Journal of Mathematics
Volume 2019 (2019), Article ID 1631979, 13 pages
https://doi.org/10.1155/2019/1631979

Research Article

Determinantal Representations of the Core Inverse and Its Generalizations with Applications

Ivan I. Kyrchei

Pidstrygach Institute for Applied Problems of Mechanics and Mathematics, NAS of Ukraine, Lviv, Ukraine

Correspondence should be addressed to Ivan I. Kyrchei; st260664@gmail.com

Received 11 June 2019; Accepted 17 August 2019; Published 1 October 2019

Academic Editor: Mehdi Ghatee

Copyright © 2019 Ivan I. Kyrchei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this paper, we give the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author. Since the Bott-Duffin inverse has close relation with the core inverse, we give its determinantal representation and its application in finding solutions of the constrained linear equations that is an analog of Cramer’s rule. A numerical example to illustrate the main result is given.

1. Introduction

In the whole article, the notations and are reserved for fields of the real and complex numbers, respectively. stands for the set of all matrices over . determines its subset of matrices with rank r. For , the symbols , , and specify the transpose, the conjugate transpose, and the rank of , respectively. or stands for its determinant. A matrix is Hermitian if .

means the Moore-Penrose inverse of , i.e., the exclusive matrix satisfying the following four equations:

For with index , i.e., the smallest positive number such that , the Drazin inverse of , denoted by , is called the unique matrix that satisfies equation (2) and the following equations:

In particular, if , then the matrix is called the group inverse and it is denoted by . If , then is nonsingular and .

It is evident that if the condition (5) is fulfilled, then (6a) and (6b) are equivalent. We put both these conditions because they will be used below independently of each other and without the obligatory fulfillment of (5).

A matrix satisfying the conditions is called an -inverse of and is denoted by . The set of matrices is denoted by . In particular, is called the inner inverse of , is called the outer inverse of , is called the reflexive inverse of , is its Moore-Penrose inverse, etc.

For an arbitrary matrix , we denote by

(i), the kernel (or the null space) of (ii) the column space (or the range space) of (iii), the row space of

and are the orthogonal projectors onto the range of and the range of , respectively.

The core inverse was introduced by Baksalary and Trenkler in [1]. Later, it was investigated by Xu et al. in [2], among others. Rakić et al. in [3] generalized the core inverse of a complex matrix to the case of an element in a ring.

Definition 1 [1]. A matrix is called the core inverse of if it satisfies the conditions

When such matrix exists, it is denoted by .
In 2014, the core inverse was extended to the core-EP inverse defined by Manjunatha Prasad and Mohana [4]. Other generalizations of the core inverse were recently introduced for complex matrices, namely, BT inverses [5], DMP inverses [6], CMP inverses [7], etc. The characterizations, computing methods, some applications of the core inverse, and its generalizations were recently investigated in complex matrices and rings (see, e.g., [3, 8–18]).
In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, for generalized inverse matrices there exist different determinantal representations as a result of the search of their more applicable explicit expressions (see, e.g., [19–25]). In this paper, we get new determinantal representations of the core inverse and its generalizations by using the author’s recently obtained determinantal representations of the Moore-Penrose inverse and the Drazin inverse over the quaternion skew field and the field of complex numbers as a special case [26, 27].
The results concerning quaternion matrices have been achieved thanks to the theory of row-column determinants introduced in [28, 29]. Within the framework of the theory of row-column determinants, determinantal representations of various kind of generalized inverses, generalized inverse solutions (analogs of Cramer’s rule) of quaternion matrix equations have been derived by the author (see, e.g., [30–38]) and by other researchers (see, e.g., [39–41]).
Note that a determinantal representation of the core-EP generalized inverse in complex matrices has been derived in [4] based on the determinantal representation of a reflexive inverse obtained in [19, 20].
One of possible applications of the determinantal representations of the inverse matrix is Cramer’s rule to find solutions of systems of linear equations. Determinantal representations of generalized inverses have similar applications. In this paper, we derive the determinantal representation of the Bott-Duffin inverse that has close relation with the core inverse, and apply it to obtain Cramer’s rule for the constrained linear equations.
The paper is organized as follows. In Section 2, we start with preliminary introduction of determinantal representations of the Moore-Penrose inverse and the Drazin inverse. In the next sections, we give determinantal representations of the core inverse and its generalizations. In particular, determinantal representations of the right and left core inverses are established in Section 3, of the right and left core-EP inverses in Section 4, and of the DMP inverse and its dual MPD inverse in Section 5. Determinantal representations of the CMP inverse are obtained in Section 6. In Section 7, we derive Cramer’s rule for the constrained linear equations by using the determinantal representation of the Bott-Duffin inverse that is same as for the right core inverse. A numerical example to illustrate the main results is considered in Section 8. Finally, in Section 9, the conclusions are drawn.

