Parametrizing W-weighted BT inverse to obtain the W-weighted q-BT inverse

Abstract

The core-EP and BT inverses for rectangular matrices were studied recently in the literature. The main aim of this paper is to unify both concepts by means of a new kind of generalized inverse called W-weighted q-BT inverse. We analyze its existence and uniqueness by considering an adequate matrix system. Basic properties and some interesting characterizations are proved for this new weighted generalized inverse. Also, we give a canonical form of the W-weighted q-BT inverse by means of the weighted core-EP decomposition.

1 Introduction and preliminaries

We denote by $\mathbb {C}^{m\times n}$ the set of all $m\times n$ complex matrices. Let $A\in \mathbb {C}^{m\times n}$. The conjugate transpose, rank, null space and column space of A are denoted by $A^*$, ${\text {rank}}(A)$, ${\mathscr {N}}(A)$, and ${\mathscr {R}}(A)$, respectively. The index of $A\in {\mathbb {C}^{n\times n}}$, denoted by ${\text {Ind}}(A)$, is the smallest nonnegative integer k such that ${\text {rank}}(A^k) = {\text {rank}}(A^{k+1})$. Moreover, $A^0=I_n$ will refer to the $n \times n$ identity matrix, and 0 will denote the null matrix of appropriate size. The standard notations $P_S$ and $P_{S,T}$ stand for the orthogonal projector onto a subspace S and a projector onto S along T, respectively, when $\mathbb {C}^n$ is equal to the direct sum of subspaces S and T.

The Drazin inverse of a matrix $A\in {\mathbb {C}^{n\times n}}$ is the unique matrix $X=A^d \in {\mathbb {C}^{n\times n}}$ that satisfies

$$\begin{aligned} X A^{k+1} =A^k, \quad XAX=X, \quad AX=XA, ~\text {where} ~k={\text {Ind}}(A). \end{aligned}$$

When ${\text {Ind}}(A)=1$, the Drazin inverse is called the group inverse of A and is denoted by $A^\#$.

The Moore–Penrose inverse of a matrix $A\in {\mathbb {C}^{m\times n}}$ is the unique matrix $X=A^\dag \in {\mathbb {C}^{n\times m}}$ that satisfies the Penrose equations

$$\begin{aligned} AXA=A, \quad XAX=X, \quad (AX)^*=AX, \quad (XA)^*=XA. \end{aligned}$$

We will denote by $P_A$ the orthogonal projector $AA^{\dag }$ onto the subspace ${\mathscr {R}}(A)$.

In 2014, Manjunatha Prassad and Mohana [13] introduced the core-EP inverse of a matrix $A\in {\mathbb {C}^{n\times n}}$ of index k as the unique matrix $X=A^{\textcircled {\dag }}\in {\mathbb {C}^{n\times n}}$ that satisfies the conditions $XAX=X$ and ${\mathscr {R}}(X)={\mathscr {R}}(X^*)={\mathscr {R}}(A^k)$. That same year, Baksalary and Trenkler [2] defined the BT inverse of A as the matrix $A^{\diamond }=(AP_A)^\dag$. When the matrix A has index 1, both inverses are reduced to the well-known core inverse $A^{\textcircled {\#}}=A^\#AA^\dag$ of A [1].

In 1980, Cline and Greville [4] extended the Drazin inverse to rectangular matrices and it was called the W-weighted Drazin inverse. Let $W\in {\mathbb {C}^{n\times m}}$ be a fixed nonzero matrix. We recall that the W-weighted Drazin inverse of $A\in {\mathbb {C}^{m\times n}}$, denoted by $A^{d,W}$, is the unique matrix $X\in {\mathbb {C}^{m\times n}}$ satisfying the three equations

$$\begin{aligned} XWAWX=X, \quad AWX=XWA, \quad XW(AW)^{k+1}=(AW)^k, \end{aligned}$$

where $k=\max \{{\text {Ind}}(AW),{\text {Ind}}(WA)\}$. If $k=1$, the W-weighted Drazin inverse of A is called the W-weighted group inverse of A and is denoted by $A^{\#,W}$. When $m=n$ and $W=I_n$, we recover the Drazin inverse, that is, $A^{d,I_n}=A^d$.

The W-weighted Drazin inverse satisfies the following two dual representations

$$\begin{aligned} A^{d,W}=A[(WA)^d]^2=[(AW)^d]^2A,\quad \text {whence} \quad A^{d,W}W=(AW)^d, \quad WA^{d,W}=(WA)^d. \end{aligned}$$

(1.1)

Interesting representations and properties of the W-weighted Drazin inverse were studied in [17].

Similarly, the core-EP inverse was extended to rectangular matrices in [5]. It was named W-weighted core-EP inverse, and defined as $A^{\textcircled {\dag },W}=(WAWP_{(AW)^k})^\dag$, which is the unique solution of

$$\begin{aligned} WAWX=P_{(WA)^k}, \quad {\mathscr {R}}(X)\subseteq {\mathscr {R}}((AW)^k). \end{aligned}$$

(1.2)

For the particular case $k=1$, the W-weighted core-EP inverse of A is known as the W-weighted core inverse of A and denoted by $A^{\textcircled {\#},W}$. Clearly, when $m=n$ and $W=I_n$, we recover the core-EP inverse, that is, $A^{\textcircled {\dag },I_n}=A^{\textcircled {\dag }}$.

The W-weighted core-EP inverse satisfies the following interesting properties [5, 12]

$$\begin{aligned} A^{\textcircled {\dag },W}=A[(WA)^{\textcircled {\dag }}]^2, \quad A^{\textcircled {\dag },W}WP_{(AW)^k}=(AW)^{\textcircled {\dag }}, \quad P_{(WA)^k}WA^{\textcircled {\dag },W}=(WA)^{\textcircled {\dag }}. \end{aligned}$$

(1.3)

Recently, the W-weighted BT inverse of A was defined in [10] as the unique matrix $X=A^{\diamond , W} \in {\mathbb {C}^{m\times n}}$ satisfying the following equations

$$\begin{aligned} XWAWX=X, \quad XWA=[W(AW)^2(AW)^\dag ]^\dag WA, \quad AWX=AW[(WA)^2W(AW)^\dag ]^\dag . \end{aligned}$$

(1.4)

It was also established that $A^{\diamond , W}=(WAWP_{AW})^\dag$.

Interesting results including different kinds of weighted generalized inverses can be found in [14,15,16].

In this paper we unify the definitions given in (1.2) and (1.4) given rise a new kind of generalized inverse called W-weighted q-BT inverse. We analyze its existence and uniqueness by considering an adequate matrix system.

