New Characterizations and Representations of the Bott–Duffin Inverse

Te paper focuses on the class of the Bott–Dufn inverses. Several original features of the class are identifed and new properties are characterized. Some of the results available in the literature are recaptured in a more general form. BD matrices are also introduced and some properties are given


Introduction
C n stands for the vector space of n-tuples over the feld of complex numbers. Te symbol C m×n denotes the set of complex m × n matrices. Te symbols R(A), N(A), A * and rank (A) represent the range space, null space, conjugate transpose, and rank of A ∈ C m×n , respectively. Te symbol Ind(A) stands for the index of A ∈ C n×n which is the smallest non-negative integer k such that rank(A k ) � rank(A k+1 ). Te symbol I n means the identity matrix in C n×n . Te symbol O means the null matrix. If L is a subspace of C n , we use the notation L ⩽ C n while L ⊥ means the orthogonal complement subspace of L. Te dimension of L is denoted by dim(L). P L,M stands for the oblique projector onto L along M, where L, M ⩽ C n and L ⊕ M � C n . P L is the orthogonal projector onto L.
A matrix X ∈ C n×m that satisfes XAX � X is called an outer inverse of A and is denoted by A (2) . Let T ⩽ C n , dim(T) � t and S ⩽ C m , dim(S) � m − t. Tere exists a unique outer inverse X of A such that R(X) � T and N(X) � S if and only if AT ⊕ S � C m . In case, there exists X, we call an outer inverse with prescribed range and null space and denote it by A (2) T,S (see [1,4]). Te symbol A D stands for the Drazin inverse of A ∈ C n×n which is the unique matrix satisfying where k � Ind(A) (see [5]). Especially, if Ind(A) � 1, then the Drazin inverse of A is called the group inverse of A and is denoted by A # .
Bott and Dufn, in their famous paper [6], introduced the "constrained inverse" of a square matrix as an important tool in the electrical network theory. Tis inverse is called in their honor the Bott-Dufn inverse (in short, BD-inverse).
Defnition 1 (see [6]). Let A ∈ C n×n and L ⩽ C n . If AP L + P L ⊥ is nonsingular, then the BD-inverse of A with respect to L, denoted by A (− 1) (L) , is defned by the following equation: in [8]. In [9], Wei studied the various norm-wise relative condition numbers that measure the sensitivity of the BDinverse and the solution of constrained linear systems. Te perturbation theory for the BD-inverse was discussed in [10].
In [11], Chen defned the generalized BD-inverse of A (denoted by A ( †) (L) ). In terms of the form of defnitions, by using Teorem A.1 (see Appendix A) and [5,Lemma 1]. It is not convenient or even difcult to study the properties of A ( †) (L) . In order to avoid this difculty and obtain more interesting properties of A ( †) (L) , all of the theorems in [11,12] restrict matrix A to an L-p.s.d matrix, which satisfes the following three conditions: However, for A (− 1) (L) , matrix A does not need to satisfy these conditions, only A (− 1) (L) needs to exist. And the necessary and sufcient condition for the existence of A (− 1) (L) is given in Lemma 2. Terefore, for the diferences in studying A (− 1) and A ( †) (L) , it is meaningful to research the properties of A (− 1) (L) . Te present paper provides a further contribution to the stream of works devoted to the BD-inverse. Several new characterizations of the BD-inverse are derived in terms of certain matrix equations and EP-property. Also, we give some new representations of the BD-inverse as well as the relationships between the BD-inverse and other generalized inverses. Te defnition and properties of the BD-matrices are also given. Several original features of the BD-inverses are identifed and new properties are characterized. In some cases, the results available in the literature are recaptured therein in a more general form.
Te rest of this paper is organised as follows. In Section 2, we introduce some lemmas and a matrix decomposition which will be used later in the paper. In Section 3, we present several characterizations of the BD-inverse in terms of certain matrix equations and EP matrix. In Section 4, we present several representations of the BD-inverse. We focus on the relationships between the BD-inverse and other generalized inverses within Section 5. In addition, we give the defnition of the BDmatrices and present some of their properties.
Henceforth, the symbol C EP n will stand for the set of n × n EP matrices, i.e., (2)

Preliminaries
Let A ∈ C n×n and L ⩽ C n . In order to discuss some properties of the BD-inverse, we will consider appropriate matrix decomposition of A with respect to L. Since there exists a unitary matrix U ∈ C n×n such that where l � dim(L), a matrix A can be written as follows: where A L ∈ C l×l , B L ∈ C l×(n− l) , C L ∈ C (n− l)×l , and D L ∈ C (n− l)×(n− l) . Now, we are ready to give the necessary and sufcient condition for the existence of A (− 1) (L) as well as the representation of A (− 1) (L) .