2. Preliminaries

Let and be subsets with . The submatrix of with rows and columns indexed by α and β, respectively, and denoted by . Then, is a principal submatrix of with rows and columns indexed by α, and is the corresponding principal minor of the determinant . Suppose that

stands for the collection of strictly increasing sequences of integers chosen from . For fixed and , put and .

Denote by and , and the jth columns and the ith rows of and , respectively. By and , we denote the matrices obtained from by replacing its ith row with the row and its jth column with the column .

Theorem 1 [21]. If , then the Moore-Penrose inverse possess the determinantal representations

Remark 1. For an arbitrary full-rank matrix , a row-vector , and a column-vector , we mean, respectively,

Corollary 1 [21]. Let . Then, the following determinantal representations can be obtained:

(i)For the projector ,

 where is the jth column and is the ith row of .(ii)For the projector ,

 where is the ith row and is the jth column of .

The following lemma gives determinantal representations of the Drazin inverse in complex matrices.

Lemma 1 [22]. Let with and . Then, the determinantal representations of the Drazin inverse are

where is the ith row and is the jth column of .

Corollary 2 [22]. Let with and . Then, the determinantal representations of the group inverse are

3. Determinantal Representations of the Core Inverses

Together with the core inverse in [3], it was introduced the dual core inverse. Since the both these core inverses are equipollent and they are different only in the position relative to the inducting matrix , we propose to call them as the right and left core inverses regarding to their positions. So, according to [1], we have the following definition that is equivalent to Definition 1.

Definition 2. A matrix is said to be the right core inverse matrix of if it satisfies the conditions

When such matrix exists, it is denoted by .
The following definition of the left core inverse can be given that is equivalent to the introduced dual core inverse [3].

Definition 3. A matrix is said to be the left core inverse matrix of if it satisfies the conditions

When such matrix exists, it is denoted by .

Remark 2. In [42], the conditions of the dual core inverse are given as follows:

Since and , then these conditions and (17) are analogous.

Remark 3. In Definitions 2 and 3, we purposely state that these definitions concern with matrices, since the right and core inverses in a ring were introduced recently in [43]. The notions of the right and left core inverse matrices, introduced here, and of the right and left core inverses in a ring, introduced in [43], have no direct connection with each other.
According to [1], we introduce the following sets of matrices:

The matrices from are called group matrices or core matrices. If then clearly . It is known that the core inverses of exist if and only if or . Moreover, if is nonsingular, , then its core inverses are the usual inverse. According to [1], we have the following representations of the right and left core inverses.

Lemma 2 [1]. Let . Then,

Remark 4. In Theorems 2 and 3, we will suppose that but . if and (in particular, is Hermitian), then from Lemma 2 and the definitions of the Moore-Penrose and group inverses it follows that .

Theorem 2. Let and . Then, its right core inverse matrix has the following determinantal representations:

where

are the row and column vectors, respectively. Here, and are the fth column and lth row of .

Proof. Taking into account (20), we have for

By substituting (15) and (12) in (25), we obtain

where and are the unit column and row vectors, respectively, such that all their components are 0, except the lth components which are 1; is the th element of the matrix .
Let

Construct the matrix . It follows that

where is the ith row of . So, we get (22).
If we first consider

and construct the matrix , then from

it follows (22).

Taking into account (21), the following theorem on the determinantal representation of the left core inverse can be proved similarly.

Theorem 3. Let and . Then, for its left core inverse matrix , we have

where

Here, and are the fth column and lth row of .

4. Determinantal Representations of the Core-EP Inverses

Similar to [4], we introduce two core-EP inverses.

Definition 4. A matrix is said to be the right core-EP inverse of if it satisfies the conditions

It is denoted by .

Definition 5. A matrix is said to be the left core-EP inverse of if it satisfies the conditions

It is denoted by .

Remark 5. Since , then the left core-EP inverse of is similar to the core inverse introduced in [4], and the dual core-EP inverse introduced in [42].
According to [4], we have the following representations the core-EP inverses of :

Thanks to [42], the following representations of the core-EP inverses will be used for their determinantal representations.

Lemma 3. Let and . Then,

Moreover, if , then we have the following representations of the right and left core inverse matrices:

Theorem 4. Suppose , , and , and there exist and . Then, and possess the determinantal representations, respectively,

where is the ith row of and is the jth column of .

Proof. Consider and . By (36),

Taking into account (10) for the determinantal representation of , we get

where is the tth row of . Since , then it follows (40).
The determinantal representation (41) can be obtained similarly by integrating (9) for the determinantal representation of in (37).

Taking into account the representations (38)–(39), we derive the determinantal representations of the right and left core inverse matrices that have simpler expressions than those obtained in Theorems 2 and 3.

Corollary 3. Let and , and there exist and . Then, and can be expressed as follows:

where is the ith row of and is the jth column of .

5. Determinantal Representations of the DMP and MPD Inverses

The concept of the DMP inverse in complex matrices was introduced in [6] by Malik and Thome.