This paper is organized as follows. In Sect. 2, we present results of existence and uniqueness of the W-weighted q-BT inverse. More precisely, the existence will be characterized as the unique solution of three matrix equations. In Sect. 3, we obtain some characterizations of the W-weighted q-BT inverse. As an interesting consequence, we present new characterizations of the W-weighted core-EP and W-weighted BT inverses. In Sect. 4, we obtain a canonical form of the W-weighted q-BT inverse by using a simultaneous decomposition of the matrices A and W called the weighted core-EP decomposition. Finally, some more properties of this new generalized inverse are investigated.

2 Existence and uniqueness

In this section, we define and investigate the W-weighted q-BT inverse for rectangular matrices $A\in {\mathbb {C}^{m\times n}}$ by considering a non-null weight $W\in {\mathbb {C}^{n\times m}}$.

We start with a result of existence and uniqueness. Before that, we need the following auxiliary lemma.

Lemma 2.1

Let $A\in {\mathbb {C}^{m\times n}}$ and $B\in \mathbb {C}^{n\times s}$. Then $P_B(AP_B)^\dag =(AP_B)^\dag$.

Proof

Since $(I_n-P_B)P_B A^*=0$ trivially holds, we have that ${\mathscr {R}}((AP_B)^\dag )\subseteq {\mathscr {N}}((I_n-P_B))$ is always valid, which in turn is equivalent to $P_B(AP_B)^\dag =(AP_B)^\dag$. $\square$

Theorem 2.2

Let $A\in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$ and $q\in \mathbb {N} \cup \{0\}$. The system of equations

$$\begin{aligned} XWAWX=X, \quad XWA=(WAWP_{(AW)^q})^\dag WA, \quad AWX=AW(WAWP_{(AW)^q})^\dag , \end{aligned}$$

(2.1)

is consistent and has a unique solution $X=(WAWP_{(AW)^q})^\dag$.

Proof

Existence. Let $X:=(WAWP_{(AW)^q})^\dag$. Clearly, X satisfies the two last equations in (2.1). Moreover, from Lemma 2.1 we have

$$\begin{aligned} XWAWX= & {} (WAWP_{(AW)^q})^\dag WAW(WAWP_{(AW)^q})^\dag \\= & {} (WAWP_{(AW)^q})^\dag WAW P_{(AW)^q}(WAWP_{(AW)^q})^\dag \\= & {} (WAWP_{(AW)^q})^\dag \\= & {} X. \end{aligned}$$

Thus, X is a solution to (2.1).

Uniqueness. Any arbitrary solution X to the system (2.1) satisfies

$$\begin{aligned} X= & {} (XWA)WX \\= & {} (WAWP_{(AW)^q})^\dag W(AWX) \\= & {} (WAWP_{(AW)^q})^\dag WAW(WAWP_{(AW)^q})^\dag \\= & {} (WAWP_{(AW)^q})^\dag WAW P_{(AW)^q}(WAWP_{(AW)^q})^\dag \\= & {} (WAWP_{(AW)^q})^\dag , \end{aligned}$$

which implies that the matrix $X=(WAWP_{(AW)^q})^\dag$ is the unique solution to (2.1). $\square$

The example below shows that the uniqueness of the solution of the system (2.1) cannot be guaranteed when the second condition is removed. Similar examples can be found by removing the first and the third conditions and mantaining the remaining two.

Example 2.3

Consider the system $XWAWX=X$ and $AWX=AW(WAWP_{AW})^\dag$, where

$$\begin{aligned} A=\left[ \begin{array}{cccc} 1 &{} \quad 1 &{} \quad 0 &{}\quad 0\\ 0 &{} \quad 0 &{}\quad 1 &{}\quad 0 \\ 0 &{}\quad 0 &{} \quad 1 &{}\quad 1\\ 0 &{}\quad 0 &{} \quad 0 &{} \quad 1 \\ 0 &{}\quad 0 &{}\quad 0 &{} \quad 0 \end{array}\right] \quad \text {and} \quad W=\left[ \begin{array}{ccccc} 1 &{} \quad 0 &{}\quad 1 &{}\quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{}\quad 1 &{} \quad 1 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 &{} \quad 1 &{}\quad 1 \\ 0 &{} \quad 0 &{} \quad 0 &{}\quad 0 &{}\quad 1 \end{array}\right] . \end{aligned}$$

It is easy to see that $k=\max \{\textrm{Ind}(AW), \textrm{Ind}(WA)\}=\max \{3,3\}=3$. Let $X_0:=(WAWP_{AW})^\dag$. By Theorem 2.2, it is clear that $X_0WAWX_0=X_0$ and $AWX_0=AW(WAWP_{AW})^\dag$.

Now, we consider the matrix $X_1:=Q_{AW}X_0+(I_m-Q_{AW})W^*$ where $Q_{AW}:=(AW)^\dag AW$. Then,

$$\begin{aligned} X_1WAWX_1= & {} [Q_{AW}X_0+(I_m-Q_{AW})W^*]WAW[Q_{AW}X_0+(I_m-Q_{AW})W^*] \\= & {} [Q_{AW}X_0+(I_m-Q_{AW})W^*]WAWX_0\\= & {} Q_{AW}X_0WAWX_0+(I_m-Q_{AW})W^*WAWX_0 \\= & {} Q_{AW}X_0+(I_m-Q_{AW})W^*WAWX_0\\= & {} Q_{AW}X_0+(I_m-Q_{AW})W^*\\= & {} X_1; \\{} & {} \\ AWX_1= & {} AW[Q_{AW}X_0+(I_m-Q_{AW})W^*]\\= & {} AWQ_{AW}X_0\\= & {} AWX_0=AW(WAWP_{AW})^\dag . \end{aligned}$$

