Determinantal Representations of the Core Inverse and Its Generalizations

*Ivan I. Kyrchei*

### **Abstract**

Generalized inverse matrices are important objects in matrix theory. In particular, 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.

**Keywords:** Moore-Penrose inverse, Drazin inverse, core inverse, core-EP inverse, **2000 AMS subject classifications:** 15A15, 16W10

#### **1. Introduction**

In the whole chapter, the notations and are reserved for fields of the real and complex numbers, respectively. *<sup>m</sup>*�*<sup>n</sup>* stands for the set of all *<sup>m</sup>* � *<sup>n</sup>* matrices over . *<sup>m</sup>*�*<sup>n</sup> <sup>r</sup>* determines its subset of matrices with a rank *r*. For **A** ∈ *<sup>m</sup>*�*<sup>n</sup>*, the symbols **<sup>A</sup>**<sup>∗</sup> and rkð Þ **<sup>A</sup>** specify the conjugate transpose and the rank of **<sup>A</sup>**, respectively, <sup>∣</sup>**A**<sup>∣</sup> or *det***<sup>A</sup>** stands for its determinant. A matrix **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* is Hermitian if **<sup>A</sup>**<sup>∗</sup> <sup>¼</sup> **<sup>A</sup>**.

**A**† means *the Moore-Penrose inverse* of **A** ∈ *<sup>n</sup>*�*<sup>m</sup>*, i.e., the exclusive matrix **X** satisfying the following four equations:

$$\mathbf{A}\mathbf{X}\mathbf{A}=\mathbf{A}\tag{1}$$

$$\mathbf{X} \mathbf{A} \mathbf{X} = \mathbf{X} \tag{2}$$

$$(\mathbf{A}\mathbf{X})^\* = \mathbf{A}\mathbf{X} \tag{3}$$

$$\left(\mathbf{X}\mathbf{A}\right)^{\*} = \mathbf{X}\mathbf{A} \tag{4}$$

For **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* with index Ind**<sup>A</sup>** <sup>¼</sup> *<sup>k</sup>*, i.e., the smallest positive number such that rk **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> <sup>¼</sup> rk **<sup>A</sup>***<sup>k</sup>* , *the Drazin inverse* of **<sup>A</sup>**, denoted by **<sup>A</sup>***<sup>d</sup>* , is called the unique matrix **X** that satisfies Eq. (2) and the following equations,

$$\mathbf{AX} = \mathbf{X}\mathbf{A};\tag{5}$$

Drazin inverse. In Section 3, we give determinantal representations of the core inverse and its generalizations, namely the right and left core inverses are

*Determinantal Representations of the Core Inverse and Its Generalizations*

the conclusions are drawn.

*DOI: http://dx.doi.org/10.5772/intechopen.89341*

<sup>1</sup>≤*k*<sup>≤</sup> min f g *<sup>m</sup>*, *<sup>n</sup>* . By **<sup>A</sup>***<sup>α</sup>*

*Jr*,*<sup>n</sup>*f g*j* ≔ f g *β* : *β* ∈*Lr*,*<sup>n</sup>*, *j*∈*β* .

and a column-vector **c**∈ *<sup>n</sup>*�<sup>1</sup>

**AA**<sup>∗</sup> ð Þ*<sup>i</sup>:* ð Þ **<sup>b</sup>** � � �

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *: <sup>j</sup>*

**Corollary 2.3.** [21] *Let* **A** ∈ *<sup>m</sup>*�*<sup>n</sup>*

*can be obtained*

**3**

� � �

ð Þ**c**

� � � <sup>¼</sup> <sup>X</sup> *β* ∈*Jn*,*n*f g*j*

columns indexed by *α* and *β*, respectively. Then, **A***<sup>α</sup>*

with rows and columns indexed by *<sup>α</sup>*, and j j **<sup>A</sup>** *<sup>α</sup>*

of the determinant ∣**A**∣. Suppose that

*possesses the determinantal representations*

*a*† *ij* ¼

> ¼ P

**Remark 2.2.** For an arbitrary full-rank matrix **A** ∈ *<sup>m</sup>*�*<sup>n</sup>*

� <sup>¼</sup> <sup>X</sup> *α* ∈*Im*,*m*f g*i*

**AA**<sup>∗</sup> j j <sup>¼</sup> <sup>X</sup>

P

**2. Preliminaries**

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,

Let *α* ≔ f g *α*1, …, *α<sup>k</sup>* ⊆f g 1, …, *m* and *β* ≔ *β*<sup>1</sup> f g , …, *β<sup>k</sup>* ⊆f g 1, …, *n* be subsets with

