$. Given Theorem 1 and [8] (Theorems 3.10 and 3.21), we obtain ${\mathbb{C}}_n^{\mathrm{CMP}}={\mathbb{C}}_n^{k-\mathrm{EP}}={\mathbb{C}}_n^{k-\mathrm{CMP}}$ directly. It is obvious that ${\mathbb{C}}_n^{\mathrm{EP}}\subseteq {\mathbb{C}}_n^{k-\mathrm{EP}}$; thus, we have ${\mathbb{C}}_n^{\mathrm{EP}}\subseteq {\mathbb{C}}_n^{\mathrm{CMP}}$. Theorem 1(2) gives the characterization of the CMP matrices, which is shown that the CMP matrices are consistent with the core-EP matrices in [4] (Definition 3.1).

4. Characterizations of the Co-CMP Matrices

In this section, we show some characterizations of the co-CMP matrices. Firstly, the following theorem gives a necessary and sufficient condition of $A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$ in terms of the core-EP decomposition.

Theorem 2.

Let $A\in {\mathbb{C}}^{n\times n}$ be given by (1). The following statements are equivalent:

$\left(1\right)$

$A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$;

$\left(2\right)$

$A_1{A}^{†}-A^{†}{A}_1$ is nonsingular, where $A_1=A{A}^{D}A$;

$\left(3\right)$

$N=0$, and $T,S\in {\mathbb{C}}^{\frac{n}{2}\times \frac{n}{2}}$ are nonsingular,

in which case,

$${(A{A}^{c,†}-A^{c,†}A)}^{-1}=U\left[\begin{array}{cc}{I}_{\frac{n}{2}}& -{\left(S^{-1}T\right)}^{*}\\ -S^{-1}T& -I_{\frac{n}{2}}\end{array}\right]U^{*}.$$

(7)

Proof. 

$\left(1\right)\iff \left(2\right)$. This is obvious from Lemma 2(1).

$\left(1\right)\iff \left(3\right)$. Using (4) and (5), we have that

$$\begin{array}{cc}\hfill A{A}^{c,†}-A^{c,†}A& =U\left[\begin{array}{cc}{I}_t-T^{*}▵T& T^{-k}\tilde{T}N{N}^{†}-T^{*}▵T^{1-k}\tilde{T}\\ -(I_{n-t}-N^{†}N)S^{*}▵T& -(I_{n-t}-N^{†}N)S^{*}▵T^{1-k}\tilde{T}\end{array}\right]U^{*}\hfill \\ \hfill & =U\left[\begin{array}{cc}{I}_t& 0\\ 0& -(I_{n-t}-N^{†}N)\end{array}\right]\left[\begin{array}{cc}{I}_t-T^{*}▵T& T^{-k}\tilde{T}N{N}^{†}-T^{*}▵T^{1-k}\tilde{T}\\ S^{*}▵T& S^{*}▵T^{1-k}\tilde{T}\end{array}\right]U^{*},\hfill \end{array}$$

which implies that if $r(A{A}^{c,†}-A^{c,†}A)=n$, then $I_{n-t}-N^{†}N$ is nonsingular, i.e., $N=0$. Moreover, if $N=0$, then $\tilde{T}=T^{k-1}S$, $▵={(T{T}^{*}+S{S}^{*})}^{-1}$,

$$\begin{array}{cc}\hfill A{A}^{c,†}-A^{c,†}A& =U\left[\begin{array}{cc}{I}_t-T^{*}▵T& -T^{*}▵S\\ -S^{*}▵T& -S^{*}▵S\end{array}\right]U^{*}\hfill \\ \hfill & =U\left[\begin{array}{cc}{I}_t& 0\\ S^{*}{\left(T^{*}\right)}^{-1}& I_{n-t}\end{array}\right]\left[\begin{array}{cc}0& -T^{*}▵S\\ -S^{*}& 0\end{array}\right]\left[\begin{array}{cc}{\left(T^{*}\right)}^{-1}& 0\\ -S^{*}{\left(T^{*}\right)}^{-1}& I_{n-t}\end{array}\right]U^{*}.\hfill \end{array}$$

(8)

Therefore, $A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$ if and only if $N=0$, and $T,S\in {\mathbb{C}}^{\frac{n}{2}\times \frac{n}{2}}$ are nonsingular, in which case, from (8), it is easy to check that

$$U\left[\begin{array}{cc}{I}_{\frac{n}{2}}-T^{*}▵T& -T^{*}▵S\\ -S^{*}▵T& -S^{*}▵S\end{array}\right]\left[\begin{array}{cc}{I}_{\frac{n}{2}}& -{\left(S^{-1}T\right)}^{*}\\ -S^{-1}T& -I_{\frac{n}{2}}\end{array}\right]U^{*}=I_n,$$

which finishes the proof. □

Example 1.