Lemma 2.
Let P L and A be given by equations (3) and (4) Proof. By equations (3) and (4), we get the following equation: Evidently, AP L + P L ⊥ is invertible if and only if A L is invertible. In this case, from equations (1) and (6), we get that equation (5) is satisfed. □ Te next lemma gives some basic properties of the BDinverse, for example, that it is an outer inverse of A with range L and null space L ⊥ , etc.
Lemma 3 (see [7]). Let A ∈ C n×n and L ⩽ C n . If AP L + P L ⊥ is invertible, then the following statements hold:

Some New Characterizations of the BDinverse
In this section, we provide several characterizations of the BD-inverse of A ∈ C n×n (in the case when it exists) mainly in terms of certain matrix equations and EP-property. By Lemma 3, we know that A (− 1) (L) is an outer inverse of A. Using this property, we present several new characterizations of the BD-inverse of A.
exists and let X ∈ C n×n . Te following statements are equivalent: Suppose that X is given by the following equation: where X 1 ∈ C l×l , X 2 ∈ C l×(n− l) , X 3 ∈ C (n− l)×l , and X 4 ∈ C (n− l)×(n− l) . Let P L and A be given by equations (3) and (4), respectively. By P L XP L � X, we have

□
In the following theorem, we present diferent characterizations of the BD-inverse in terms of two matrix equations.
exists and let X ∈ C n×n . Te following statements are equivalent: c) P L AX � P L and P L X � X (d) XAP L � P L and P L XP L � X (e) P L AX � P L and P L XP L � X (f ) AX � P AL,L ⊥ and P L X � X (g) XA � P L,(A * L) ⊥ and XP L � X Proof. Item (a) implies any of the assertions (b) − (g)which follows directly by Lemma 3 and Teorem 3. For the converse implications, we will only give the proof that (b) implies (a) since other proofs are similar.
(b)⟹(a): Let P L , A, and X be given by equations (3), (4), and (7), respectively. Te condition XP L � X implies In the following theorem, we discuss other characterizations of the BD-inverse using this fact.
exists and let X ∈ C n×n . Te following statements are equivalent: Since P L X � X, multiplying XAP L � P L by X from the right, we get XAX � X. Ten, R(XA) � R(X), N(AX) � N(X), and AX is idempotent. From XAP L � P L and P L X � X, we get L ⩽ R(XA) and R(X) ⩽ L, respectively, which implies (e) ⟹(a): Let P L , A, and X be given by equations (3), (4), and (7), respectively. From XP L � X and X ∈ C EP n , we have

Different Representation of the BD-inverse
Theorem 7. Let A ∈ C n×n and L ⩽ C n and let a, b, c, d ∈ C be such that a + b ≠ 0 and cd ≠ 0. If A (− 1) (L) exists, then

Journal of Mathematics
Proof. Let P L and A be given by equations (3) and (4), respectively. We have the following equation: Evidently, aAP L + dP L ⊥ + bP L AP L is nonsingular and using equation (5) and the facts that a + b ≠ 0 and cd ≠ 0, we get the following equation: Similar, we have the following equation: Ten, Te rest of the proof follows similarly.
□ Remark 8. Under the hypotheses of Teorem 7 and additional assumption a � 0, we have the following equation: while if b � 0, we have the following equation: If we take a � 1 and d � 1 or b � 1 and d � 1 in Teorem 7, we get results from the paper of Chen [6, Lemma 4 (a)].
In [13], Yuan and Zuo presented several limit expressions for the BD-inverse. Motivated by this result, in the following theorem we give some similar expressions.

Theorem 9.
Let A ∈ C n×n and L ⩽ C n such that AP L + P L ⊥ is nonsingular. Ten, Proof. (a): Let P L and A be given by equations (3) and (4), respectively. By Lemma 2, we know that A L is invertible, so A * L A L is positive defnite. For λ small enough, we have that Let S � P L (λI n +(P L A) * AP L ) − 1 A * P L . By equations (3) and (4), we have the following equation: Hence, from equation (5) and (15), we have the following equation: Assertions (b), (c), and (d) can be proved similarly. □ We use an example to verify Teorem 9 (a).

Example 1. Let
By simple calculation, we have the following equation: Tus, In the next theorem, we present representations for the BD-inverse, using the projectors P � P (A * L) ⊥ ,L and Q � P L ⊥ ,AL .