*Lk*,*<sup>n</sup>* ≔ f g *α* : *α* ¼ ð Þ *α*1, …, *α<sup>k</sup>* , 1≤*α*<sup>1</sup> < ⋯ < *α<sup>k</sup>* ≤*n*

stands for the collection of strictly increasing sequences of 1≤ *k*≤*n* integers chosen from 1, f g …, *n* . For fixed *i*∈*α* and *j*∈*β*, put *Ir*,*<sup>m</sup>*f g*i* ≔ f g *α* : *α*∈*Lr*,*<sup>m</sup>*, *i* ∈*α* and

respectively. By **A***<sup>i</sup>:*ð Þ **b** and **A***: <sup>j</sup>*ð Þ**c** , we denote the matrices obtained from **A** by replacing its *i*th row with the row **b**, and its *j*th column with the column **c**. **Theorem 2.1.** [28] *If* **<sup>A</sup>** <sup>∈</sup> *<sup>m</sup>*�*<sup>n</sup> <sup>r</sup> , then the Moore-Penrose inverse* **<sup>A</sup>**† <sup>¼</sup> *<sup>a</sup>*†

*<sup>β</sup>* <sup>∈</sup>*Jr*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>a</sup>** <sup>∗</sup>

*<sup>α</sup>*∈*Ir*,*m*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>a</sup>** <sup>∗</sup>

*<sup>β</sup>* <sup>∈</sup>*Jr*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

�

� �

, we put, respectively,

**AA**<sup>∗</sup> ð Þ*<sup>i</sup>:* ð Þ **<sup>b</sup>** � � �

*α*∈*Im*,*<sup>m</sup>*

� � �

**<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** <sup>¼</sup> <sup>X</sup>

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *: <sup>j</sup>*

*β* ∈*Jn*,*<sup>n</sup>*

P

� � � �

� � �

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*∈*Ir*,*<sup>m</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

*: j*

*β*

*α*

*i:*

�*α*

**AA**<sup>∗</sup> j j*<sup>α</sup>*

ð Þ**c**

**<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

� � � *β β*

*<sup>α</sup>*, *i* ¼ 1, …, *m*,

*<sup>α</sup>*, when *r* ¼ *m*;

, *j* ¼ 1, …, *n*,

*<sup>β</sup>*, when *r* ¼ *n:*

*<sup>r</sup> . Then, the following determinantal representations*

� � � *β β*

� � � *α*

The *j*th columns and the *i*th rows of **A** and **A**<sup>∗</sup> denote **a***: <sup>j</sup>* and **a** <sup>∗</sup>

*<sup>β</sup>*, we denote a submatrix of **A** ∈ *<sup>m</sup>*�*<sup>n</sup>* with rows and

*<sup>α</sup>* is a principal submatrix of **A**

*: <sup>j</sup>* and **a***<sup>i</sup>:* and **a** <sup>∗</sup>

¼ (8)

*:* (9)

*<sup>r</sup>* , a row vector **b**∈ <sup>1</sup>�*<sup>m</sup>*,

*ij* � �<sup>∈</sup> *<sup>n</sup>*�*<sup>m</sup>*

*i:* ,

*<sup>α</sup>* is the corresponding principal minor

$$\mathbf{X} \mathbf{A}^{k+1} = \mathbf{A}^k \tag{6}$$

$$\mathbf{A}^{k+1}\mathbf{X}=\mathbf{A}^{k}.\tag{7}$$

In particular, if Ind**A** ¼ 1, then the matrix **X** is called *the group inverse*, and it is denoted by **<sup>X</sup>** <sup>¼</sup> **<sup>A</sup>**#. If Ind**<sup>A</sup>** <sup>¼</sup> 0, then **<sup>A</sup>** is nonsingular and **<sup>A</sup>***<sup>d</sup>* <sup>¼</sup> **<sup>A</sup>**† <sup>¼</sup> **<sup>A</sup>**�<sup>1</sup> .

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 ð Þ*i* ,ð Þ*j* , … is called an f g *i*, *j*, … -inverse of **A**, and is denoted by **<sup>A</sup>**ð Þ *<sup>i</sup>*, *<sup>j</sup>*,… . The set of matrices **<sup>A</sup>**ð Þ *<sup>i</sup>*, *<sup>j</sup>*,… is denoted **<sup>A</sup>**f g *<sup>i</sup>*, *<sup>j</sup>*, … . In particular, **A**ð Þ<sup>1</sup> is called the inner inverse, **A**ð Þ<sup>2</sup> is called the outer inverse, **A**ð Þ 1,2 is called the reflexive inverse, **A**ð Þ 1,2,3,4 is the Moore-Penrose inverse, etc.

For an arbitrary matrix **A** ∈ *<sup>m</sup>*�*<sup>n</sup>*, we denote by


**P***<sup>A</sup>* ≔ **AA**† and **Q***<sup>A</sup>* ≔ **A**† **A** are the orthogonal projectors onto the range of **A** and the range of **A**<sup>∗</sup> , respectively.