Let $A=U\left[\begin{array}{cc}T& S\\ 0& N\end{array}\right]U^{*}$, where

$$\begin{array}{cc}\hfill T& =\left[\begin{array}{ccc}1& 1& 1\\ 1& 2& 1\\ 1& 0& 2\end{array}\right],S=\left[\begin{array}{ccc}2& 1& 1\\ 1& 1& 0\\ 0& 1& 2\end{array}\right],N=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right],\hfill \\ \hfill U& =\left[\begin{array}{cccccc}-0.48565& -0.41286& -0.54083& 0.29637& -0.46191& 0\\ -0.63339& -0.37922& 0.21457& -0.27421& 0.57774& 0\\ -0.10752& 0.31226& -0.4842& -0.8022& -0.11383& 0\\ -0.55742& 0.52966& 0.48088& 0.049989& -0.41832& 0\\ -0.2017& 0.5547& -0.44246& 0.43697& 0.5147& 0\\ 0& 0& 0& 0& 0& 1\end{array}\right].\hfill \end{array}$$

It is easy to verify that U is unitary, and both T and S are nonsingular, which, from Theorem 2, imply that $A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$ is nonsingular. Moreover,

$$\begin{array}{cc}\hfill & {(A{A}^{c,†}-A^{c,†}A)}^{-1}=\hfill \\ \hfill & \left[\begin{array}{cccccc}-1.587& 1.4305& 0.17633& -0.6998& 1.2869& -0.23263\\ 1.4305& 1.4205& -0.37319& -0.65265& 0.10218& -0.45073\\ 0.17633& -0.37319& -0.083255& -0.075269& 0.96838& 0.4737\\ -0.6998& -0.65265& -0.075269& 1.6175& -0.0059163& 0.3694\\ 1.2869& 0.10218& 0.96838& -0.0059163& -0.36771& 0.7022\\ -0.23263& -0.45073& 0.4737& 0.3694& 0.7022& -1\end{array}\right].\hfill \end{array}$$

Theorem 3.

Let $A\in {\mathbb{C}}^{n\times n}$ be given by (1). Then, $A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$ if and only if $A{\left(A^{c,†}\right)}^{2}A-A^{c,†}{A}^2{A}^{c,†}$ is nonsingular, in which case,

$${(A{\left(A^{c,†}\right)}^{2}A-A^{c,†}{A}^2{A}^{c,†})}^{-1}=U\left[\begin{array}{cc}0& -T^{-1}S-S^{-1}{T}^{*}\\ S^{-1}T+S^{*}{\left(T^{*}\right)}^{-1}& 0\end{array}\right]U^{*}.$$

(9)

Proof. 

If $A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$, applying Theorem 2 to (4) and (5), we have

$$I_n-A{A}^{c,†}-A^{c,†}A=U\left[\begin{array}{cc}-T^{*}▵T& -T^{*}▵S\\ -S^{*}▵T& I_{\frac{n}{2}}-S^{*}▵S\end{array}\right]U^{*},$$

in which case, it is easy to directly verify that

$${(I_n-A{A}^{c,†}-A^{c,†}A)}^{-1}=U\left[\begin{array}{cc}-I_{\frac{n}{2}}& -T^{-1}S\\ -S^{*}{\left(T^{*}\right)}^{-1}& I_{\frac{n}{2}}\end{array}\right]U^{*},$$

(10)

implying that $I_n-A{A}^{c,†}-A^{c,†}A$ is nonsingular. Note that

$$A{\left(A^{c,†}\right)}^{2}A-A^{c,†}{A}^2{A}^{c,†}=(I_n-A{A}^{c,†}-A^{c,†}A)(A{A}^{c,†}-A^{c,†}A).$$

(11)

Hence, $A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$ if and only if $A{\left(A^{c,†}\right)}^{2}A-A^{c,†}{A}^2{A}^{c,†}$ is nonsingular. Moreover, it follows from (9)–(11) that

$$\begin{array}{cc}\hfill {(A{\left(A^{c,†}\right)}^{2}A-A^{c,†}{A}^2{A}^{c,†})}^{-1}=& {(A{A}^{c,†}-A^{c,†}A)}^{-1}{(I_n-A{A}^{c,†}-A^{c,†}A)}^{-1}\hfill \\ \hfill =& U\left[\begin{array}{cc}{I}_{\frac{n}{2}}& -{\left(S^{-1}T\right)}^{*}\\ -S^{-1}T& -I_{\frac{n}{2}}\end{array}\right]\left[\begin{array}{cc}-I_{\frac{n}{2}}& -T^{-1}S\\ -S^{*}{\left(T^{*}\right)}^{-1}& I_{\frac{n}{2}}\end{array}\right]U^{*}\hfill \\ \hfill =& U\left[\begin{array}{cc}0& -T^{-1}S-S^{-1}{T}^{*}\\ S^{-1}T+S^{*}{\left(T^{*}\right)}^{-1}& 0\end{array}\right]U^{*},\hfill \end{array}$$