Thus, $X_0$ and $X_1$ both satisfy $XWAWX=X$ and $AWX=AW(WAWP_{AW})^\dag$. Finally, we observe that, due to Theorem 2.2, the matrix $X_0$ is also a solution of the equation $XWA=(WAWP_{AW})^\dag WA$. However, $X_1$ does not satisfy such an equation. In fact,

$$\begin{aligned} {X_1WA}=\left[ \begin{array}{ccrr} \frac{3}{5} &{} \quad \frac{3}{5} &{} \quad -\frac{1}{5} &{}\quad -1\\ 0 &{} \quad 0 &{}\quad 0 &{} \quad 0 \\ \frac{1}{5} &{}\quad \frac{1}{5} &{} \quad -\frac{2}{5} &{} \quad -1\\ 0 &{} \quad 0 &{} \quad 1 &{}\quad 2 \\ 0 &{} \quad 0 &{}\quad 0 &{}\quad 0 \end{array}\right] \ne \left[ \begin{array}{ccrr} \frac{1}{3} &{} \quad \frac{1}{3} &{}\quad -\frac{2}{3} &{}\quad -\frac{5}{3}\\ \frac{1}{3} &{} \quad \frac{1}{3} &{}\quad -\frac{7}{6} &{}\quad -\frac{8}{3} \\ \frac{1}{3} &{} \quad \frac{1}{3} &{}\quad -\frac{1}{6} &{}\quad -\frac{2}{3}\\ 0 &{} \quad \quad 0 &{}\quad 1 &{} \quad 2 \\ 0 &{} \quad 0 &{}\quad 0 &{} \quad 0 \end{array}\right] =X_0WA={(WAWP_{AW})^\dag WA}. \end{aligned}$$

Definition 2.4

Let $A \in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$, $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}$, and $q\in \mathbb {N}\cup \{0\}$. The unique matrix $X \in {\mathbb {C}^{m\times n}}$ that satisfies the system (2.1) is called the W-weighted q-BT inverse of A, and is denoted by $A^{\diamond _q,W}$.

Remark 2.5

Note that when $m=n$ and $W=I_n$, the W-weighted q-BT inverse of A gives rise a new generalized inverse for square matrices. For simplicity, it will be denoted as $A^{\diamond _q}:=(AP_{A^q})^\dag$ and will be called the q-BT inverse of A.

The motivation for the study of this new kind of generalized inverse is stated in the following result by showing that it extends certain inverses known in the literature.

Corollary 2.6

Let $A \in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$, $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}$, and $q\in \mathbb {N}\cup \{0\}$. Then

• (i) $A^{\diamond _q,W}=(WAW)^{\dag }$ if $q=0$;

• (ii) $A^{\diamond _q,W}=A^{\diamond ,W}$ if $q=1$;

• (iii) $A^{\diamond _q,W}=A^{\textcircled {\dag },W}$ if either $q={\text {Ind}}(AW)$ or $q\ge k$.

Proof

• (i) Follows from Theorem 2.2 with $q=0$.

• (ii) It is a consequence from Theorem 2.2 and the expression of $A^{\diamond ,W}$ recalled below (1.4).

• (iii) It follows from Theorem 2.2, the definition of $A^{\textcircled {\dag },W}$ and the fact that $P_{(AW)^q}=P_{(AW)^k}$ when either $q={\text {Ind}}(AW)$ or $q\ge k$.

$\square$

Remark 2.7

When $WAW=A$, from the above corollary it follows that the W-weighted q-BT inverse of A reduces to the Moore–Penrose inverse of A. Note that the condition $WAW=A$ is a Stein equation (in A). We recall that this equation has important applications in system theory, among them, the stability analysis of discrete-time systems [11].

Remark 2.8

If $m=n$ and $W=I_n$, from Corollary 2.6 we deduce that the W-weighted q-BT inverse concides with the BT inverse and core-EP inverse, when $q=1$ and $q\ge k={\text {Ind}}(A)$, respectively.

An interesting relationship between the products AW and WA is

$$\begin{aligned} (AW)^{\ell -1} A= A(WA)^{\ell -1}, \quad \ell \in \mathbb {N}. \end{aligned}$$

(2.2)

Corollary 2.9

Let $A\in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$ and $q\in \mathbb {N} \cup \{0\}$. Then

$$\begin{aligned} A^{\diamond _q,W}=[W(AW)^{(q+1)} [(AW)^q]^\dag ]^\dag =[(WA)^{q+1} W[(AW)^q]^\dag ]^\dag . \end{aligned}$$

Proof

Follows from Theorem 2.2 and (2.2). $\square$

In the following example we show that when $1<q<k$ (eventually with $q \ne {\text {Ind}}(AW)$), this new inverse is different from other known ones.

Example 2.10

Consider the matrices

$$\begin{aligned} A=\left[ \begin{array}{ccc} 1 &{} \quad 1 &{}\quad 0 \\ 0 &{} \quad 1 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 1 \\ 0 &{} \quad 0 &{}\quad 0 \end{array}\right] \quad \text {and} \quad W=\left[ \begin{array}{cccc} 1 &{} \quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{} \quad 1 &{} \quad 0 \\ 0 &{}\quad 0 &{} \quad 0 &{} \quad 1 \end{array}\right] . \end{aligned}$$

Since ${\text {Ind}}(AW)=3$ and ${\text {Ind}}(WA)=2$, we have $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}=3$. Therefore, we must consider $q=2$. Thus, the W-weighted core-EP inverse, the W-weighted BT inverse, and the W-weighted 2-BT inverse are given by

$$\begin{aligned} A^{\textcircled {\dag },W} = \left[ \begin{array}{ccc} 1 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \end{array}\right] , \quad A^{\diamond ,W}= \left[ \begin{array}{ccc} \frac{1}{6} &{}\quad 0 &{}\quad 0 \\ \frac{1}{6} &{}\quad 0 &{}\quad 0 \\ \frac{1}{3} &{} \quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \end{array}\right] , \quad A^{\diamond _2,W}= \left[ \begin{array}{ccc} \frac{1}{2} &{} \quad 0 &{}\quad 0 \\ \frac{1}{2} &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \\ 0 &{}\quad 0 &{} \quad 0 \end{array}\right] . \end{aligned}$$

Some properties of the W-weighted q-BT inverse are established below. For example, the W-weighted q-BT inverse can be expressed in terms of the q-BT inverse. In particular, the q-BT inverse provides the range and null space of the W-weighted BT inverse.

Theorem 2.11

Let $A\in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$ and $q\in \mathbb {N} \cup \{0\}$. Then the following statements hold:

• (i) $A^{\diamond _q, W}=(W[(AW)^{\diamond _q}]^\dag )^\dag$.

• (ii) ${\mathscr {R}}(A^{\diamond _q,W})={\mathscr {R}}(P_{(AW)^q}(WAW)^*)$ and ${\mathscr {N}}(A^{\diamond _q,W})={\mathscr {N}}(P_{(AW)^q}(WAW)^*)$.

• (iii) ${\mathscr {R}}(A^{\diamond , W})= {\mathscr {R}}([(AW)^{\diamond _q}]^\dag )^* W^*)$ and ${\mathscr {N}}(A^{\diamond ,W})={\mathscr {N}}([(AW)^{\diamond _q}]^\dag )^* W^*)$.