The core inverse was introduced by Baksalary and Trenkler in [1]. Later, it was investigated by S. Malik in [2] and S.Z. Xu et al. in [3], among others.

**Definition 1.1.** [1] A matrix **X** ∈ *<sup>n</sup>*�*<sup>n</sup>* is called the core inverse of **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* if it satisfies the conditions

$$\mathbf{A}\mathbf{X} = \mathbf{P}\_A,\\
and\ \mathcal{C}(\mathbf{X}) = \mathcal{C}(\mathbf{A}).$$

When such matrix **X** exists, it is denoted as *A*○#.

In 2014, the core inverse was extended to the core-EP inverse defined by K. Manjunatha Prasad and K.S. Mohana [4]. Other generalizations of the core inverse were recently introduced for *n* � *n* complex matrices, namely BT inverses [5], DMP inverses [2], CMP inverses [6], etc. The characterizations, computing methods, and some applications of the core inverse and its generalizations were recently investigated in complex matrices and rings (see, e.g., [7–18]).

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 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 [26–34]. 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 *Determinantal Representations of the Core Inverse and Its Generalizations DOI: http://dx.doi.org/10.5772/intechopen.89341*

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.

### **2. Preliminaries**

**AX** ¼ **XA**; (5) **XA***<sup>k</sup>*þ<sup>1</sup> <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>* (6)

*:* (7)

.

**A***<sup>k</sup>*þ<sup>1</sup>

and is denoted by **<sup>A</sup>**ð Þ *<sup>i</sup>*, *<sup>j</sup>*,… . The set of matrices **<sup>A</sup>**ð Þ *<sup>i</sup>*, *<sup>j</sup>*,… is denoted **<sup>A</sup>**f g *<sup>i</sup>*, *<sup>j</sup>*, … . In particular, **A**ð Þ<sup>1</sup> is called the inner inverse, **A**ð Þ<sup>2</sup> is called the outer inverse, **A**ð Þ 1,2 is

called the reflexive inverse, **A**ð Þ 1,2,3,4 is the Moore-Penrose inverse, etc.

• <sup>N</sup> ð Þ¼ **<sup>A</sup> <sup>x</sup>**<sup>∈</sup> *<sup>n</sup>*�<sup>1</sup> : **Ax** <sup>¼</sup> <sup>0</sup> , the kernel (or the null space) of **<sup>A</sup>**;

• Rð Þ¼ **<sup>A</sup> <sup>y</sup>** <sup>∈</sup> <sup>1</sup>�*<sup>n</sup>* : **<sup>y</sup>** <sup>¼</sup> **xA**, **<sup>x</sup>**<sup>∈</sup> <sup>1</sup>�*<sup>m</sup>* , the row space of **<sup>A</sup>**.

each other and without the obligatory fulfillment of (5).

For an arbitrary matrix **A** ∈ *<sup>m</sup>*�*<sup>n</sup>*, we denote by

of **A**; and

*Functional Calculus*

the range of **A**<sup>∗</sup>

satisfies the conditions

obtained in [19, 20].

**2**

**P***<sup>A</sup>* ≔ **AA**† and **Q***<sup>A</sup>* ≔ **A**†

, respectively.

When such matrix **X** exists, it is denoted as *A*○#.

investigated in complex matrices and rings (see, e.g., [7–18]).

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 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 [26–34]. 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

The chapter is organized as follows: in Section 2, we start with preliminary introduction of determinantal representations of the Moore-Penrose inverse and the

**<sup>X</sup>** <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>*

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

A matrix **A** satisfying the conditions ð Þ*i* ,ð Þ*j* , … is called an f g *i*, *j*, … -inverse of **A**,

• Cð Þ¼ **<sup>A</sup> <sup>y</sup>**<sup>∈</sup> *<sup>m</sup>*�<sup>1</sup> : **<sup>y</sup>** <sup>¼</sup> **Ax**, **<sup>x</sup>**<sup>∈</sup> *<sup>n</sup>*�<sup>1</sup> , the column space (or the range space)

The core inverse was introduced by Baksalary and Trenkler in [1]. Later, it was investigated by S. Malik in [2] and S.Z. Xu et al. in [3], among others.

