New Characterizations and Representations of the Bott–Duffin Inverse

Wu, Jiabao; Zuo, Kezheng; Cvetković-Ilić, Dragana S.; Zou, Honglin

doi:https://doi.org/10.1155/2023/7623837

Journal of Mathematics

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Abstract Introduction Preliminaries Conclusion Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 7623837 | https://doi.org/10.1155/2023/7623837

New Characterizations and Representations of the Bott–Duffin Inverse

Jiabao Wu,¹Kezheng Zuo,¹Dragana S. Cvetković-Ilić,²and Honglin Zou¹

Academic Editor: Xian-Ming Gu

Received01 Mar 2023

Revised29 May 2023

Accepted06 Jun 2023

Published01 Aug 2023

Abstract

The paper focuses on the class of the Bott–Duffin inverses. Several original features of the class are identified 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.

1. Introduction

stands for the vector space of n-tuples over the field of complex numbers. The symbol denotes the set of complex matrices. The symbols and rank represent the range space, null space, conjugate transpose, and rank of , respectively. The symbol stands for the index of which is the smallest non-negative integer such that . The symbol means the identity matrix in . The symbol means the null matrix. If is a subspace of , we use the notation while means the orthogonal complement subspace of . The dimension of is denoted by . stands for the oblique projector onto along , where and . is the orthogonal projector onto .

Additionally, the Moore–Penrose inverse of is the unique matrix verifying (see [1–4]).

A matrix that satisfies is called an outer inverse of and is denoted by . Let , and , . There exists a unique outer inverse of such that and if and only if . In case, there exists , we call an outer inverse with prescribed range and null space and denote it by (see [1, 4]).

The symbol stands for the Drazin inverse of which is the unique matrix satisfying , where (see [5]). Especially, if , then the Drazin inverse of is called the group inverse of and is denoted by .

Bott and Duffin, in their famous paper [6], introduced the “constrained inverse” of a square matrix as an important tool in the electrical network theory. This inverse is called in their honor the Bott–Duffin inverse (in short, BD-inverse).

Definition 1 (see [6]). Let and . If is nonsingular, then the BD-inverse of with respect to , denoted by , is defined by the following equation:There are huge literatures on the BD-inverse and here we will mention only the part. Some important applications of the BD-inverse can be found in the monograph [1]. Chen [7] presented several properties and different representations of the BD-inverse. Also, certain relationships between a class of nonsingular bordered matrices and the BD-inverse are given in [8]. In [9], Wei studied the various norm-wise relative condition numbers that measure the sensitivity of the BD-inverse and the solution of constrained linear systems. The perturbation theory for the BD-inverse was discussed in [10].
In [11], Chen defined the generalized BD-inverse of (denoted by ). In terms of the form of definitions, is a natural extension of . However, if is arbitrary, we have by using Theorem A.1 (see Appendix A) and [5, Lemma 1]. It is not convenient or even difficult to study the properties of . In order to avoid this difficulty and obtain more interesting properties of , all of the theorems in [11, 12] restrict matrix to an -p.s.d matrix, which satisfies the following three conditions:(i)(ii) for all (iii) for implies However, for , matrix does not need to satisfy these conditions, only needs to exist. And the necessary and sufficient condition for the existence of is given in Lemma 2. Therefore, for the differences in studying and , it is meaningful to research the properties of .
The 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. The definition and properties of the BD-matrices are also given. Several original features of the BD-inverses are identified and new properties are characterized. In some cases, the results available in the literature are recaptured therein in a more general form.
The 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 definition of the BD-matrices and present some of their properties.
Henceforth, the symbol will stand for the set of EP matrices, i.e.,

2. Preliminaries

Let and . In order to discuss some properties of the BD-inverse, we will consider appropriate matrix decomposition of with respect to . Since there exists a unitary matrix such thatwhere , a matrix can be written as follows:where , , , and .

Now, we are ready to give the necessary and sufficient condition for the existence of as well as the representation of .

Lemma 2. Let and be given by equations (3) and (4), respectively. exists if and only if is invertible. In this case,

Proof. By equations (3) and (4), we get the following equation:Evidently, is invertible if and only if is invertible. In this case, from equations (1) and (6), we get that equation (5) is satisfied.

