Determinantal Representations of the Core Inverse and Its Generalizations

Generalized inverse matrices are important objects in matrix theory. In particu-lar, they are useful tools in solving matrix equations. The most famous generalized inverses are the Moore-Penrose inverse and the Drazin inverse. Recently, it was introduced new generalized inverse matrix, namely the core inverse, which was late extended to the core-EP inverse, the BT, DMP, and CMP inverses. In contrast to the inverse matrix that has a definitely determinantal representation in terms of cofactors, even for basic generalized inverses, there exist different determinantal representations as a result of the search of their more applicable explicit expressions. In this chapter, we give new and exclusive determinantal representations of the core inverse and its generalizations by using determinantal representations of the Moore-Penrose and Drazin inverses previously obtained by the author.


Introduction
In the whole chapter, the notations  and  are reserved for fields of the real and complex numbers, respectively.  mÂn stands for the set of all m Â n matrices over .  mÂn r determines its subset of matrices with a rank r. For A ∈  mÂn , the symbols A * and rk A ð Þ specify the conjugate transpose and the rank of A, respectively, |A| or detA stands for its determinant. A matrix A ∈  nÂn is Hermitian if A * ¼ A.
A † means the Moore-Penrose inverse of A ∈  nÂm , i.e., the exclusive matrix X satisfying the following four equations: For A ∈  nÂn with index Ind A ¼ k, i.e., the smallest positive number such that rk A kþ1 ¼ rk A k , the Drazin inverse of A, denoted by A d , is called the unique matrix X that satisfies Eq. (2) and the following equations, AX ¼ XA; In particular, if Ind A ¼ 1, then the matrix X is called the group inverse, and it is denoted by X ¼ A # . If Ind A ¼ 0, then A is nonsingular and It is evident that if the condition (5) is fulfilled, then (6) and (7) 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 A satisfying the conditions i ð Þ, j ð Þ, … is called an i, j, … f g-inverse of A, and is denoted by A i, j,… ð Þ . The set of matrices A i, j,… ð Þ is denoted A i, j, … f g. In particular, A 1 ð Þ is called the inner inverse, A 2 ð Þ is called the outer inverse, A 1,2 ð Þ is called the reflexive inverse, A 1,2,3,4 ð Þ is the Moore-Penrose inverse, etc. For an arbitrary matrix A ∈  mÂn , we denote by , the kernel (or the null space) of A; , the column space (or the range space) of A; and , the row space of A. P A ≔ AA † and Q A ≔ A † A are the orthogonal projectors onto the range of A and the range of A * , respectively.
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][20][21][22][23][24][25]). In this chapter, we get new determinantal representations of the core inverse and its generalizations using recently obtained by the author determinantal representations of the Moore-Penrose inverse and the Drazin inverse over the quaternion skew field, and over the field of complex numbers as a special case [26][27][28][29][30][31][32][33][34]. 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 an reflexive inverse obtained in [19,20].
The chapter 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 Section 3, we give determinantal representations of the core inverse and its generalizations, namely the right and left core inverses are established in Section 3.1, the core-EP inverses in Section 3.2, the core DMP inverse and its dual in Section 3.3, and finally the CMP inverse in Section 3.4. A numerical example to illustrate the main results is considered in Section 4. Finally, in Section 5, the conclusions are drawn.

Preliminaries
we denote a submatrix of A ∈  mÂn with rows and columns indexed by α and β, respectively. Then, A α α is a principal submatrix of A with rows and columns indexed by α, and A j j α α is the corresponding principal minor of the determinant |A|. Suppose that stands for the collection of strictly increasing sequences of 1 ≤ k ≤ n integers chosen from 1, …, n f g. For fixed i ∈ α and j ∈ β, put I r,m i f g ≔ α : α ∈ L r,m , i ∈ α f g and J r,n j f g ≔ β : β ∈ L r,n , j ∈ β f g . The jth columns and the ith rows of A and A * denote a : j and a * : j and a i: and a * i: , respectively. By A i: b ð Þ and A : j c ð Þ, we denote the matrices obtained from A by replacing its ith row with the row b, and its jth column with the column c.
Remark 2.2. For an arbitrary full-rank matrix A ∈  mÂn r , a row vector b ∈  1Âm , and a column-vector c ∈  nÂ1 , we put, respectively, i. for the projector where _ a : j is the jth column and _ a i: is the ith row of A * A; and ii. for the projector where € a i: is the ith row and € a : j is the jth column of AA * . The following lemma gives determinantal representations of the Drazin inverse in complex matrices.
where a k ð Þ i: is the ith row and a

Determinantal representations of the core inverses
Together with the core inverse in [35], the dual core inverse was to be introduced. Since the both these core inverses are equipollent and they are different only in the position relative to the inducting matrix A, we propose called them as the right and left core inverses regarding to their positions. So, from [1], we have the following definition that is equivalent to Definition 1.1.