**Definition 1.1.** [1] A matrix **X** ∈ *<sup>n</sup>*�*<sup>n</sup>* is called the core inverse of **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* if it

**AX** ¼ **P***A*, *and* Cð Þ¼C **X** ð Þ **A** *:*

In 2014, the core inverse was extended to the core-EP inverse defined by K. Manjunatha Prasad and K.S. Mohana [4]. Other generalizations of the core inverse were recently introduced for *n* � *n* complex matrices, namely BT inverses [5], DMP inverses [2], CMP inverses [6], etc. The characterizations, computing methods, and some applications of the core inverse and its generalizations were recently

**A** are the orthogonal projectors onto the range of **A** and

In particular, if Ind**A** ¼ 1, then the matrix **X** is called *the group inverse*, and it is denoted by **<sup>X</sup>** <sup>¼</sup> **<sup>A</sup>**#. If Ind**<sup>A</sup>** <sup>¼</sup> 0, then **<sup>A</sup>** is nonsingular and **<sup>A</sup>***<sup>d</sup>* <sup>¼</sup> **<sup>A</sup>**† <sup>¼</sup> **<sup>A</sup>**�<sup>1</sup>

> Let *α* ≔ f g *α*1, …, *α<sup>k</sup>* ⊆f g 1, …, *m* and *β* ≔ *β*<sup>1</sup> f g , …, *β<sup>k</sup>* ⊆f g 1, …, *n* be subsets with <sup>1</sup>≤*k*<sup>≤</sup> min f g *<sup>m</sup>*, *<sup>n</sup>* . By **<sup>A</sup>***<sup>α</sup> <sup>β</sup>*, we denote a submatrix of **A** ∈ *<sup>m</sup>*�*<sup>n</sup>* with rows and columns indexed by *α* and *β*, respectively. Then, **A***<sup>α</sup> <sup>α</sup>* is a principal submatrix of **A** with rows and columns indexed by *<sup>α</sup>*, and j j **<sup>A</sup>** *<sup>α</sup> <sup>α</sup>* is the corresponding principal minor of the determinant ∣**A**∣. Suppose that

$$L\_{k,n} := \{ a \, : \, a = (a\_1, \ldots, a\_k), \, 1 \le a\_1 < \cdots < a\_k \le n \}$$

stands for the collection of strictly increasing sequences of 1≤ *k*≤*n* integers chosen from 1, f g …, *n* . For fixed *i*∈*α* and *j*∈*β*, put *Ir*,*<sup>m</sup>*f g*i* ≔ f g *α* : *α*∈*Lr*,*<sup>m</sup>*, *i* ∈*α* and *Jr*,*<sup>n</sup>*f g*j* ≔ f g *β* : *β* ∈*Lr*,*<sup>n</sup>*, *j*∈*β* .

The *j*th columns and the *i*th rows of **A** and **A**<sup>∗</sup> denote **a***: <sup>j</sup>* and **a** <sup>∗</sup> *: <sup>j</sup>* and **a***<sup>i</sup>:* and **a** <sup>∗</sup> *i:* , respectively. By **A***<sup>i</sup>:*ð Þ **b** and **A***: <sup>j</sup>*ð Þ**c** , we denote the matrices obtained from **A** by replacing its *i*th row with the row **b**, and its *j*th column with the column **c**.

**Theorem 2.1.** [28] *If* **<sup>A</sup>** <sup>∈</sup> *<sup>m</sup>*�*<sup>n</sup> <sup>r</sup> , then the Moore-Penrose inverse* **<sup>A</sup>**† <sup>¼</sup> *<sup>a</sup>*† *ij* � �<sup>∈</sup> *<sup>n</sup>*�*<sup>m</sup> possesses the determinantal representations*

$$a\_{ij}^{\dagger} = \frac{\sum\_{\beta \in I\_{r\mu}\{i\}} \left| \left(\mathbf{A}^{\*}\mathbf{A}\right)\_{i} \left(\mathbf{a}\_{j}^{\*}\right) \right|\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{r\mu}} \left| \mathbf{A}^{\*}\mathbf{A}\right|\_{\beta}^{\beta}} = \tag{8}$$

$$=\frac{\sum\_{a\in I\_{r,m}\{j\}}\left| (\mathbf{A}\mathbf{A}^\*)\_{j.}\left(\mathbf{a}\_{i.}^\*\right)\right|\_{a}^{a}}{\sum\_{a\in I\_{r,m}}|\mathbf{A}\mathbf{A}^\*|\_{a}^{a}}.\tag{9}$$