which completes the proof. □

Zuo [11], Groß et al. [12] and Koliha et al. [13] established some results on the nonsingularity of the difference and sum of two idempotent matrices by the range and null spaces. Based on their work, the following theorem further characterizes the co-CMP matrices in a similar way. Moreover, for $A\in {\mathbb{C}}^{m\times n}$ and a subspace $\mathcal{W}\subseteq {\mathbb{C}}^{n\times 1}$, we denote $A\mathcal{W}=\{Ax\mid x\in \mathcal{W}\}$.

Theorem 4.

Let $A\in {\mathbb{C}}_k^{n\times n}$. Then, the following statements are equivalent:

$\left(1\right)$

$A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$;

$\left(2\right)$

$\mathcal{R}\left(A^k\right)\cap \mathcal{R}\left(A^{†}{A}^k\right)=\left\{0\right\}$ and $\mathcal{N}\left(A^k\right)\cap \mathcal{N}\left(A^k{A}^{†}\right)=\left\{0\right\}$;

$\left(3\right)$

$A{A}^D\mathcal{N}\left(A^{*}\right)\oplus (I_n-A^D{A}^2{A}^{†})\mathcal{R}\left(A^{†}{A}^k\right)={\mathbb{C}}^{n\times 1}$;

$\left(4\right)$

$\mathcal{N}\left(A^k{A}^{†}(I_n-A^{†}{A}^2{A}^D)\right)\oplus \mathcal{N}\left((I_n-A^{†}A)A{A}^D\right)={\mathbb{C}}^{n\times 1}$.

Proof. 

$\left(1\right)\iff \left(2\right)$. It is a direct corollary of Lemma 2(2) and Lemma 3(1),(3).

$\left(1\right)\iff \left(3\right)$. Using $A{A}^D=P_{\mathcal{R}\left(A^k\right),\mathcal{N}\left(A^k\right)}$ and $A{A}^{†}=P_{\mathcal{R}\left(A\right),\mathcal{N}\left(A^{*}\right)}$ from Lemma 2(1),(2), we have that

$$\begin{array}{cc}\hfill \mathcal{R}\left(A{A}^{c,†}(I_n-A^{c,†}A)\right)& =A{A}^{c,†}\mathcal{R}(I_n-A{A}^D)=\mathcal{R}(A{A}^{c,†}-A{A}^D)\hfill \\ \hfill =\mathcal{R}(A{A}^{D}A{A}^{†}-A{A}^D)& =\mathcal{R}\left(A{A}^D(I_n-A{A}^{†})\right)=A{A}^D\mathcal{N}\left(A^{*}\right)\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill \mathcal{R}\left((I_n-A{A}^{c,†})A^{c,†}A\right)& =\mathcal{R}\left((I_n-A{A}^{c,†})A^{†}{A}^k\right)\hfill \\ \hfill & =\mathcal{R}(A^{†}{A}^k-A{A}^{D}A{A}^{†}{A}^{†}{A}^k)=(I_n-A^D{A}^2{A}^{†})\mathcal{R}\left(A^{†}{A}^k\right),\hfill \end{array}$$

(13)

which, together with Lemma 3(1),(4), implies that $\left(1\right)$ is equivalent to $\left(3\right)$.

$\left(1\right)\iff \left(4\right)$. Using Lemma 2(2), i.e., $A{A}^{c,†}=P_{\mathcal{R}\left(A^k\right),\mathcal{N}\left(A^k{A}^{†}\right)}$ and $A^{c,†}A=P_{\mathcal{R}\left(A^{†}{A}^k\right),\mathcal{N}\left(A^k\right)}$, we obtain that

$${\left(A^{c,†}A\right)}^{*}=P_{{\mathcal{N}}^{\perp }\left(A^k\right),{\mathcal{R}}^{\perp }\left(A^{†}{A}^k\right)}=P_{\mathcal{R}\left({\left(A^k\right)}^{*}\right),\mathcal{N}\left({\left(A^k\right)}^{*}{\left(A^{†}\right)}^{*}\right)}=P_{\mathcal{R}\left({\left(A^{*}\right)}^k\right),\mathcal{N}\left({\left(A^{*}\right)}^k{\left(A^{*}\right)}^{†}\right)}=A^{*}{\left(A^{*}\right)}^{c,†},$$