Definition 3.1.
A matrix X ∈  nÂn is said to be the right core inverse of A ∈  nÂn if it satisfies the conditions When such matrix X exists, it is denoted as A ○ # .
The following definition of the left core inverse can be given that is equivalent to the introduced dual core inverse [35].
Definition 3.2 A matrix X ∈  nÂn is said to be the left core inverse of A ∈  nÂn if it satisfies the conditions When such matrix X exists, it is denoted as A ○ # . Remark 3.3. In [35], the conditions of the dual core inverse are given as follows: Þ, then these conditions and (15) are analogous.
Due to [1], we introduce the following sets of quaternion matrices The matrices from  CM n are called group matrices or core matrices. If A ∈  EP n , then clearly A † ¼ A # . It is known that the core inverses of A ∈  nÂn exist if and only if A ∈  CM n or Ind A ¼ 1. Moreover, if A is nonsingular, Ind A ¼ 0, then its core inverses are the usual inverse. Due to [1], we have the following representations of the right and left core inverses.
Lemma 3.4. [1] Let A ∈  CM n . Then, Remark 3.5. In Theorems 3.6 and 3.7, we will suppose that A ∈  CM n but A ∉  EP n . Because, if A ∈  CM n and A ∈  EP n (in particular, A is Hermitian), then from Lemma 3.4 and the definitions of the Moore-Penrose and group inverses, it follows that Theorem 3.6. Let A ∈  CM n and rk A 2 ¼ rk A ¼ s. Then, its right core inverse has the following determinantal representations where are the row and column vectors, respectively. Hereã :f andã l: are the fth column and lth row ofÃ ≔ A 2 A * .
Proof. Taking into account (16), we have for # A, By substituting (14) and (15) in (20), we obtain where e :l and e l: are the unit column and row vectors, respectively, such that all their components are 0, except the lth components which are 1;ã lf is the (lf)th element of the matrixÃ ≔ A 2 A * . Let and construct the matrix

Functional Calculus
Taking into account (17), the following theorem on the determinantal representation of the left core inverse can be proved similarly.
Theorem 3.7. Let A ∈  CM n and rk A 2 ¼ rk A ¼ s. Then for its left core inverse Here a :f and a l: are the fth column and lth row of A ≔ A * A 2 .

Determinantal representations of the core-EP inverses
Similar as in [4], we introduce two core-EP inverses. Definition 3.8. A matrix X ∈  nÂn is said to be the right core-EP inverse of A ∈  nÂn if it satisfies the conditions It is denoted as A ○ † . Definition 3.9. A matrix X ∈  nÂn is said to be the left core-EP inverse of A ∈  nÂn if it satisfies the conditions It is denoted as A ○ † .

then the left core inverse A ○ † of
A ∈  nÂn is similar to the * core inverse introduced in [4], and the dual core-EP inverse introduced in [35]. Due to [4], we have the following representations the core-EP inverses of A ∈  nÂn , Thanks to [35], the following representations of the core-EP inverses will be used for their determinantal representations. Lemma 3.11. Let A ∈  nÂn and Ind A ¼ k. Then Moreover, if Ind A ¼ 1, then we have the following representations of the right and left core inverses possess the determinantal representations, respectively, whereâ i: is the ith row ofÂ ¼ A k A kþ1 * and a : j is the jth column of . By (21), Taking into account (9) for the determinantal representation of A kþ1 † , we get ¼â i: , then it follows (25).
The determinantal representation (26) can be obtained similarly by integrating (8) for the determinantal representation of A kþ1 † in (22). □ Taking into account the representations (23)- (24), we obtain the determinantal representations of the right and left core inverses that have more simpler expressions than they are obtained in Theorems 3.6 and 3.7.

Determinantal representations of the DMP and MPD inverses
The concept of the DMP inverse in complex matrices was introduced in [2] by S. Malik and N. Thome.
[2] Suppose A ∈  nÂn and Ind A ¼ k. A matrix X ∈  nÂn is said to be the DMP inverse of A if it satisfies the conditions It is denoted as A d, † . Due to [2], if an arbitrary matrix satisfies the system of Eq. (27), then it is unique and has the following representation where :iã :f À Á Here,ã :f andâ l: are the f th column and the lth row ofÃ ≔ A kþ1 A * .
Proof. Taking into account (28) for A d, † , we get By substituting (12) and (9) for the determinantal representations of A d and A † in (31), we get where e :l and e l: are the lth unit column and row vectors, andã lf is the lf ð Þth as the fth component of the row vector u it follows (29). If we initially obtain as the lth component of the column vector u it follows (30). □ 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 3.16. Suppose A ∈  nÂn and Ind A ¼ k. A matrix X ∈  nÂn is said to be the MPD inverse of A if it satisfies the conditions The matrix A †,d is unique, and it can be represented as Here,â l: andâ :f are the lth row and the fth column ofÂ ≔ A * A kþ1 . Proof. The proof is similar to the proof of Theorem 3.15. □

Determinantal representations of the CMP inverse
Definition 3.18. [6] Suppose A ∈  nÂn has the core-nilpotent decomposition Lemma 3.19. [6] Let A ∈  nÂn . The matrix X ¼ A c, † is the unique matrix that satisfies the following system of equations: Taking into account (34), it follows the next theorem about determinantal representations of the quaternion CMP inverse.
a. Taking into account the expressions (13), (10), and (11) for the determinantal representations of A d , Q A , and P A , respectively, we have where _ a :t is the tth column of A * A, € a l: is the lth row of AA * , and a m ð Þ t: is the tth row of A m . So, it is clear that we have where u tl is given by (42), and denotê U ≔ UAA * . Then, taking into account that