**Remark 2.2.** For an arbitrary full-rank matrix **A** ∈ *<sup>m</sup>*�*<sup>n</sup> <sup>r</sup>* , a row vector **b**∈ <sup>1</sup>�*<sup>m</sup>*, and a column-vector **c**∈ *<sup>n</sup>*�<sup>1</sup> , we put, respectively,

$$\left| (\mathbf{A} \mathbf{A}^\*)\_{i.} (\mathbf{b}) \right| = \sum\_{a \in I\_{n,n}} \left| (\mathbf{A} \mathbf{A}^\*)\_{i.} (\mathbf{b}) \right|\_{a}^{a}, \quad i = 1, \ldots, m,$$

$$\left| \mathbf{A} \mathbf{A}^\* \right| = \sum\_{a \in I\_{m,n}} \left| \mathbf{A} \mathbf{A}^\* \right|\_{a}^{a}, \quad \text{when } r = m;$$

$$\left| (\mathbf{A}^\* \mathbf{A})\_{.j} (\mathbf{c}) \right| = \sum\_{\beta \in I\_{n,n} \{j\}} \left| (\mathbf{A}^\* \mathbf{A})\_{.j} (\mathbf{c}) \right|\_{\beta}^{\beta}, \quad j = 1, \ldots, n,$$

$$\left| \mathbf{A}^\* \mathbf{A} \right| = \sum\_{\beta \in I\_{n,n}} \left| \mathbf{A}^\* \mathbf{A} \right|\_{\beta}^{\beta}, \quad \text{when } r = n.$$

**Corollary 2.3.** [21] *Let* **A** ∈ *<sup>m</sup>*�*<sup>n</sup> <sup>r</sup> . Then, the following determinantal representations can be obtained*

i. for the projector **<sup>Q</sup>***<sup>A</sup>* <sup>¼</sup> *qij* � � *n*�*n* ,

$$q\_{ij} = \frac{\sum\_{\boldsymbol{\beta} \in I\_{r,n}\{i\}} \left| (\mathbf{A}^\* \,\mathbf{A})\_{\boldsymbol{i}} (\dot{\mathbf{a}}\_{\cdot j}) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}}}{\sum\_{\boldsymbol{\beta} \in I\_{r,n}} |\mathbf{A}^\* \,\mathbf{A}|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}}} = \frac{\sum\_{a \in I\_{r,n}\{j\}} \left| (\mathbf{A}^\* \,\mathbf{A})\_{j} (\dot{\mathbf{a}}\_{\cdot i}) \right|\_{a}^{a}}{\sum\_{a \in I\_{r,n}} |\mathbf{A}^\* \,\mathbf{A}|\_{a}^{a}},\tag{10}$$

**Definition 3.1.** A matrix **X** ∈ *<sup>n</sup>*�*<sup>n</sup>* is said to be the right core inverse of **A** ∈ *<sup>n</sup>*�*<sup>n</sup>*

**AX** ¼ **P***A*, *and* Cð Þ¼C **X** ð Þ **A** *:*

The following definition of the left core inverse can be given that is equivalent to

**Definition 3.2** A matrix **X** ∈ *<sup>n</sup>*�*<sup>n</sup>* is said to be the left core inverse of **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* if

**Remark 3.3.** In [35], the conditions of the dual core inverse are given as follows:

**<sup>A</sup>** <sup>¼</sup> **AA**† � � ¼ Cð Þ¼C **<sup>A</sup> <sup>A</sup>**<sup>∗</sup> f g ð Þ *:*

then clearly **<sup>A</sup>**† <sup>¼</sup> **<sup>A</sup>**#. It is known that the core inverses of **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* exist if and only

**<sup>A</sup>**○# <sup>¼</sup> **<sup>A</sup>**#**AA**†

Lemma 3.4 and the definitions of the Moore-Penrose and group inverses, it follows

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>u</sup>**ð Þ<sup>1</sup>

�

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>2</sup> � � � � *β β* P

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>2</sup> � � � � *β β* P

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>2</sup> � �

�

� � � �

*:<sup>i</sup>* **<sup>u</sup>**ð Þ<sup>2</sup> *: j* � � � �

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

**<sup>A</sup>**○# <sup>¼</sup> **<sup>A</sup>**†

**Remark 3.5.** In Theorems 3.6 and 3.7, we will suppose that **A** ∈ CM

*<sup>n</sup>* 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

*<sup>A</sup>*○#**<sup>A</sup>** <sup>¼</sup> **<sup>P</sup>***A*<sup>∗</sup> , *and* Cð Þ *<sup>A</sup>*○# <sup>⊆</sup><sup>C</sup> **<sup>A</sup>**<sup>∗</sup> ð Þ*:*

**<sup>A</sup>** � � <sup>∗</sup> <sup>¼</sup> **<sup>A</sup>**†

Due to [1], we introduce the following sets of quaternion matrices

CM

*<sup>n</sup> . Then*,

*<sup>n</sup>* and **A** ∈ EP

P

P

P

P

.