$${\left(A{A}^{c,†}\right)}^{*}=P_{{\mathcal{N}}^{\perp }\left(A^k{A}^{†}\right),{\mathcal{R}}^{\perp }\left(A^k\right)}=P_{\mathcal{R}\left({\left(A^{†}\right)}^{*}{\left(A^k\right)}^{*}\right),\mathcal{N}\left({\left(A^k\right)}^{*}\right)}=P_{\mathcal{R}\left({\left(A^{*}\right)}^{†}{\left(A^{*}\right)}^k\right),\mathcal{N}\left({\left(A^{*}\right)}^k\right)}={\left(A^{*}\right)}^{c,†}{A}^{*}.$$

From the above conclusions, (12) and (13), we have

$$\begin{array}{cc}\hfill \mathcal{R}\left({(I_n-A^{c,†}A)}^{*}{\left(A{A}^{c,†}\right)}^{*}\right)& =\mathcal{R}\left((I_n-A^{*}{\left(A^{*}\right)}^{c,†}){\left(A^{*}\right)}^{c,†}{A}^{*}\right)\hfill \\ & =(I_n-{\left(A^{*}\right)}^D{\left(A^{*}\right)}^2{\left(A^{*}\right)}^{†})\mathcal{R}\left({\left(A^{*}\right)}^{†}{\left(A^{*}\right)}^k\right),\hfill \\ \hfill \mathcal{R}\left({\left(A^{c,†}A\right)}^{*}{(I_n-A{A}^{c,†})}^{*}\right)& =\mathcal{R}\left(A^{*}{\left(A^{*}\right)}^{c,†}(I_n-{\left(A^{*}\right)}^{c,†}{A}^{*})\right)\hfill \\ & =A^{*}{\left(A^{*}\right)}^D\mathcal{N}\left(A\right).\hfill \end{array}$$

Thus, we have that

$$\begin{array}{cc}\hfill \mathcal{N}\left(A{A}^{c,†}(I_n-A^{c,†}A)\right)& ={\mathcal{R}}^{\perp }\left({(I_n-A^{c,†}A)}^{*}{\left(A{A}^{c,†}\right)}^{*}\right)\hfill \\ \hfill & ={\left((I_n-{\left(A^{*}\right)}^D{\left(A^{*}\right)}^2{\left(A^{*}\right)}^{†})\mathcal{R}\left({\left(A^{*}\right)}^{†}{\left(A^{*}\right)}^k\right)\right)}^{\perp }\hfill \\ \hfill & =\mathcal{N}\left(A^k{A}^{†}(I_n-A^{†}{A}^2{A}^D)\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathcal{N}\left((I_n-A{A}^{c,†})A^{c,†}A\right)& ={\mathcal{R}}^{\perp }(\left(A^{*}{\left(A^{*}\right)}^{c,†}(I_n-{\left(A^{*}\right)}^{c,†}{A}^{*})\right)={\left(A^{*}{\left(A^{*}\right)}^D\mathcal{N}\left(A\right)\right)}^{\perp }\hfill \\ \hfill & =\mathcal{N}\left((I_n-A^{†}A)A{A}^D\right),\hfill \end{array}$$

(14)

which, together with Lemma 3(1),(5), implies that $\left(1\right)$ is equivalent to $\left(4\right)$. □

Zuo et al. [15] (Lemma 2.1) established an important assertion for a co-EP matrix; that is, for $A\in {\mathbb{C}}^{n\times n}$ and $r=r\left(A\right)$, it follows that $A\in {\mathbb{C}}_n^{co-\mathrm{EP}}$ if and only if there exist a unitary matrix $V\in {\mathbb{C}}^{n\times n}$, $W\in {\mathbb{C}}^{r\times (n-r)}$ and a nonsingular matrix $R\in {\mathbb{C}}^{r\times r}$ such that

$$A=V\left[\begin{array}{cc}W& R\\ 0& 0\end{array}\right]V^{*}.$$

(15)

In view of the above conclusion and Theorem 2, it is interesting to remark that

$$A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}\Rightarrow A\in {\mathbb{C}}_n^{co-\mathrm{EP}},$$

but its converse is not necessarily true, which will be illustrated using a simple example.

Example 2.

Let $A=\left[\begin{array}{cc}1& 1\\ -1& -1\end{array}\right].$ Then $Ind\left(A\right)=2$, $A^{†}=\left[\begin{array}{cc}\frac{1}{4}& \frac{-1}{4}\\ \frac{1}{4}& \frac{-1}{4}\end{array}\right]$ and $A^{c,†}=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$. Thus, $A{A}^{†}-A^{†}A=\left[\begin{array}{cc}0& -1\\ -1& 0\end{array}\right]$ and $A{A}^{c,†}-A^{c,†}A=\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right],$ so $A\in {\mathbb{C}}_n^{co-\mathrm{EP}}$ but $A\notin {\mathbb{C}}_n^{co-\mathrm{CMP}}$.