Definition 6. [6] Suppose and . A matrix is said to be the DMP inverse of if it satisfies the conditions

It is denoted by .
According to [6], if an arbitrary matrix satisfies the system of equations (45), then it is unique and has the following representation:

Theorem 5. Let , , and . Then, its DMP inverse has the following determinantal representations:

where

Here, and are the fth column and the lth row of .

Proof. Taking into account (46) for , we get

By substituting (14) and (10) for the determinantal representations of and in (49), we get

where and are the lth unit column and row vectors and is the th element of the matrix . If we put

as the fth component of the row vector . Then from

it follows (47). If we initially obtain

the lth component of the column vector . Then from

it follows (47).

The name of the DMP inverse is in accordance with the order of using the Drazin inverse (D) and the Moore-Penrose (MP) inverse. In that connection, it would be logical to consider the following definition.

Definition 7. Suppose and . A matrix is said to be the MPD inverse of if it satisfies the conditions

It is denoted by .
Similar as for the DMP inverse, it can be proved that the matrix is unique, and it can be represented as

Theorem 6. Let , , and . Then, its MPD inverse has the following determinantal representations:

where

Here, and are the lth row and the fth column of .

Proof. The proof is similar to the proof of Theorem 6.

6. Determinantal Representations of the CMP Inverse

Definition 8 [7]. Suppose has the core-nilpotent decomposition , where , is nilpotent, and . The CMP inverse of is called the matrix .

Lemma 4 [7]. Let . The matrix is the unique matrix that satisfies the following system of equations:

Moreover,

Taking into account (60), it follows the next theorem about determinantal representations of the quaternion CMP inverse.

Theorem 7. Let , , and . Then, the determinantal representations of its CMP inverse can be expressed as

for all , where

Here, is the tth row and is the kth column of , is the tth row and is the kth column of , and the matrices and are such that

where is the ith row of and is the jth column of .

Proof. Suppose , , and . Taking into account (60), we get

(a)Taking into account the expressions (14), (12), and (13) for the determinantal representations of , , and , respectively, we have

 where is the tth column of , is the lth row of , and is the tth row of . So, it is clear that

 where is the tth unit column vector, is the kth unit row vector, and is the th element of . Denote

 the tth component of a column-vector . Since

 then

 Construct the matrix , where is given by (70), and denote . Then, taking into account that , we have

 If we put that

 is the tth component of a column vector, . Then from

 it follows (61) with given by (62). If we initially put

 as the kth component of the row vector, . Then from

 it follows (61) with given by (64).(b)By using the determinantal representation (14) for in (67), we have

 Therefore,

 where is the kth unit column vector, is the lth unit row vector, and is the th element of .Denote by

 the lth component of a row-vector . Then,

 From this, it follows that

 Construct the matrix , where is given by (80). Denote . Then

 Similar to the above, if we denote by

 the tth component of a column vector , then

 Thus, we have (61) with given by (64). If, now, we denote by

 the kth component of the row vector . Then, from

So, finally, we have (61) with given by (65).

Remark 6. Concerning the possible existence of dual to the CMP inverse, we note the following. Taking into account ,

7. Cramer’s Rule for Some Constrained Linear Equations

Cramer’s rules for special solutions to systems of linear equations stand as possible applications of determinantal representations of the core inverse and its generalizations. For instance, consider Cramer’s rule for the system which has the important applied significance.

First, note that the core inverse has close relation with the Bott-Duffin inverse. According to [19], the Bott-Duffin inverse of with respect to can be given by

where needs to be nonsingular. According to [1], coincides with the right core inverse . One can find more details regarding the Bott-Duffin inverse in [44].

Consider the constrained linear equations

where and . According to [45], this equation arises in electrical networks and its solution is determined by the following lemma.

Lemma 5. Let be nonsingular. Then, equation (90) has for every the unique solution

The following theorem gives Cramer’s rule of finding the solution (91).

Theorem 8. Suppose that exists. Then,

where , is such that

and are the ith row of .

Proof. Suppose and . Since the Bott-Duffin inverse coincides with the right core inverse , then its determinantal representation can be obtained by Theorem 2. We use the determinantal representation (22). So, for all , we have

Construct the matrix with columns determined by (24) and denote . Then, from using this denotation in (95), it follows (93).

It is evident that the solution from (92) by the components can be expressed as

where is the ith component of the column vector .

8. An Example

Consider the matrix

Since

then and , then and . So, we shall find and by (40) and (41), respectively. Since

then by (40)

By similarly continuing, we get

By analogy, according to (41), we have

The DMP inverse can be find by Theorem 5. Since

and , then

Furthermore, by (47),

By similarly continuing, we get

Similarly by Theorem 6, we get

Finally, by Theorem, we find the CMP inverse . Since , then and

Furthermore, by (65),

By similar calculations, we get

So, by (61), we get

By similarly continuing, we derive

9. Conclusions

In this chapter, we get the direct method to find of the core inverse and its generalizations that is based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, and the DMP, MPD, and CMP inverses are derived. Determinantal representation the Bott-Duffin inverse and its application to get Cramer’s rule of the solution to the constrained linear equations are obtained.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest.

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