• (iv) ${\mathscr {R}}(A^{\diamond _q,W})={\mathscr {R}}([((AW)^q)^\dag ]^*[(AW)^{q+1}]^* W^*)$ and ${\mathscr {N}}(A^{\diamond _q,W})={\mathscr {N}}([(AW)^{q+1}]^* W^*)$.

• (v) ${\mathscr {R}}(A^{\diamond _q,W})\subseteq {\mathscr {R}}((AW)^q)$.

• (vi) $P_{(AW)^q} A^{\diamond _q,W}=A^{\diamond _q,W}$.

Proof

• (i) By Theorem 2.2 we have $A^{\diamond _q,W}=(W[AWP_{(AW)^q}])^\dag$. Now, by Remark 2.5 we deduce $AWP_{(AW)^q}=((AW)^{\diamond _q})^\dag$, whence the statement is clear.

• (ii) By Theorem 2.2 we have $A^{\diamond _q,W}=(WAWP_{(AW)^q})^\dag$. Now, the statement follows of the properties ${\mathscr {R}}(B^\dag )={\mathscr {R}}(B^*)$ and ${\mathscr {N}}(B^\dag )={\mathscr {N}}(B^*)$.

• (iii) It follows immediately from part (i).

• (iv) By Corollary 2.9 and the property ${\mathscr {R}}(B^\dag )={\mathscr {R}}(B^*)$ we get

  $$\begin{aligned} {\mathscr {R}}(A^{\diamond _q,W})={\mathscr {R}}([W(AW)^{(q+1)} ((AW)^q)^\dag ]^*)={\mathscr {R}}([((AW)^q)^\dag ]^*[(AW)^{q+1}]^* W^*). \end{aligned}$$

  Similarly, Corollary 2.9 and the property ${\mathscr {N}}(B^\dag )={\mathscr {N}}(B^*)$ imply

  $$\begin{aligned} {\mathscr {N}}(A^{\diamond _q,W})= & {} {\mathscr {N}}([((AW)^q)^\dag ]^*[(AW)^{q+1}]^* W^*)\\= & {} {\mathscr {N}}([((AW)^q)^\dag ]^*[(AW)^q]^* (AW)^* W^*) \\\subseteq & {} {\mathscr {N}}([(AW)^q]^*[((AW)^q)^\dag ]^*[(AW)^q]^* (AW)^* W^*)\\= & {} {\mathscr {N}}([(AW)^{q+1}]^* W^*)\\\subseteq & {} {\mathscr {N}}([((AW)^q)^\dag ]^*[(AW)^{q+1}]^* W^*)\\= & {} {\mathscr {N}}(A^{\diamond _q,W}). \end{aligned}$$

  Thus, ${\mathscr {N}}(A^{\diamond ,W})={\mathscr {N}}([(AW)^{q+1}]^* W^*)$.

• (v) It directly follows from (ii) and the fact that ${\mathscr {R}}(P_{(AW)^k})={\mathscr {R}}((AW)^k)$.

• (vi) It is sufficient to note that $P_{(AW)^q} A^{\diamond _q,W}=A^{\diamond _q,W}$ holds if and only if ${\mathscr {R}}(A^{\diamond _q,W})\subseteq {\mathscr {N}}(I_m-P_{(AW)^q})={\mathscr {R}}(P_{(AW)^q})={\mathscr {R}}((AW)^q)$, which is true due to part (v).

$\square$

We finish this section by showing that the W-weighted q-BT inverse can be written as a generalized inverse with prescribed range and null space. Moreover, some idempotent matrices related to the W-weighted q-BT inverse are found.

Proposition 2.12

Let $A\in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$ and $q\in \mathbb {N} \cup \{0\}$. Then the following representations are valid:

• (i) $A^{\diamond _q,W}=(WAW)^{(2)}_{{\mathscr {R}}(P_{(AW)^q}(WAW)^*),\,{\mathscr {N}}([(AW)^{q+1}]^* W^*)}$;

• (ii) $WAWA^{\diamond _q,W}=P_{{\mathscr {R}}(W [(AW)^{\diamond _q}]^\dag (WAW)^*),\,{\mathscr {N}}([(AW)^{q+1}]^* W^*)}$;

• (iii) $A^{\diamond _q,W}WAW=P_{{\mathscr {R}}(P_{(AW)^q}(WAW)^*),\,{\mathscr {N}}([(AW)^{q+1}]^* W^* WAW)}$.

Proof

(i) By definition of the W-weighted q-BT inverse we know that $A^{\diamond , W}WAWA^{\diamond , W}=A^{\diamond , W}$. Now, parts (ii) and (iv) of Theorem 2.11 imply ${\mathscr {R}}(A^{\diamond _q,W})={\mathscr {R}}(P_{(AW)^q}(WAW)^*)$ and ${\mathscr {N}}(A^{\diamond , W})={\mathscr {N}}([(AW)^{q+1}]^* W^*)$, respectively. Thus, the statement follows by definition of an outer inverse with prescribed range and null space.

(ii) Since $A^{\diamond , W}WAWA^{\diamond , W}=A^{\diamond , W}$ by definition, we have that $WAWA^{\diamond _q, W}$ is idempotent. Also, from Theorem 2.11 (ii) we obtain

$$\begin{aligned} {\mathscr {R}}(WAWA^{\diamond _q, W})= & {} WAW{\mathscr {R}}(A^{\diamond _q, W})\\= & {} WAW{\mathscr {R}}(P_{(AW)^q}(WAW)^*) \\= & {} W{\mathscr {R}}(AWP_{(AW)^q}(WAW)^*)\\= & {} W{\mathscr {R}}([(AW)^{\diamond _q}]^\dag (WAW)^*)\\= & {} {\mathscr {R}}(W[(AW)^{\diamond _q}]^\dag (WAW)^*). \end{aligned}$$

On the other hand, note that ${\mathscr {N}}(WAWA^{\diamond , W})={\mathscr {N}}(A^{\diamond , W})$ because $A^{\diamond , W}$ is an outer inverse of WAW. Thus, from Theorem 2.11 (iv) we have ${\mathscr {N}}(WAWA^{\diamond , W})={\mathscr {N}}([(AW)^{q+1}]^* W^*)$.

(iii) By Theorem 2.11 (ii) we know that ${\mathscr {R}}(A^{\diamond _q,W})={\mathscr {R}}(P_{(AW)^q}(WAW)^*)$. Thus, as $A^{\diamond , W}WAWA^{\diamond , W}=A^{\diamond , W}$, clearly ${\mathscr {R}}(A^{\diamond _q,W}WAW)={\mathscr {R}}(A^{\diamond _q,W})={\mathscr {R}}(P_{(AW)^q}(WAW)^*)$.

