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

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.


Introduction and preliminaries
We denote by C m×n the set of all m × n complex matrices.Let A ∈ C m×n .The conjugate transpose, rank, null space and column space of A are denoted by A * , rank(A), N (A), and R(A), respectively.The index of A ∈ C n×n , denoted by Ind(A), is the smallest nonnegative integer k such that rank(A k ) = rank(A k+1 ).Moreover, A 0 = I n will refer to the n × 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 C n is equal to the direct sum of subspaces S and T .
The Drazin inverse of a matrix A ∈ C n×n is the unique matrix X = A d ∈ C n×n that satisfies XA k+1 = A k , XAX = X, AX = XA, where k = Ind(A).
When 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 ∈ C m×n is the unique matrix X = A † ∈ C n×m that satisfies the Penrose equations AXA = A, XAX = X, (AX) * = AX, (XA) * = XA.
We will denote by P A the orthogonal projector AA † onto the subspace R(A).
In 2014, Manjunatha Prassad and Mohana [13] introduced the core-EP inverse of a matrix A ∈ C n×n of index k as the unique matrix X = A † ○ ∈ C n×n that satisfies the conditions XAX = X and R(X) = R(X * ) = R(A k ).That same year, Baksalary and Trenkler [2] defined the BT inverse of A as the matrix A ⋄ = (AP A ) † .When the matrix A has index 1, both inverses are reduced to the well-known In 1980, Cline and Greville [4] extended the Drazin inverse to rectangular matrices and it was called the W -weighted Drazin inverse.Let W ∈ C n×m be a fixed nonzero matrix.We recall that the W -weighted Drazin inverse of A ∈ C m×n , denoted by A d,W , is the unique matrix X ∈ C m×n satisfying the three equations XW AW X = X, AW X = XW A, XW (AW ) k+1 = (AW ) k , where k = max{Ind(AW ), Ind(W A)}.
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, The W -weighted Drazin inverse satisfies the following two dual representations 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 Wweighted core-EP inverse, and defined as A † ○,W = (W AW P (AW ) k ) † , which is the unique solution of 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 # ○,W .Clearly, when m = n and W = I n , we recover the core-EP inverse, that is, The W -weighted core-EP inverse satisfies the following interesting properties [5,12] It was also established that A ⋄,W = (W AW P AW ) † .
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 ( 2) and ( 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 Section 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 Section 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 Section 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.