The following theorem gives a relationship among the co-EP, co-CMP, and core matrices.

Theorem 5.

${\mathbb{C}}_n^{co-\mathrm{CMP}}={\mathbb{C}}_n^{co-\mathrm{EP}}\cap {\mathbb{C}}_n^{\mathrm{CM}}.$

Proof. 

Since N is nilpotent, using (1), we have

$$\begin{array}{cc}\hfill A\in {\mathbb{C}}_n^{\mathrm{CM}}& \iff r\left(A^2\right)=r\left(A\right)\iff r\left(\left[\begin{array}{cc}{T}^2& TS+SN\\ 0& N^2\end{array}\right]\right)=r\left(\left[\begin{array}{cc}T& S\\ 0& N\end{array}\right]\right)\hfill \\ \hfill & \iff r\left(N^2\right)=r\left(N\right)\iff N=0.\hfill \end{array}$$

Then, it follows from [15] (Lemma 2.1) and Theorem 2 that ${\mathbb{C}}_n^{co-\mathrm{CMP}}\subseteq {\mathbb{C}}_n^{co-\mathrm{EP}}\cap {\mathbb{C}}_n^{\mathrm{CM}}$. Conversely, put $A\in {\mathbb{C}}_n^{co-\mathrm{EP}}\cap {\mathbb{C}}_n^{\mathrm{CM}}$. Then, using (15), from the nonsingularity of R, we have

$$\begin{array}{cc}\hfill r\left(A\right)=r\left(A^2\right)& \iff r\left(\left[\begin{array}{cc}W& R\\ 0& 0\end{array}\right]\right)=r\left(\left[\begin{array}{cc}{W}^2& WR\\ 0& 0\end{array}\right]\right)\iff r\left(W\right)=r\left(R\right),\hfill \end{array}$$

which implies that $W,R\in {\mathbb{C}}^{\frac{n}{2}\times \frac{n}{2}}$ are nonsingular. Clearly, $A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$ per Theorem 2. □

In particular, the following corollary provides a necessary and sufficient condition for an idempotent matrix to be co-CMP by directly applying [15] (Theorem 2.9) and Theorem 5.

Corollary 1.

Let $A\in {\mathbb{C}}_n^{\mathrm{P}}$. Then, $A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$ if and only if $r\left(A\right)=\frac{n}{2}$ and $r(A-A^{†}A)=r\left(A\right)$.

5. The Nonsingularity of $\mathit{A}{\mathit{A}}^{\mathit{c}\mathbf{,}\mathbf{†}}\mathbf{+}{\mathit{A}}^{\mathit{c}\mathbf{,}\mathbf{†}}\mathit{A}$ and $\mathit{I}\mathbf{-}\mathit{A}{\mathbf{\left(}{\mathit{A}}^{\mathit{c}\mathbf{,}\mathbf{†}}\mathbf{\right)}}^{\mathbf{2}}\mathit{A}$

An interesting result proved by Groß and Trenkler [13] (Corollary 5) reveals a close relationship among the nonsingularity of $P-Q$, $P+Q$ and $I_n-PQ$ for $P,Q\in {\mathbb{C}}_n^{\mathrm{P}}$, which was given in Lemma 3(1),(2). Note that $A{A}^{c,†}$ and $A^{c,†}A$ are two special idempotent matrices. In Section 4, we characterized the nonsingularity of $A{A}^{c,†}-A^{c,†}A$ and gave its inverse, so this section mainly shows a few characterizations of the nonsingularity of $A{A}^{c,†}+A^{c,†}A$ and $I_n-A{\left(A^{c,†}\right)}^{2}A$ and their inverses, which is essentially an enrichment of the characterization of co-CMP matrices.

Theorem 6.

Let $A\in {\mathbb{C}}^{n\times n}$ be given by (1). Then, $A{A}^{c,†}+A^{c,†}A$ is nonsingular if and only if $N=0$ and $r\left(S\right)=n-t$, in which case,

$${(A{A}^{c,†}+A^{c,†}A)}^{-1}=U\left[\begin{array}{cc}\frac{1}{2}{I}_t+\frac{1}{2}{T}^{-1}S{(S^{*}▵S)}^{-1}{S}^{*}▵T& T^{-1}S-T^{-1}S{(S^{*}▵S)}^{-1}\\ -{(S^{*}▵S)}^{-1}{S}^{*}▵T& 2{(S^{*}▵S)}^{-1}-I_{n-t}\end{array}\right]U^{*}.$$

Proof. 