Similarly, from Theorem 2.11 (iv) we know that ${\mathscr {N}}(A^{\diamond _q,W})={\mathscr {N}}([(AW)^{q+1}]^* W^*)$. On the other hand, it is easy to see that ${\mathscr {N}}(B)={\mathscr {N}}(C)$ implies ${\mathscr {N}}(BD)={\mathscr {N}}(CD)$, where B, C, and D are complex rectangular matrices of adequate sizes. Therefore, ${\mathscr {N}}(A^{\diamond ,W}WAW)={\mathscr {N}}([(AW)^{q+1}]^* W^* WAW)$. $\square$

Recall that the Moore–Penroe inverse [3], the core-EP inverse [6, Theorem 3.2] and the BT inverse [7, Theorem 4.7] of a matrix $A\in {\mathbb {C}^{n\times n}}$ of index k, are outer inverses that can be represented as outer inverse with prescribed range and null spaces as:

$$\begin{aligned} A^\dag =A^{(2)}_{{\mathscr {R}}(A^*),\,{\mathscr {N}}(A^*)},\quad A^{\textcircled {\dag }}=A^{(2)}_{{\mathscr {R}}(A^k),\,{\mathscr {N}}((A^k)^*)} \quad \text {and} \quad A^\diamond =A^{(2)}_{{\mathscr {R}}(P_A A^*),\,{\mathscr {N}}((A^2)^*)}.\qquad \end{aligned}$$

(2.3)

Our next theorem shows that the representations given in (2.3) are particular cases of the following expression for the q-BT inverse.

Corollary 2.13

Let $A\in {\mathbb {C}^{n\times n}}$ and $q\in \mathbb {N} \cup \{0\}$. Then the following statements hold:

• (i) $A^{\diamond _q}=A^{(2)}_{{\mathscr {R}}(P_{A^q}A^*),\, {\mathscr {N}}([A^{q+1}]^*)}$

• (ii) $AA^{\diamond _q}=P_{{\mathscr {R}}([A^{\diamond _q}]^\dag A^*),\, {\mathscr {N}}([A^{q+1}]^*)}$.

• (iii) $A^{\diamond _q}A=P_{{\mathscr {R}}(P_{A^q}A^*),\, {\mathscr {N}}([A^{q+1}]^*A)}$.

Proof

Items (i)–(iii) immediately follow from Proposition 2.12 by taking $m=n$ and $W=I_n$. $\square$

Remark 2.14

From Corollary 2.13 (i), it is clear that when $q=0$ and $q=1$, we recover the expressions given in (2.3) for the Moore–Penrose inverse and the BT inverse, respectively.

On the other hand, if $q\ge k={\text {Ind}}(A)$ we have that ${\mathscr {R}}(P_{A^q}A^*)={\mathscr {R}}((AP_{A^q})^*)={\mathscr {R}}((AP_{A^k})^\dag )={\mathscr {R}}(A^{\textcircled {\dag }})={\mathscr {R}}(A^k)$. Also, by definition of index, we obtain ${\mathscr {N}}((A^{q+1})^*)={\mathscr {N}}((A^{k+1})^*)={\mathscr {N}}((A^k)^*)$. In consequence, $A^{\diamond _q}=A^{(2)}_{{\mathscr {R}}(A^k),\, {\mathscr {N}}((A^k)^*)}=A^{\textcircled {\dag }}$.

3 Algebraic characterizations

In this section we give some algebraic characterizations of the W-weighted q-BT inverse.

Theorem 3.1

Let $A \in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$, $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}$, and $q\in \mathbb {N}\cup \{0\}$. There exists a unique matrix X satisfying the conditions

$$\begin{aligned} P_{(AW)^q}X=(WAWP_{(AW)^q})^\dag \quad \text {and} \quad {\mathscr {R}}(X)\subseteq {\mathscr {R}}((AW)^q) \end{aligned}$$

(3.1)

and is given by $X=A^{\diamond _q,W}$.

Proof

Existence. Let $X:=A^{\diamond _q,W}$. From parts (v) and (vi) of Theorem 2.11 it is clear that X is a solution to (3.1).

Uniqueness. Any matrix X satisfying conditions (3.1), in particular satisfies ${\mathscr {R}}(X)\subseteq {\mathscr {R}}((AW)^q)$ which is equivalent to $P_{(AW)^q}X=X$. Thus, from the condition $P_{(AW)^q}X=(WAWP_{(AW)^q})^\dag$, we get $X=(WAWP_{(AW)^q})^\dag$, which gives the conclusion. $\square$

Theorem 3.2

Let $A \in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$, $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}$, and $q\in \mathbb {N}\cup \{0\}$. The unique matrix X satisfying the conditions

$$\begin{aligned} AWX=AW(WAWP_{(AW)^q})^\dag \quad \text {and} \quad {\mathscr {R}}(X)\subseteq {\mathscr {R}}(P_{(AW)^q}(WAW)^*) \end{aligned}$$

(3.2)

is given by $X=A^{\diamond _q,W}$.

Proof

Existence. Let $X:=A^{\diamond _q,W}$. By Definition 2.4 and Theorem 2.11 (ii) it is clear that X satisfies both conditions in (3.2).

Uniqueness. Let X be an arbitrary matrix satisfying both conditions in (3.2). Since ${\mathscr {R}}((WAWP_{(AW)^q})^\dag )={\mathscr {R}}(P_{(AW)^q}(WAW)^*)$, the second condition in (3.2) implies $X=(WAWP_{(AW)^q})^\dag Z$ for some matrix Z. Now, from Lemma 2.1 and the first equation in (3.2) we obtain

$$\begin{aligned} X= & {} (WAWP_{(AW)^q})^\dag Z \\= & {} (WAWP_{(AW)^q})^\dag WAW P_{(AW)^q}(WAWP_{(AW)^q})^\dag Z\\= & {} (WAWP_{(AW)^q})^\dag W AW [(WAWP_{(AW)^q})^\dag Z] \\= & {} (WAWP_{AW})^\dag W (AWX) \\= & {} (WAWP_{AW})^\dag WAW(WAWP_{(AW)^q})^\dag \\= & {} (WAWP_{(AW)^q})^\dag WAW P_{(AW)^q}(WAWP_{(AW)^q})^\dag \\= & {} (WAWP_{(AW)^q})^\dag \\= & {} A^{\diamond _q,W}, \end{aligned}$$

which gives the uniqueness. $\square$

A similar result can be obtained using the null space.