The next lemma gives some basic properties of the BD-inverse, for example, that it is an outer inverse of with range and null space , etc.

Lemma 3 (see [7]). Let and . If is invertible, then the following statements hold:(i)(ii)(iii)(iv)(v)(vi)

3. Some New Characterizations of the BD-inverse

In this section, we provide several characterizations of the BD-inverse of (in the case when it exists) mainly in terms of certain matrix equations and EP-property. By Lemma 3, we know that is an outer inverse of . Using this property, we present several new characterizations of the BD-inverse of .

Theorem 4. Let and be such that exists and let . The following statements are equivalent:(a)(b)(c)(d)(e)(f).

Proof. : This follows directly by , , and of Lemma 3. : From , we have . Since , it follows that and by , we get . Hence, . By , we have which together with gives . Thus, . : Since and , we have , which implies . Since and , it follows that . Thus, , which implies . : The proof is similar to the implication . : By and , we have , which gives . From and , we get , which implies . Thus, . : Suppose that is given by the following equation: where , , , and . Let and be given by equations (3) and (4), respectively. By , we have , , and . Also, is equivalent to . Since , it follows that is invertible and . Thus, by equation (5), we have .

In the following theorem, we present different characterizations of the BD-inverse in terms of two matrix equations.

Theorem 5. Let and be such that exists and let . The following statements are equivalent:(a)(b)(c)(d)(e)(f)(g)

Proof. Item implies any of the assertions which follows directly by Lemma 3 and Theorem 3. For the converse implications, we will only give the proof that implies since other proofs are similar.
: Let , , and be given by equations (3), (4), and (7), respectively. The condition implies . By , we have that and . Hence, from equation (5), we have .

Using the representation of given in Lemma 2, we can easily conclude that . In the following theorem, we discuss other characterizations of the BD-inverse using this fact.

Theorem 6. Let and be such that exists and let . The following statements are equivalent:(a)(b)(c)(d)(e)

Proof. : It is obvious by , , and of Lemma 3. : Evidently, implies . : Since , multiplying by from the right, we get . Then, , , and is idempotent. From and , we get and , respectively, which implies . Hence, . Since and , we have . So and . Thus, . : From , we easily get . : Let , , and be given by equations (3), (4), and (7), respectively. From and , we have , , and . By , it follows that . Now, by equation (5) from Lemma 2, we get .

4. Different Representation of the BD-inverse

Theorem 7. Let and and let be such that and . If exists, then

Proof. Let and be given by equations (3) and (4), respectively. We have the following equation:Evidently, is nonsingular and using equation (5) and the facts that and , we get the following equation:Similar, we have the following equation:Then,The rest of the proof follows similarly.

Remark 8. Under the hypotheses of Theorem 7 and additional assumption , we have the following equation:while if , we have the following equation:If we take and or and in Theorem 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 and such that is nonsingular. Then,(a)(b)(c)(d)

Proof. : Let and be given by equations (3) and (4), respectively. By Lemma 2, we know that is invertible, so is positive definite. For small enough, we have that is invertible; so,Let . By equations (3) and (4), we have the following equation:Hence, from equation (5) and (15), we have the following equation:Assertions , and can be proved similarly.

We use an example to verify Theorem 9.

Example 1. LetBy simple calculation, we have the following equation:Thus,In the next theorem, we present representations for the BD-inverse, using the projectors .

Theorem 10. Let and be such that is nonsingular and let and . For any such that and , the following statements hold:(a)(b)(c)(d)

Proof. (a)Let . In terms of Lemma 3, we have the following equation: Then, we only need to prove that is invertible. From and in Lemma 3, it is easy to derive the following equation: Let and be given by equations (3) and (4), respectively. Then, From , , and Lemma 2, we can verify the invertibility of .(b)The proof can be given as for item .(c)By Lemma 3, we have the following equation: Next, we need to prove the invertibility of . Let and be given by equations (3) and (4), respectively. Then, so is invertible.(d)The proof can be given as for item (c).

We provide the following example to calculate by using Theorem 10 .

Example 2. Let the matrix and the subspace be given as in Example 1. Then,By direct calculation,

Remark 11. Under the hypotheses of Theorem 10 and additional assumptions and , we have the following equation:

5. Relations of the BD-inverse with Other Generalized Inverses

First of all, we will present the connection between BD-inverse and -inverse. Recall that Drazin [14] introduced the -inverse in a semigroup. In [15], the -inverse of matrices was studied by Benítez et al.