Applying (4) and (5), we have

$$A{A}^{c,†}+A^{c,†}A=U\left[\begin{array}{cc}{I}_t+T^{*}▵T& T^{-k}\tilde{T}N{N}^{†}+T^{*}▵T^{1-k}\tilde{T}\\ (I_{n-t}-N^{†}N)S^{*}▵T& (I_{n-t}-N^{†}N)S^{*}▵T^{1-k}\tilde{T}\end{array}\right]U^{*},$$

(16)

implying that $N=0$ if $A{A}^{c,†}+A^{c,†}A$ is nonsingular. Then, if $N=0$; then $\tilde{T}=T^{k-1}S$, $▵={(T{T}^{*}+S{S}^{*})}^{-1}$ and

$$\begin{array}{cc}\hfill & A{A}^{c,†}+A^{c,†}A=U\left[\begin{array}{cc}{I}_t+T^{*}▵T& T^{*}▵S\\ S^{*}▵T& S^{*}▵S\end{array}\right]U^{*}\hfill \\ \hfill & =U\left[\begin{array}{cc}{I}_t& 0\\ S^{*}{\left(T^{*}\right)}^{-1}& I_{n-t}\end{array}\right]\left[\begin{array}{cc}{I}_t& 0\\ \frac{-1}{2}{S}^{*}{\left(T^{*}\right)}^{-1}& I_{n-t}\end{array}\right]\left[\begin{array}{cc}2T^{*}& T^{*}▵S\\ 0& \frac{1}{2}{S}^{*}▵S\end{array}\right]\left[\begin{array}{cc}{\left(T^{*}\right)}^{-1}& 0\\ -S^{*}{\left(T^{*}\right)}^{-1}& I_{n-t}\end{array}\right]U^{*}.\hfill \end{array}$$

(17)

Applying the Cholesky decomposition to $▵={(T{T}^{*}+S{S}^{*})}^{-1}$ shows that there exists a lower triangular matrix with positive elements on the diagonal such that $▵=L^{*}L$, which implies that

$$r(S^{*}▵S)=r\left({\left(LS\right)}^{*}LS\right)=r\left(LS\right)=r\left(S\right),$$

showing that $S^{*}▵S$ is nonsingular if and only if $r\left(S\right)=n-r$. Hence, we can conclude that $A{A}^{c,†}+A^{c,†}A$ is nonsingular if and only if $N=0$ and $r\left(S\right)=n-t$. In this case, it follows from (17) that

$$\begin{array}{cc}\hfill {(A{A}^{c,†}+A^{c,†}A)}^{-1}=U& \left[\begin{array}{cc}{T}^{*}& 0\\ S^{*}& I_{n-t}\end{array}\right]\left[\begin{array}{cc}\frac{1}{2}{\left(T^{*}\right)}^{-1}& 2▵S{(S^{*}▵S)}^{-1}\\ 0& 2{(S^{*}▵S)}^{-1}\end{array}\right]\hfill \\ \hfill & \left[\begin{array}{cc}{I}_t& 0\\ \frac{1}{2}{S}^{*}{\left(T^{*}\right)}^{-1}& I_{n-t}\end{array}\right]\left[\begin{array}{cc}{I}_t& 0\\ -S^{*}{\left(T^{*}\right)}^{-1}& I_{n-t}\end{array}\right]U^{*}\hfill \\ \hfill =U& \left[\begin{array}{cc}\frac{1}{2}{I}_t+\frac{1}{2}{T}^{-1}S{(S^{*}▵S)}^{-1}{S}^{*}▵T& T^{-1}S-T^{-1}S{(S^{*}▵S)}^{-1}\\ -{(S^{*}▵S)}^{-1}{S}^{*}▵T& 2{(S^{*}▵S)}^{-1}-I_{n-t}\end{array}\right]U^{*}.\hfill \end{array}$$

This completes the proof. □

Example 3.