Theorem 3.3

Let $A \in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$, $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}$, and $q\in \mathbb {N}\cup \{0\}$. The unique matrix X that satisfies both conditions

$$\begin{aligned} XWA=(WAWP_{(AW)^q})^\dag W A \quad \text {and}\quad {\mathscr {N}}(P_{(AW)^q}(WAW)^*)\subseteq {\mathscr {N}}(X) \end{aligned}$$

(3.3)

is given by $X=A^{\diamond ,W}$.

As a consequence of above results we obtain some characterizations of the q-BT inverse of a square matrix.

Theorem 3.4

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

• (i) X is the q-BT inverse of A;

• (ii) $XAX=X$, $AX=A(AP_{A^q})^\dag$, and $XA=(AP_{A^q})^\dag A$;

• (iii) $P_{A^q}X=(AP_{A^q})^\dag$ and ${\mathscr {R}}(X)\subseteq {\mathscr {R}}(A^q)$;

• (iv) $AX=A(AP_{A^q})^\dag$ and ${\mathscr {R}}(X)\subseteq {\mathscr {R}}(P_{A^q}A^*)$;

• (v) $XA=(AP_{A^q})^\dag A$ and ${\mathscr {N}}(P_{A^q}A^*)\subseteq {\mathscr {N}}(X)$.

4 Canonical form of the W-weighted q-BT inverse

In [5] the authors introduced a simultaneous unitary block upper triangularization of a pair of rectangular matrices, called the weighted core-EP decomposition of the pair (A, W). More precisely, we have the following result:

Theorem 4.1

Let $A\in {\mathbb {C}^{m\times n}}$ and $0\ne W\in {\mathbb {C}^{n\times m}}$ with $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}$. Then there exist two unitary matrices $U \in {\mathbb {C}^{m\times m}}$, $V \in {\mathbb {C}^{n\times n}}$, two nonsingular matrices $A_1, W_1 \in \mathbb {C}^{t \times t}$, and two matrices $A_3 \in \mathbb {C}^{(m-t)\times (n-t)}$ and $W_3 \in \mathbb {C}^{(n-t)\times (m-t)}$ such that $A_3W_3$ and $W_3A_3$ are nilpotent of indices ${\text {Ind}}(AW)$ and ${\text {Ind}}(WA)$, respectively, with

$$\begin{aligned} A = U\left[ \begin{array}{cc} A_1 &{}\quad A_2 \\ 0 &{}\quad A_3 \end{array}\right] V^* \quad \text {and} \quad W = V\left[ \begin{array}{cc} W_1 &{}\quad W_2 \\ 0 &{} \quad W_3 \end{array}\right] U^*. \end{aligned}$$

(4.1)

The following lemma allows us to find the Moore–Penrose inverse of a partitioned matrix with some of its diagonal block nonsingular.

Lemma 4.2

[10] Let $A=U\begin{bmatrix} A_1 &{} A_2 \\ 0 &{} A_3 \\ \end{bmatrix} V^* \in {\mathbb {C}^{m\times n}}$ be such that $A_1 \in \mathbb {C}^{t\times t}$ is nonsingular and $U \in {\mathbb {C}}^{m \times m}$ and $V \in {\mathbb {C}}^{n \times n}$ are unitary. Then

$$\begin{aligned} A^\dag = V\left[ \begin{array}{cc} A_1^*\varOmega &{} -A_1^*\varOmega A_2 A_3^\dagger \\ (I_{n-t}-Q_{A_3})A_2^*\varOmega &{} A_3^\dagger -(I_{n-t}-Q_{A_3})A_2^*\varOmega A_2 A_3^\dagger \end{array}\right] U^*, \end{aligned}$$

(4.2)

where $\varOmega =[A_1 A_1^*+ A_2(I_{n-t}-Q_{A_3})A_2^*]^{-1}$. In consequence,

$$\begin{aligned} P_A= U\left[ \begin{array}{cc} I_t &{} 0\\ 0 &{} P_{A_3} \end{array}\right] U^*. \end{aligned}$$

(4.3)

Now, we present a representation for the W-weighted q-BT inverses by using the weighted core-EP decomposition.

Theorem 4.3

Let $A \in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$, $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}$, and $q\in \mathbb {N}\cup \{0\}$. If A and W are written as in (4.1), then the W-weighted q-BT inverse of A is given by

$$\begin{aligned} A^{\diamond _q,W} = U\left[ \begin{array}{cc} (W_1A_1W_1)^* \varOmega _W &{} -(W_1A_1W_1)^* \varOmega _W M A_3^{\diamond _q,W_3} \\ (P_{(A_3W_3)^q}-P_{A_3^{\diamond _q,W_3}})M^*\varOmega _W &{} A_3^{\diamond _q,W_3}-(P_{(A_3W_3)^q}-P_{A_3^{\diamond _q,W_3}})M^*\varOmega _W M A_3^{\diamond _q,W_3} \end{array}\right] V^*, \end{aligned}$$

(4.4)

where

$$\begin{aligned} M:=W_1A_1W_2+W_1A_2W_3+W_2A_3W_3 \end{aligned}$$

and

$$\begin{aligned} \varOmega _W:=[W_1A_1W_1 (W_1A_1W_1)^*+ M (P_{(A_3W_3)^q}-P_{A_3^{\diamond _q,W_3}})M^*]^{-1}. \end{aligned}$$

Proof

We assume that A and W are written as in (4.1). Applying Theorem 2.2, we have $A^{\diamond _q,W} = (WAWP_{(AW)^q})^\dag$. It can be easily obtained that

$$\begin{aligned} WAW=V \left[ \begin{array}{cc} W_1A_1W_1 &{} W_1A_1W_2+(W_1A_2+W_2A_3)W_3 \\ 0 &{} W_3A_3W_3 \end{array} \right] U^*=V \left[ \begin{array}{cc} W_1A_1W_1 &{} M \\ 0 &{} W_3A_3W_3 \end{array} \right] U^*, \end{aligned}$$

where $M:= W_1A_1W_2+W_1A_2W_3+W_2A_3W_3$, and

$$\begin{aligned} P_{(AW)^q} = U\left[ \begin{array}{cc} I_t &{} 0 \\ 0 &{} P_{(A_3W_3)^q} \end{array}\right] U^*. \end{aligned}$$