Definition 12 (see [15]). Let and . If there exist a matrix satisfying the following equation:then is called the -inverse of , denoted by .
The next theorem shows that the BD-inverse of a matrix is a special case of -inverse.

Theorem 13. Let and such that is nonsingular. Then,

Proof. Let . By and of Lemma 3, we have the following equation:Thus, .

From and of Lemma 3, we have that and are oblique projectors. Next, we will discuss the necessary and sufficient conditions for and to be the orthogonal projector onto , which can easily be derived by Lemma 3.

Theorem 14. Let and such that exists. The following statements hold:(a) if and only if (b) if and only if

Proof. Evidently.
In the following theorem, we give the relationships between the BD-inverse and other generalized inverses such as Moore–Penrose inverse , Drazin inverse , core-EP inverse , DMP inverse , generalized Moore–Penrose inverse , dual DMP inverse , BT-inverse , and weak group inverse .

Theorem 15. Let , , and be such that exists. The following statements hold:(a)(b)(c)(d)(e)(f)(g)(h)

Proof. : Since and , we have that if and only if which are equivalent to and . : Since (see [1]) and (see [16]), the proof can be obtained directly by . : Since (see [17]), it follows that if and only if and . However, , so if and only if . : From (see [18]), , and , we have if and only if and which are further equivalent to and . : It is well-known that (see [19]) and . Therefore, if and only if and i.e., . : Since (see [20]), the proof follows directly. : Note the fact that in [21], by and , we have if and only if and .

In the next result, we consider the cases when the BD-inverse is equal to the core inverse and group inverse , respectively, which means that we must consider these properties on the set of matrices of index 1 that will be denoted by , i.e.,

Remark 16. Let and be such that exists. Then,(a)(b)In [22], we have investigated the properties of . Naturally, we investigate the properties of (i.e., ). More than that in [23], Pearl introduced the set of EP-matrices and in [24], weak group matrices were defined by . Motivated by the above, we introduce the definition of the BD-matrices as follows:

Definition 17. Let and be such that exists. Then, is called a BD-matrix with respect to if and only if . The set of all BD-matrices with respect to is denoted by the following equation:In the following theorem, we give some characterizations of the BD-matrices.

Theorem 18. Let and such that exists. The following statements are equivalent:(a)(b)(c) is given by equation (4), where and ;(d);(e);(f);(g);(h);(i).

Proof. : By Lemma 3, we have the following equation: : Let and be given by equations (4) and (5). Then, : This follows directly from equations (4) and (5). : It is clear that : If is given by equation (4), where and , we have the following equation:Therefore, it is easy to verify . On the converse, if , then , where and . Evidently, . Hence, , , , and .
Equivalences , , and can easily be verified.

From Definition 17, we use to characterize BD-matrices, and in the following theorem, we provide other equivalent characterizations of BD-matrices by and .

Theorem 19. Let be given in equation (4), , and be such that exists. Let and . Then, the following statements are equivalent:(a)(b)(c)(d) is idempotent

Proof. . Since , it directly follows from (vi) in Lemma 3. . From equations (4) and (5), it is easy to obtain and by simple calculation. In terms of the equivalence between and in Theorem 18, item holds. . Since and and are idempotent, it follows that . . Using (i) in Lemma 3 and multiplying by from the left side, we have . Hence, we have . . From equations (4) and (5), we have the following equation:Thus, if , by simple calculation, we can verify and . By the equivalence between and in Theorem 18, item holds.

6. 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 definition 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)The solution of the restricted matrix equation(2)The iterative algorithm for computing the BD-inverse according to [25](3)The perturbation analysis for the solution of restricted linear systems

Appendix

A Theorem Used in the Proof

Theorem A.1. Let . Then,

Proof. It is clear that . Let . Then, there exists such that . Since , then we have , which means . This prove that . Let , where and . It is clear that . Therefore, . Since , it follows that . Thus, . Let . Since , we have . From , we can obtain , where is denoted by the inner product of two vectors. The proof is completed.

Data Availability

No underlying data were collected or produced in this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11961076).

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