Let $A=U\left[\begin{array}{cc}T& S\\ 0& N\end{array}\right]U^{*}$, where

$$\begin{array}{cc}\hfill T& =\left[\begin{array}{ccc}1& 0& 0\\ 0& 2& 1\\ 0& 1& 2\end{array}\right],S=\left[\begin{array}{cc}0& 1\\ 1& 1\\ 0& 0\end{array}\right],N=\left[\begin{array}{cc}0& 0\\ 0& 0\\ 0& 0\end{array}\right],\hfill \\ \hfill U& =\left[\begin{array}{ccccc}-0.15066& 0.52044& -0.22514& -0.80979& 0\\ -0.61339& -0.62916& 0.29789& -0.37305& 0\\ -0.68685& 0.15686& -0.59097& 0.39291& 0\\ -0.35956& 0.5556& 0.71507& 0.22516& 0\\ 0& 0& 0& 0& 1\end{array}\right].\hfill \end{array}$$

It is easy to verify that U is unitary, T is nonsingular, and $r\left(S\right)=2$. Then, from Theorem 6, we see that $A{A}^{c,†}+A^{c,†}A$ is nonsingular and

$$\begin{array}{c}\hfill {(A{A}^{c,†}+A^{c,†}A)}^{-1}=\left[\begin{array}{ccccc}6.146& 1.7721& -1.1365& -0.75633& 1.7702\\ 1.7721& 1.5402& -0.23965& -0.16883& 1.3595\\ -1.1365& -0.23965& 0.79314& 0.18744& -0.098965\\ -0.75633& -0.16883& 0.18744& 0.62064& -0.090768\\ 1.7702& 1.3595& -0.098965& -0.090768& 3\end{array}\right].\end{array}$$

Theorem 7.

Let $A\in {\mathbb{C}}^{n\times n}$. Then, the following statements are equivalent:

$\left(1\right)$

$A{A}^{c,†}+A^{c,†}A$ is nonsingular;

$\left(2\right)$

$A^{†}{A}^2{A}^D\mathcal{N}\left(A^{*}\right)\cap \mathcal{R}\left(A^k\right)=\left\{0\right\}$ and $\mathcal{N}\left((I_n-A^{†}A)A{A}^D\right)\cap \mathcal{N}\left(A^k{A}^{†}\right)=\left\{0\right\}$;

$\left(3\right)$

$A^{†}{A}^2{A}^D\mathcal{N}\left(A^{*}\right)+\mathcal{R}\left(A^k\right)={\mathbb{C}}^{n\times 1}$ and $\mathcal{N}\left((I_n-A^{†}A)A{A}^D\right)+\mathcal{N}\left(A^k{A}^{†}\right)={\mathbb{C}}^{n\times 1}$.

Proof. 

Using $A{A}^{†}=P_{\mathcal{R}\left(A\right),\mathcal{N}\left(A^{*}\right)}$ and Lemma 2(1), we obtain

$$\begin{array}{cc}\hfill \mathcal{R}\left(A^{c,†}A(I_n-A{A}^{c,†})\right)& =\mathcal{R}\left(A^{†}A{A}^{D}A{A}^{†}A(I_n-A{A}^{†}A{A}^{D}A{A}^{†})\right)\hfill \\ \hfill & =\mathcal{R}\left(A^{†}A{A}^{D}A(I_n-A{A}^{D}A{A}^{†})\right)\hfill \\ \hfill & =\mathcal{R}(A^{†}A{A}^{D}A-A^{†}AA{A}^{D}A{A}^{D}A{A}^{†})\hfill \\ \hfill & =\mathcal{R}(A^{†}{A}^2{A}^D-A^{†}{A}^2{A}^{D}A{A}^{†})\hfill \\ \hfill & =A^{†}{A}^2{A}^D\mathcal{R}(I_n-A{A}^{†})\hfill \\ \hfill & =A^{†}{A}^2{A}^D\mathcal{N}\left(A{A}^{†}\right)=A^{†}{A}^2{A}^D\mathcal{N}\left(A^{*}\right).\hfill \end{array}$$

(18)

From (14), (18), and Lemma 4, the proof is finished. □

Theorem 8.

Let $A\in {\mathbb{C}}^{n\times n}$ be written as in (1). Then, $I_n-A{\left(A^{c,†}\right)}^{2}A$ is nonsingular if and only if $r\left(S\right)=t$ and $N=0$, in which case

$${(I_n-A{\left(A^{c,†}\right)}^{2}A)}^{-1}=U\left[\begin{array}{cc}{I}_t+T^{*}{\left(S{S}^{*}\right)}^{-1}T& T^{*}{\left(S^{*}\right)}^{-1}\\ 0& I_{n-t}\end{array}\right]U^{*}.$$

(19)

Proof. 