Thus, we have that

$$\begin{aligned} A^{\diamond _q,W} = (WAWP_{(AW)^q})^\dag = U\left[ \begin{array}{cc} W_1 A_1 W_1 &{} M P_{(A_3W_3)^q}\\ 0 &{} W_3A_3W_3 P_{(A_3W_3)^q} \end{array}\right] ^\dag V^* = U \left[ \begin{array}{cc} B_1 &{} B_2\\ B_3 &{} B_4 \end{array}\right] V^*, \end{aligned}$$

where we are considering the partition given by the blocks $B_1,B_2,B_3$ and $B_4$ having appropriate sizes induced by the central matrix in the previous step. By Theorem 4.1, $W_1A_1W_1$ is nonsingular. In order to determine the blocks $B_1$, $B_2$, $B_3$, and $B_4$ we will use Lemma 4.2. Taking $Z:=P_{(A_3W_3)^q}(I_{n-t}-Q_{W_3A_3W_3P_{(A_3W_3)^q}})P_{(A_3W_3)^q}$, we get

$$\begin{aligned} \varOmega _W=[W_1A_1W_1 (W_1A_1W_1)^*+MZM^*]^{-1}. \end{aligned}$$

(4.5)

Moreover, from Lemma 2.1 and Theorem 2.2, it follows

$$\begin{aligned} Z= & {} (P_{(A_3W_3)^q})^2-[P_{(A_3W_3)^q} (W_3A_3W_3P_{(A_3W_3)^q})^\dag ] W_3A_3W_3(P_{(A_3W_3)^q})^2 \nonumber \\= & {} P_{(A_3W_3)^q}-[W_3A_3W_3P_{(A_3W_3)^q}]^\dag W_3A_3W_3P_{(A_3W_3)^q} \nonumber \\= & {} P_{(A_3W_3)^q}-A_3^{\diamond _q,W_3} [A_3^{\diamond _q,W_3}]^\dag \nonumber \\= & {} P_{(A_3W_3)^q}-P_{A_3^{\diamond _q,W_3}}. \end{aligned}$$

(4.6)

From (4.6) and (4.5) we have

$$\begin{aligned} \varOmega _W=[W_1A_1W_1 (W_1A_1W_1)^*+ M (P_{(A_3W_3)^q}-P_{A_3^{\diamond _q,W_3}})M^*]^{-1}. \end{aligned}$$

Finally,

$$\begin{aligned} B_1= & {} (W_1A_1W_1)^* \varOmega _W, \\ B_2= & {} -(W_1A_1W_1)^* \varOmega _W M P_{(A_3W_3)^q} (W_3A_3W_3P_{(A_3W_3)^q})^\dag \\= & {} -(W_1A_1W_1)^* \varOmega _W M A_3^{\diamond _q,W_3},\\ B_3= & {} (I_{n-t}-Q_{W_3A_3W_3P_{(A_3W_3)^q}})(MP_{(A_3W_3)^q})^*\varOmega _W \\= & {} (P_{(A_3W_3)^q}-P_{A_3^{\diamond _q,W_3}})M^*\varOmega _W,\\ B_4= & {} A_3^{\diamond _q,W_3}- B_3 M P_{(A_3W_3)^q} (W_3A_3W_3P_{(A_3W_3)^q})^\dag \\= & {} A_3^{\diamond _q,W_3}- (P_{(A_3W_3)^q}-P_{A_3^{\diamond _q,W_3}})M^*\varOmega _W M A_3^{\diamond _q,W_3}, \end{aligned}$$

which completes the proof. $\square$

Corollary 4.4

Let $A \in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$, and $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}$. If A and W are written as in (4.1), then the W-weighted BT inverse of A is given by

$$\begin{aligned} A^{\diamond _1,W}=A^{\diamond ,W} = U\left[ \begin{array}{cc} (W_1A_1W_1)^* \varOmega _W &{} -(W_1A_1W_1)^* \varOmega _W M A_3^{\diamond ,W_3} \\ (P_{A_3W_3}-P_{A_3^{\diamond ,W_3}})M^*\varOmega _W &{} A_3^{\diamond ,W_3}-(P_{A_3W_3}-P_{A_3^{\diamond ,W_3}})M^*\varOmega _W M A_3^{\diamond ,W_3} \end{array}\right] V^*, \end{aligned}$$

(4.7)

where

$$\begin{aligned} M= & {} W_1A_1W_2+(W_1A_2+W_2A_3)W_3, \quad \text {and} \\ \varOmega _W= & {} [W_1A_1W_1 (W_1A_1W_1)^*+ M(P_{A_3W_3}- P_{A_3^{\diamond ,W_3}})M^*]^{-1}, \end{aligned}$$

and the W-weighted core-EP inverse of A is given by

$$\begin{aligned} A^{\diamond _q,W}=A^{\textcircled {\dag },W}= U\left[ \begin{array}{cc} (W_1A_1W_1)^{-1} &{} 0 \\ 0 &{} 0 \end{array}\right] V^*, \quad \text { for } q\ge k. \end{aligned}$$

(4.8)

Proof

By Corollary 2.6 we know that $A^{\diamond _q,W}=A^{\diamond ,W}$ if $q=1$ and $A^{\diamond _q,W}=A^{\textcircled {\dag },W}$ if $q\ge k$. Clearly, if $q=1$, (4.4) reduces to the expression given in (4.7). On the other hand, if $q\ge k$ we obtain $(A_3W_3)^q=0$. In fact, since $A_3W_3$ is nilpotent of index at most k, we have $P_{(A_3W_3)^q}=0$. Hence, $A_3^{\diamond _q,W_3}=(W_3A_3W_3P_{(A_3W_3)^q})^\dag =0$. Now, from (4.5), it follows that $\varOmega _W=[W_1A_1W_1 (W_1A_1W_1)^*]^{-1}$. In this way, (4.4) reduces to (4.8). $\square$

Remark 4.5

When $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}=1$, the above representations coincide with the W-weighted core inverse, that is, $A^{\diamond ,W}=A^{\textcircled {\dag },W}=A^{\textcircled {\#},W}$.

If $A\in {\mathbb {C}^{n\times n}}$ has index k, by applying Theorem 4.1 with $m=n$ and $W=I_n$, we obtain the following canonical form of A

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

(4.9)

where $U\in {\mathbb {C}^{n\times n}}$ is unitary, T is nonsingular, ${\text {rank}}(T)={\text {rank}}(A^k)$, and N is nilpotent of index k. This representation of A is called the core-EP decomposition of A [18].

By using (4.9) we can give a canonical form for the q-BT inverse of a square matrix.