Existence and Uniqueness
In this section, we define and investigate the W -weighted q-BT inverse for rectangular matrices A ∈ C m×n by considering a non-null weight W ∈ C n×m .
We start with a result of existence and uniqueness.Before that, we need the following auxiliary lemma.Theorem 2.2.Let A ∈ C m×n , 0 = W ∈ C n×m and q ∈ N ∪ {0}.The system of equations is consistent and has a unique solution X = (W AW P (AW ) q ) † .
Moreover, from Lemma 2.1 we have Thus, X is a solution to (5).
We obtain that X := Q AW X 0 + (I m − Q AW )W * satisfies the first and third equations in (5).However, we can observe that the second equation does not hold.In fact, The unique matrix X ∈ C m×n that satisfies the system ( 5) is called the W -weighted q-BT inverse of A, and is denoted by A ⋄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 ⋄q := (AP A q ) † 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.
(ii) It is a consequence from Theorem 2.2 and the expression of A ⋄,W recalled below (4).
(iii) It follows from Theorem 2.2, the definition of A † ○,W and the fact that P (AW ) q = P (AW ) k when either q = Ind(AW ) or q ≥ k.
Remark 2.7.When W AW = 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 W AW = 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 ≥ k = Ind(A), respectively.
An interesting relationship between the products AW and W A is Corollary 2.9.Let A ∈ C m×n , 0 = W ∈ C n×m and q ∈ N ∪ {0}.Then Proof.Follows from Theorem 2.2 and (6).
In the following example we show that when 1 < q < k (eventually with q = Ind(AW )), this new inverse is different from other known ones.
Example 2.10.Consider the matrices Since Ind(AW ) = 3 and Ind(W A) = 2, we have k = max{Ind(AW ), Ind(W A)} = 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 0 0 . Some properties of the W -weighted q-BT inverse are established below.For example, the Wweighted 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.
(ii) By Theorem 2.2 we have A ⋄q,W = (W AW P (AW ) q ) † .Now, the statement follows of the properties (iii) It follows immediately from part (i).
(iv) By Corollary 2.9 and the property R(B † ) = R(B * ) we get Similarly, Corollary 2.9 and the property N (B † ) = N (B * ) imply (v) It directly follows from (ii) and the fact that (vi) It is sufficient to note that P (AW ) q A ⋄q ,W = A ⋄q ,W holds if and only if R(A ⋄q ,W ) ⊆ N (I m − P (AW ) q ) = R(P (AW ) q ) = R((AW ) q ), which is true due to part (v).
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 Wweighted q-BT inverse are found.
Proposition 2.12.Let A ∈ C m×n , 0 = W ∈ C n×m and q ∈ N ∪ {0}.Then the following representations are valid: Proof.(i) By definition of the W -weighted q-BT inverse we know that A ⋄,W W AW A ⋄,W = A ⋄,W .Now, parts (ii) and (iv) of Theorem 2.11 imply R(A ⋄q,W ) = R(P (AW ) q (W AW ) * ) and N (A ⋄,W ) = 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 ⋄,W W AW A ⋄,W = A ⋄,W by definition, we have that W AW A ⋄q ,W is idempotent.Also, from Theorem 2.11 (ii) we obtain On the other hand, note that N (W AW A ⋄,W ) = N (A ⋄,W ) because A ⋄,W is an outer inverse of W AW .
Thus, from Theorem 2.11 (iv) we have (iii) By Theorem 2.11 (ii) we know that R(A ⋄q ,W ) = R(P (AW ) q (W AW ) * ).Thus, as Similarly, from Theorem 2.11 (iv) we know that N (A ⋄q ,W ) = N ([(AW ) q+1 ] * W * ).On the other hand, it is easy to see that N (B) = N (C) implies N (BD) = N (CD), where B, C, and D are complex rectangular matrices of adequate sizes.Therefore, 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 ∈ C n×n of index k, are outer inverses that can be represented as outer inverse with prescribed range and null spaces as: Our next theorem shows that the representations given in (7) are particular cases of the following expression for the q-BT inverse.
Corollary 2.13.Let A ∈ C n×n and q ∈ N ∪ {0}.Then the following statements hold: Proof.Items (i)-(iii) immediately follow from Proposition 2.12 by taking m = n and W = I n .
Remark 2.14.From Corollary 2.13 (i), it is clear that when q = 0 and q = 1, we recover the expressions given in (7) for the Moore-Penrose inverse and the BT inverse, respectively.
On the other hand, if q ≥ k = Ind(A) we have that R(P