Using (4) and (5), we can directly obtain that

$$I_n-A{\left(A^{c,†}\right)}^{2}A=U\left[\begin{array}{cc}{T}^{-1}H▵T& T^{-1}(H▵-I_t)T^{1-k}\tilde{T}\\ 0& I_{n-t}\end{array}\right]U^{*},$$

where $H=(S-T^{1-k}\tilde{T}N{N}^{†})(I_{n-t}-N^{†}N)S^{*}$. Thus, $I_n-A{\left(A^{c,†}\right)}^{2}A$ is nonsingular if and only if

$$\begin{array}{cc}\hfill r(T^{-1}H▵T)=t& \iff r\left(H\right)=t\iff N=0,r\left(S\right)=t,\hfill \end{array}$$

in which case, $H=S{S}^{*}$ and

$$\begin{array}{cc}\hfill & (I_n-A{\left(A^{c,†}\right)}^{2}A)U\left[\begin{array}{cc}{I}_t+T^{*}{\left(S{S}^{*}\right)}^{-1}T& T^{*}{\left(S^{*}\right)}^{-1}\\ 0& I_{n-t}\end{array}\right]U^{*}\hfill \\ \hfill =& U\left[\begin{array}{cc}{T}^{-1}S{S}^{*}▵T& -T^{*}▵S\\ 0& I_{n-t}\end{array}\right]\left[\begin{array}{cc}{I}_t+T^{*}{\left(S{S}^{*}\right)}^{-1}T& T^{*}{\left(S^{*}\right)}^{-1}\\ 0& I_{n-t}\end{array}\right]U^{*}=I_n,\hfill \end{array}$$

which completes the proof. □

Remark 2.

In view of Theorems 6 and 8, we can conclude that $A\in {\mathbb{C}}_n^{co-\mathrm{CMP}}$ if and only if $N=0$ and $T,S\in {\mathbb{C}}^{\frac{n}{2}\times \frac{n}{2}}$ are nonsingular, which gives another method to prove Theorem 2.

Example 4.

Let $A=U\left[\begin{array}{cc}T& S\\ 0& N\end{array}\right]U^{*}$, where

$$\begin{array}{cc}\hfill T& =\left[\begin{array}{ccc}1& 0& 1\\ 0& 2& 0\\ 0& 1& 2\end{array}\right],S=\left[\begin{array}{cccc}0& 0& 1& 1\\ 0& 1& 0& 1\\ 1& 0& 0& 0\end{array}\right],N=\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right],\hfill \\ \hfill U& =\left[\begin{array}{ccccccc}-0.43271& 0.41754& 0.36924& 0.25828& -0.14448& 0.64382& 0\\ -0.55299& -0.45425& -0.2462& -0.56474& 0.10075& 0.3133& 0\\ -0.26717& 0.51191& -0.66918& -0.045914& -0.41872& -0.20332& 0\\ -0.18275& 0.51063& 0.37517& -0.53299& 0.37949& -0.37017& 0\\ -0.45718& -0.055044& -0.20828& 0.55927& 0.61206& -0.23912& 0\\ -0.43952& -0.30577& 0.41367& 0.12405& -0.52443& -0.50179& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right].\hfill \end{array}$$

It is easy to verify that U is unitary, T is nonsingular, and $r\left(S\right)=3$. From Theorem 8, we have that $I_n-A{\left(A^{c,†}\right)}^{2}A$ is nonsingular and

$$\begin{array}{cc}\hfill & {(I_n-A{\left(A^{c,†}\right)}^{2}A)}^{-1}=\hfill \\ \hfill & \left[\begin{array}{ccccccc}2.8479& -2.0582& -0.29333& 1.708& 0.68151& 0.25909& 0.2572\\ -1.7066& 2.8165& 0.88261& -1.1235& -0.045289& -0.10444& -0.56923\\ -1.0423& 0.91478& 3.4232& 0.96324& 1.3002& -1.154& 0.029156\\ 2.0861& -2.3009& -0.48509& 2.8195& 0.61202& 0.22549& 0.40456\\ -0.80064& 0.7906& 0.96499& -0.063071& 1.4013& -0.25836& -0.25851\\ 0.57035& -0.55662& -0.90915& -0.21248& -0.3234& 1.6916& -0.21246\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right].\hfill \end{array}$$

6. Conclusions

The purpose of this paper is to provide different characterizations of the CMP and co-CMP matrices via the core-EP decomposition and subspace operationsand to show the inverses of $A{A}^{c,†}-A^{c,†}A$, $A{A}^{c,†}+A^{c,†}A$, and $I-A{\left(A^{c,†}\right)}^{2}A$. We are convinced that research on the CMP and co-CMP matrices will remain popular for years to come. Some perspectives for further study are proposed as follows:

$\left(1\right)$

Benítez et al. [16] considered when the limit of an EP (resp. co-EP) matrix sequence still satisfies the EP (resp. co-EP) property. So, we can discuss the continuity of the CMP and co-CMP matrices.

$\left(2\right)$

Also, it is possible to further characterize the co-CMP matrices using the Hartwig–Spindelböck decomposition [21], which is also a powerful tool for studying generalized inverses.

$\left(3\right)$

The applications of CMP and co-CMP matrices to mathematics and physics are also worth studying.