Corollary 4.6

Let $A \in {\mathbb {C}^{n\times n}}$, $k={\text {Ind}}(A)$, and $q\in \mathbb {N}\cup \{0\}$. If A is written as in (4.9), then the q-BT inverse of A is given by

$$\begin{aligned} A^{\diamond _q} = U\left[ \begin{array}{cc} T^*\Delta &{} -T^*\Delta S N^{\diamond _q}\\ (P_N-P_{N^{\diamond _q}})S^*\Delta &{} N^{\diamond _q}-(P_N-P_{N^{\diamond _q}})S^*\Delta S N^{\diamond _q} \end{array}\right] U^*, \end{aligned}$$

(4.10)

where $\Delta =(T T^*+ S(P_N-P_{N^{\diamond _q}})S^*)^{-1}$.

Corollary 4.7

Let $A \in {\mathbb {C}^{m\times n}}$, $0\ne W\in {\mathbb {C}^{n\times m}}$, and $k=\max \{{\text {Ind}}(AW), {\text {Ind}}(WA)\}$. If A and W are written as in (4.1), then it results that

$$\begin{aligned} (AW)^{\diamond _q}=U\left[ \begin{array}{cc} (A_1W_1)^*\Delta &{} -(A_1W_1)^*\Delta S (A_3W_3)^{\diamond _q}\\ (P_{A_3W_3}-P_{(A_3W_3)^{\diamond _q}})S^*\Delta &{} (A_3W_3)^{\diamond _q}-(P_{A_3W_3}-P_{(A_3W_3)^{\diamond _q}})S^*\Delta S (A_3W_3)^{\diamond _q} \end{array}\right] U^*, \end{aligned}$$

with $\Delta =(A_1W_1(A_1W_1)^*+ S(P_{A_3W_3}-P_{(A_3W_3)^{\diamond _q}})S^*)^{-1}$ and $S=A_1W_2+A_2W_3$, and

$$\begin{aligned} (WA)^{\diamond _q}=U\left[ \begin{array}{cc} (W_1A_1)^*\Delta &{} -(W_1A_1)^*\Delta S (W_3A_3)^{\diamond _q}\\ (P_{W_3A_3}-P_{(W_3A_3)^{\diamond _q}})S^*\Delta &{} (W_3A_3)^{\diamond _q}-(P_{W_3A_3}-P_{(W_3A_3)^{\diamond _q}})S^*\Delta S (W_3A_3)^{\diamond _q} \end{array}\right] U^*, \end{aligned}$$

with $\Delta =(W_1A_1(W_1A_1)^*+ S(P_{W_3A_3}-P_{(W_3A_3)^{\diamond _q}})S^*)^{-1}$ and $S=W_1A_2+W_2A_3$.

Proof

From Theorem 4.1 we obtain

$$\begin{aligned} AW= U\left[ \begin{array}{cc} A_1W_1 &{} A_1W_2 + A_2 W_3\\ 0 &{} A_3W_3 \end{array}\right] U^*, \end{aligned}$$

(4.11)

where U is unitary, $A_1W_1$ is nonsingular, and $A_3W_3$ is nilpotent of index ${\text {Ind}}(AW)$.

Clearly, (4.11) is a core-EP decomposition of AW. Thus, the the expression for $(AW)^{\diamond _q}$ follows from Corollary 4.6.

The expression for $(WA)^{\diamond _q}$ can be found in a similar way. $\square$

We recall that the W-weighted Drazin inverse and the W-weighted core-EP inverse of A satisfy the interesting identities $A^{d,W}=[(AW)^d]^2A=A[(WA)^d]^2$ and $A^{\textcircled {\dag },W}=A[(WA)^{\textcircled {\dag }}]^2$, from (1.1) and (1.3), respectively.

However, these equalities do not remain valid for the W-weighted q-BT inverse whenever $1\le q <k=\max \{\textrm{Ind}(AW), \textrm{Ind}(WA)\}$, as we can check with the following example.

Example 4.8

Let

$$\begin{aligned} A=\left[ \begin{array}{ccc} 1 &{} \quad 1 &{} \quad 0 \\ 0 &{}\quad 1 &{}\quad 0 \\ 0 &{} \quad 0 &{} \quad 1 \\ 0 &{} \quad 0 &{}\quad 0 \end{array}\right] \quad \text {and} \quad W=\left[ \begin{array}{cccc} 1 &{}\quad 1 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 &{}\quad 1 \end{array}\right] . \end{aligned}$$

Note that $k=\max \{\textrm{Ind}(AW), \textrm{Ind}(WA)\}=\max \{3,2\}=3$. For $1\le q<3$ we obtain

$$\begin{aligned} A^{\diamond _1,W}= \left[ \begin{array}{ccc} \frac{1}{6} &{} \quad 0 &{}\quad 0 \\ \frac{1}{6} &{} \quad 0 &{} \quad 0 \\ \frac{1}{3} &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \end{array}\right] ,\quad [(AW)^{\diamond _1}]^2A=\left[ \begin{array}{ccc} 0 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \\ \frac{1}{2} &{}\quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \end{array}\right] \quad \text {and}\quad A[(WA)^{\diamond _1}]^2= \left[ \begin{array}{ccc} \frac{3}{25} &{} \quad 0 &{} \quad 0 \\ \frac{2}{25} &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \end{array}\right] , \end{aligned}$$

$$\begin{aligned} A^{\diamond _2,W}= \left[ \begin{array}{ccc} \frac{1}{2} &{} \quad 0 &{}\quad 0 \\ \frac{1}{2} &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \\ 0 &{}\quad 0 &{} \quad 0 \end{array}\right] ,\quad [(AW)^{\diamond _2}]^2A= \left[ \begin{array}{ccc} \frac{1}{4} &{}\quad \frac{1}{4} &{} \quad 0 \\ \frac{1}{4} &{}\quad \frac{1}{4} &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \end{array}\right] \quad \text {and}\quad A[(WA)^{\diamond _2}]^2=\left[ \begin{array}{ccc} 1 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \end{array}\right] . \end{aligned}$$

Remark 4.9

If we take $q=3$ in the above example (i.e., $q=k=3$), from Corollary 4.4 we have that $A^{\diamond _3,W}=A^{\textcircled {\dag },W}$. Thus, from (1.3) we obtain $A^{\diamond _3,W}=A[(WA)^{\diamond _3}]^2$, which can be verified in the example given above, that is,

$$\begin{aligned} A^{\diamond _3,W}= \left[ \begin{array}{ccc} 1 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \end{array}\right] ,\quad [(AW)^{\diamond _3}]^2A= \left[ \begin{array}{ccc} 1 &{} \quad 1 &{} \quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \end{array}\right] , \quad \text {and}\quad A[(WA)^{\diamond _3}]^2=\left[ \begin{array}{ccc} 1 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{}\quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \\ 0 &{} \quad 0 &{} \quad 0 \end{array}\right] . \end{aligned}$$