*a*○#,*<sup>r</sup> ij* ¼

¼

**XA** ¼ **Q***A*, *and* Rð Þ¼R **X** ð Þ **A** *:* (15)

*<sup>n</sup>* <sup>¼</sup> **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* : rk**A**<sup>2</sup> <sup>¼</sup> rk**<sup>A</sup>** � �,

*<sup>n</sup>* are called group matrices or core matrices. If **A** ∈ EP

**<sup>A</sup>** <sup>¼</sup> **<sup>Q</sup>***A*, and Rð Þ¼C **<sup>A</sup> <sup>A</sup>**<sup>∗</sup> ð Þ, then these

, (16)

*<sup>n</sup>* but

¼ (18)

, (19)

**AA**# (17)

*<sup>n</sup>* (in particular, **A** is Hermitian), then from

*<sup>n</sup> and* rk**A**<sup>2</sup> <sup>¼</sup> rk**<sup>A</sup>** <sup>¼</sup> *s. Then, its right core inverse has the*

� � � *α α*

*α*

*i:*

� � � *β β*

*α*

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

*n* ,

When such matrix **X** exists, it is denoted as *A*○#.

*Determinantal Representations of the Core Inverse and Its Generalizations*

When such matrix **<sup>X</sup>** exists, it is denoted as *<sup>A</sup>*○#.

*<sup>n</sup>* <sup>¼</sup> **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* : **<sup>A</sup>**†

if it satisfies the conditions

it satisfies the conditions

the introduced dual core inverse [35].

*DOI: http://dx.doi.org/10.5772/intechopen.89341*

Since **<sup>P</sup>***A*<sup>∗</sup> <sup>¼</sup> **<sup>A</sup>**<sup>∗</sup> **<sup>A</sup>**<sup>∗</sup> ð Þ† <sup>¼</sup> **<sup>A</sup>**†

conditions and (15) are analogous.

EP

The matrices from CM

the right and left core inverses. **Lemma 3.4.** [1] *Let* **A** ∈ CM

*<sup>n</sup>* . Because, if **A** ∈ CM

**Theorem 3.6.** *Let* **A** ∈ CM

*following determinantal representations*

that **<sup>A</sup>**○# <sup>¼</sup> **<sup>A</sup>**○# <sup>¼</sup> **<sup>A</sup>**# <sup>¼</sup> **<sup>A</sup>**†

if **A** ∈ CM

**A** ∉ EP

**5**

where **a**\_*: <sup>j</sup>* is the *j*th column and **a**\_ *<sup>i</sup>:* is the *i*th row of **A**<sup>∗</sup> **A**; and

ii. for the projector **<sup>P</sup>***<sup>A</sup>* <sup>¼</sup> *pij* � � *m*�*m* ,

$$p\_{\vec{\eta}} = \frac{\sum\_{a \in I\_{r\mathfrak{u}}\{j\}} \left| (\mathbf{A}\mathbf{A}^\*)\_{j\_\cdot} (\ddot{\mathbf{a}}\_{i\cdot}) \right|\_{a}^{a}}{\sum\_{a \in I\_{r\mathfrak{u}}} |\mathbf{A}\mathbf{A}^\*|\_{a}^{a}} = \frac{\sum\_{\beta \in I\_{r\mathfrak{u}}\{i\}} \left| (\mathbf{A}\mathbf{A}^\*)\_{j\_\cdot} (\ddot{\mathbf{a}}\_{\cdot j}) \right|\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{r\mathfrak{u}}} |\mathbf{A}\mathbf{A}^\*|\_{\beta}^{\beta}},\tag{11}$$

where **a**€*<sup>i</sup>:* is the *i*th row and **a**€*: <sup>j</sup>* is the *j*th column of **AA**<sup>∗</sup> .

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

**Lemma 2.4.** [21] *Let* **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup> with* Ind**<sup>A</sup>** <sup>¼</sup> *k and* rk**A***<sup>k</sup>*þ<sup>1</sup> <sup>¼</sup> rk**A***<sup>k</sup>* <sup>¼</sup> *r. Then, the determinantal representations of the Drazin inverse* **<sup>A</sup>***<sup>d</sup>* <sup>¼</sup> *ad ij* � � <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup> are*

$$\mathbf{a}\_{ij}^{d} = \frac{\sum\_{\beta \in I\_{r, \mathbf{z}}\{i\}} \left| \left( \mathbf{A}^{k+1} \right)\_{\cdot, i} \left( \mathbf{a}\_{\cdot, j}^{(k)} \right) \right|\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{r, \mathbf{z}}\{j\}} \left| \mathbf{A}^{k+1} \right|\_{\beta}^{\beta}} = \tag{12}$$
 
$$= \frac{\sum\_{a \in I\_{r, \mathbf{z}}\{j\}} \left| \left( \mathbf{A}^{k+1} \right)\_{j\_{\cdot}} \left( \mathbf{a}\_{i\cdot}^{(k)} \right) \right|\_{a}^{a}}{\sum\_{a \in I\_{r, \mathbf{z}}} \left| \mathbf{A}^{k+1} \right|\_{a}^{a}}, \tag{13}$$

where **a** ð Þ*k <sup>i</sup>:* is the *i*th row and **a** ð Þ*k : <sup>j</sup>* is the *j*th column of **A***<sup>k</sup>* .