Theorem 10.
Let A ∈ C n×n and L ⩽ C n be such that AP L + P L ⊥ is nonsingular and let P � P (A * L) ⊥ ,L and Q � P L ⊥ ,AL . For any a, b, c, d ∈ C such that cd ≠ 0 and a + b ≠ 0, the following statements hold:

Proof
(a) Let B � aAP L + bP L AP L + cP L ⊥ P. In terms of Lemma 3, we have the following equation: Ten, we only need to prove that B is invertible. From (iii) and (vi) in Lemma 3, it is easy to derive the following equation: Let P L and A be given by equations (3) and (4), respectively. Ten,

Journal of Mathematics
From a + b ≠ 0, c ≠ 0, and Lemma 2, we can verify the invertibility of B. (b) Te proof can be given as for item (a). (c) By Lemma 3, we have the following equation: Next, we need to prove the invertibility of cP L AP L + dP. Let P L and A be given by equations (3) and (4), respectively. Ten, so cP L AP L + dP is invertible. (d) Te proof can be given as for item (c).

□
We provide the following example to calculate A (− 1) (L) by using Teorem 10 (a).
Example 2. Let the matrix A and the subspace L be given as in Example 1. Ten, By direct calculation, Remark 11. Under the hypotheses of Teorem 10 and additional assumptions b � 0 and a � c � 1, we have the following equation:

Relations of the BD-inverse with Other Generalized Inverses
First of all, we will present the connection between BDinverse and (B, C)-inverse. Recall that Drazin [14] introduced the (b, c)-inverse in a semigroup. In [15], the (B, C)-inverse of matrices was studied by Benítez et al.

Journal of Mathematics
Defnition 12 (see [15]). Let A ∈ C m×n and B, C ∈ C n×m . If there exist a matrix X ∈ C n×m satisfying the following equation: then X is called the (B, C)-inverse of A, denoted by A (B,C) . Te next theorem shows that the BD-inverse of a matrix A ∈ C n×n is a special case of (B, C)-inverse.
Theorem 13. Let A ∈ C n×n and L ⩽ C n such that AP L + P L ⊥ is nonsingular. Ten, Proof. Let B � C � P L . By (i) and (iii) of Lemma 3, we have the following equation: (31) □ From (v) and (vi) of Lemma 3, we have that AA (− 1) (L) and A (− 1) (L) A are oblique projectors. Next, we will discuss the necessary and sufcient conditions for AA (− 1) (L) and A (− 1) (L) A to be the orthogonal projector onto L, which can easily be derived by Lemma 3.

Theorem 14.
Let A ∈ C n×n and L ⩽ C n such that A (− 1) (L) exists. Te following statements hold: In the following theorem, we give the relationships between the BD-inverse and other generalized inverses such as Moore-Penrose inverse A † , Drazin inverse A D , core-EP inverse A ◯ † , DMP inverse A D, † , generalized Moore-Penrose inverse A ⊗ , dual DMP inverse A †,D , BT-inverse A ◇ , and weak group inverse A ⓦ .
More than that in [23], Pearl introduced the set of EP-matrices and in [24], weak group matrices were defned by AA ⓦ � A ⓦ A. Motivated by the above, we introduce the defnition of the BD-matrices as follows: Defnition 17. Let A ∈ C n×n and L ⩽ C n be such that A (− 1) exists. Ten, A is called a BD-matrix with respect to L if and only if AA (− 1) Te set of all BD-matrices with respect to L is denoted by the following equation: In the following theorem, we give some characterizations of the BD-matrices.

Theorem 18.
Let A ∈ C n×n and L ⩽ C n such that A (− 1) (L) exists. Te following statements are equivalent:
(d)⟺(f): It is clear that (c)⟺(g): If A is given by equation (4), where B L � O and C L � O, we have the following equation: Terefore, it is easy to verify A † P L � P L A † . On the

Theorem 19.
Let A ∈ C n×n be given in equation (4), L ⩽ C n , and dim(L) � l be such that A (− 1) (L) exists. Let P � AA (− 1) (L) and Q � A (− 1) (L) A. Ten, the following statements are equivalent: (a)⟹(b). Since P � Q, it directly follows from (vi) in Lemma 3.  (4) and (5), we have the following equation: Tus, if (P − Q) 2 � P − Q, by simple calculation, we can verify B L � O and C L � O. By the equivalence between (a) and (c) in Teorem 18, item (a) holds.

Conclusion
In this paper, some characterizations of the BD-inverse are derived from certain matrices and EP matrix. Some representations of the BD-inverse are also given. Finally, we show the relationships between BD-inverse and other generalized inverse and give the defnition of the BD-matrix.
It is interesting to remark that analogous results can also be given in the case of generalized BD-inverse (see [11]) as well as in the setting of bounded linear operators. On a basis of the current research background, there are many topics on the BD-inverse which can be discussed. Some ideas are given as follows: (1) Te solution of the restricted matrix equation (2) Te iterative algorithm for computing the BD-inverse according to [25] (3) Te perturbation analysis for the solution of restricted linear systems