Algebraic characterizations
In this section we give some algebraic characterizations of the W -weighted q-BT inverse.
There exists a unique matrix X satisfying the conditions and is given by X = A ⋄q,W .
Proof.Existence.Let X := A ⋄q,W .From parts (v) and (vi) of Theorem 2.11 it is clear that X is a solution to (8).
Uniqueness.Any matrix X satisfying conditions (8), in particular satisfies R(X) ⊆ R((AW ) q ) which is equivalent to P (AW ) q X = X.Thus, from the condition P (AW ) q X = (W AW P (AW ) q ) † , we get X = (W AW P (AW ) q ) † , which gives the conclusion.
The unique matrix X satisfying the conditions is given by X = A ⋄q ,W .
Proof.Existence.Let X := A ⋄q ,W .By Definition 2.4 and Theorem 2.11 (ii) it is clear that X satisfies both conditions in (9).
Uniqueness.Let X be an arbitrary matrix satisfying both conditions in (9).Since R((W AW P (AW ) q ) † ) = R(P (AW ) q (W AW ) * ), the second condition in ( 9) implies X = (W AW P (AW ) q ) † Z for some matrix Z. Now, from Lemma 2.1 and the first equation in ( 9) we obtain which gives the uniqueness.
A similar result can be obtained using the null space.
The unique matrix X that satisfies both conditions XW A = (W AW P (AW ) q ) † W A and N (P (AW ) q (W AW ) * ) ⊆ N (X) (10) is given by X = A ⋄,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 ∈ C n×n .The following statements are equivalent: (i) X is the q-BT inverse of A; (ii) XAX = X, AX = A(AP A q ) † , and XA = (AP A q ) † A; 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 ∈ C m×n and 0 = W ∈ C n×m with k = max{Ind(AW ), Ind(W A)}.Then there exist two unitary matrices U ∈ C m×m , V ∈ C n×n , two nonsingular matrices A 1 , W 1 ∈ C t×t , and indices Ind(AW ) and Ind(W A), respectively, with 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
where In consequence, Now, we present a representation for the W -weighted q-BT inverses by using the weighted core-EP decomposition.
Theorem 4.3.Let A ∈ C m×n , 0 = W ∈ C n×m , k = max{Ind(AW ), Ind(W A)}, and q ∈ N ∪ {0}.If A and W are written as in (11), then the W -weighted q-BT inverse of A is given by 3 3 where and 3 Proof.We assume that A and W are written as in (11).Applying Theorem 2.2, we have A ⋄q,W = (W AW P (AW ) q ) † .It can be easily obtained that where Thus, we have that where we are considering the partition given by the blocks B 1 , B Moreover, from Lemma 2.1 and Theorem 2.2, it follows From ( 16) and ( 15) we have 3 Finally, , 3 3 , which completes the proof.are written as in (11), then the W -weighted BT inverse of A is given by 3 3 where 3 and the W -weighted core-EP inverse of A is given by Proof.By Corollary 2.6 we know that A ⋄q,W = A ⋄,W if q = 1 and A ⋄q,W = A † ○,W if q ≥ k.Clearly, if q = 1, (20) reduces to the expression given in (17).
where U ∈ C n×n is unitary, T is nonsingular, rank(T ) = 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 (19) we can give a canonical form for the q-BT inverse of a square matrix.
Corollary 4.6.Let A ∈ C n×n , k = Ind(A), and q ∈ N ∪ {0}.If A is written as in (19), then the q-BT inverse of A is given by where ∆ = (T T * + S(P N − P N ⋄q )S * ) −1 .
Corollary 4.7.Let A ∈ C m×n , 0 = W ∈ C n×m , and k = max{Ind(AW ), Ind(W A)}.If A and W are written as in (11), then it results that
Proof.From Theorem 4.1 we obtain where U is unitary, A 1 W 1 is nonsingular, and A 3 W 3 is nilpotent of index Ind(AW ).
Clearly, ( 21) is a core-EP decomposition of AW .Thus, the the expression for (AW ) ⋄q follows from Corollary 4.6.
The expression for (W A) ⋄q can be found in a similar way.
We recall that the W -weighted Drazin inverse and the W -weighted core-EP inverse of A satisfy the However, these equalities do not remain valid for the W -weighted q-BT inverse whenever 1 ≤ q < k = max{Ind(AW ), Ind(W A)}, as we can check with the following example.Note that k = max{Ind(AW ), Ind(W A)} = max{3, 2} = 3.For 1 ≤ q < 3 we obtain 0 0

Lemma 2 . 1 .
Let A ∈ C m×n and B ∈ C n×s .Then P B (AP B ) † = (AP B ) † .Proof.Since (I n − P B )P B A * = 0 trivially holds, we have that R((AP B ) † ) ⊆ N ((I n − P B )) is always valid, which in turn is equivalent to P B (AP B ) † = (AP B ) † .

Remark 4 . 9 .
If we take q = 3 in the above example (i.e., q = k = 3), from Corollary 4.4 we have thatA ⋄3,W = A † ○,W .Thus, from (3) we obtain A ⋄3,W = A[(W A) ⋄3 ] 2 ,which can be verified in the examplegiven above, that is, 2 , B 3 and B 4 having appropriate sizes induced by the central matrix in the previous step.By Theorem 4.1, W 1 A 1 W 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