**Corollary 2.5.** [21] *Let* **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup> with* Ind**<sup>A</sup>** <sup>¼</sup> <sup>1</sup> *and* rk**A**<sup>2</sup> <sup>¼</sup> rk**<sup>A</sup>** <sup>¼</sup> *r. Then, the determinantal representations of the group inverse* **<sup>A</sup>**# <sup>¼</sup> *<sup>a</sup>*# *ij* � �<sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup> are*

$$a\_{\vec{v}}^{\
u} = \frac{\sum\_{\beta \in I\_{r,\mathfrak{a}}\{i\}} \left| \left(\mathbf{A}^2\right)\_{.i} \left(\mathbf{a}\_{.j}\right) \right|\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{r,\mathfrak{a}}} \left| \mathbf{A}^2 \right|\_{\beta}^{\beta}} = \frac{\sum\_{a \in I\_{r,\mathfrak{a}}\{j\}} \left| \left(\mathbf{A}^2\right)\_{.j} \left(\mathbf{a}\_{i}\right) \right|\_{a}^{a}}{\sum\_{a \in I\_{r,\mathfrak{a}}} \left| \mathbf{A}^2 \right|\_{a}^{a}}.\tag{14}$$

�

*α*

### **3. Determinantal representations of the core inverse and its generalizations**

#### **3.1 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.

*Determinantal Representations of the Core Inverse and Its Generalizations DOI: http://dx.doi.org/10.5772/intechopen.89341*

**Definition 3.1.** A matrix **X** ∈ *<sup>n</sup>*�*<sup>n</sup>* is said to be the right core inverse of **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* if it satisfies the conditions

$$\mathbf{A}\mathbf{X} = \mathbf{P}\_A,\\
and\ \mathcal{C}(\mathbf{X}) = \mathcal{C}(\mathbf{A}).$$

When such matrix **X** exists, it is denoted as *A*○#.

i. for the projector **Q***<sup>A</sup>* ¼ *qij*

P

ii. for the projector **P***<sup>A</sup>* ¼ *pij*

P

P

*qij* ¼

*Functional Calculus*

*pij* ¼

in complex matrices.

where **a**

ð Þ*k*

**generalizations**

**4**

*a*# *ij* ¼ � �

*β*

where **a**\_*: <sup>j</sup>* is the *j*th column and **a**\_ *<sup>i</sup>:* is the *i*th row of **A**<sup>∗</sup> **A**; and

*m*�*m* ,

> � � � *α*

¼ P

The following lemma gives determinantal representations of the Drazin inverse

**Lemma 2.4.** [21] *Let* **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup> with* Ind**<sup>A</sup>** <sup>¼</sup> *k and* rk**A***<sup>k</sup>*þ<sup>1</sup> <sup>¼</sup> rk**A***<sup>k</sup>* <sup>¼</sup> *r. Then, the*

*<sup>β</sup>* <sup>∈</sup>*Jr*,*<sup>n</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � � � � *β β*

*<sup>β</sup>* <sup>∈</sup>*Jr*,*n*f g*<sup>i</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

�

*<sup>α</sup>* <sup>∈</sup>*Ir*,*n*f g*<sup>j</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

� � �

P

ð Þ*k*

*:<sup>i</sup>* **a***: <sup>j</sup>* � � � � �

� *β β*

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

¼ P

ð Þ **a**€*i:*

*α*

where **a**€*<sup>i</sup>:* is the *i*th row and **a**€*: <sup>j</sup>* is the *j*th column of **AA**<sup>∗</sup> .

� �

*<sup>β</sup>* <sup>∈</sup>*Jr*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>a</sup>**\_*: <sup>j</sup>* � � � � �

*<sup>α</sup>*∈*Ir*,*m*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*j:*

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*∈*Ir*,*<sup>m</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

*determinantal representations of the Drazin inverse* **<sup>A</sup>***<sup>d</sup>* <sup>¼</sup> *ad*

P

P

*ad ij* ¼

¼

*determinantal representations of the group inverse* **<sup>A</sup>**# <sup>¼</sup> *<sup>a</sup>*#

*<sup>β</sup>* <sup>∈</sup>*Jr*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>2</sup> � �

**3.1 Determinantal representations of the core inverses**

following definition that is equivalent to Definition 1.1.

*<sup>β</sup>* <sup>∈</sup>*Jr*,*<sup>n</sup>* **<sup>A</sup>**<sup>2</sup> � � � � *β β*

**3. Determinantal representations of the core inverse and its**

P

*<sup>i</sup>:* is the *i*th row and **a**

P

� � �

*<sup>β</sup>* <sup>∈</sup>*Jr*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

*n*�*n* ,

> � *β β*

¼ P

*<sup>α</sup>*∈*Ir*,*n*f g*<sup>j</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *<sup>j</sup>:* **<sup>a</sup>**\_ ð Þ*i:* � � �

*<sup>β</sup>* <sup>∈</sup>*Jr*,*m*f g*<sup>i</sup>* **AA**<sup>∗</sup> ð Þ*:<sup>i</sup>* **<sup>a</sup>**€*: <sup>j</sup>* � � � � �

*<sup>β</sup>* <sup>∈</sup>*Jr*,*<sup>m</sup>* **AA**<sup>∗</sup> j j*<sup>β</sup>*

*ij* � �

� � � *β β*

� � � �

*α*

.

∈ *<sup>n</sup>*�*<sup>n</sup> are*

� � � *α*

*:* (14)

*j:* ð Þ **a***<sup>i</sup>:*

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*∈*Ir*,*<sup>n</sup>* **<sup>A</sup>**<sup>2</sup> � � � � *α α*

*ij* � �

*<sup>α</sup>*∈*Ir*,*n*f g*<sup>j</sup>* **<sup>A</sup>**<sup>2</sup> � �

� � �

P

*:i* **a** ð Þ*k : j*

*j:* **a** ð Þ*k i:*

� � � �

� � �

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*∈*Ir*,*<sup>n</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � � � � *α α*

*: <sup>j</sup>* is the *j*th column of **A***<sup>k</sup>*

**Corollary 2.5.** [21] *Let* **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup> with* Ind**<sup>A</sup>** <sup>¼</sup> <sup>1</sup> *and* rk**A**<sup>2</sup> <sup>¼</sup> rk**<sup>A</sup>** <sup>¼</sup> *r. Then, the*

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*∈*Ir*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>α</sup>*

*α*

� � � *α*

> � *β β*

∈ *<sup>n</sup>*�*<sup>n</sup> are*

¼ (12)

, (13)

*β*

, (10)

, (11)

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** ∈ *<sup>n</sup>*�*<sup>n</sup>* is said to be the left core inverse of **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* if it satisfies the conditions

$$\mathbf{X} \mathbf{A} = \mathbf{Q}\_A,\\
and \; \mathcal{R}(\mathbf{X}) = \mathcal{R}(\mathbf{A}).\tag{15}$$

When such matrix **<sup>X</sup>** exists, it is denoted as *<sup>A</sup>*○#.

**Remark 3.3.** In [35], the conditions of the dual core inverse are given as follows:

$$A\_{\otimes} \mathbf{A} = \mathbf{P}\_{A^\*} \text{, and } \mathcal{C}(A\_{\otimes}) \subseteq \mathcal{C}(\mathbf{A}^\*).$$

Since **<sup>P</sup>***A*<sup>∗</sup> <sup>¼</sup> **<sup>A</sup>**<sup>∗</sup> **<sup>A</sup>**<sup>∗</sup> ð Þ† <sup>¼</sup> **<sup>A</sup>**† **<sup>A</sup>** � � <sup>∗</sup> <sup>¼</sup> **<sup>A</sup>**† **<sup>A</sup>** <sup>¼</sup> **<sup>Q</sup>***A*, and Rð Þ¼C **<sup>A</sup> <sup>A</sup>**<sup>∗</sup> ð Þ, then these conditions and (15) are analogous.

Due to [1], we introduce the following sets of quaternion matrices

$$\mathbb{C}\_{n}^{\complement\mathbf{M}} = \{ \mathbf{A} \in \mathbb{C}^{n \times n} : \mathbf{r} \mathbf{A} \mathbf{A}^{2} = \mathbf{r} \mathbf{k} \mathbf{A} \},$$

$$\mathbb{C}\_{n}^{\text{EP}} = \{ \mathbf{A} \in \mathbb{C}^{n \times n} : \mathbf{A}^{\dagger} \mathbf{A} = \mathbf{A} \mathbf{A}^{\dagger} \} = \{ \mathcal{C}(\mathbf{A}) = \mathcal{C}(\mathbf{A}^{\*}) \}.$$

The matrices from CM *<sup>n</sup>* are called group matrices or core matrices. If **A** ∈ EP *n* , then clearly **<sup>A</sup>**† <sup>¼</sup> **<sup>A</sup>**#. It is known that the core inverses of **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* exist if and only if **A** ∈ CM *<sup>n</sup>* 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 *<sup>n</sup> . Then*,

$$\mathbf{A}^{\otimes} = \mathbf{A}^{\ast} \mathbf{A} \mathbf{A}^{\dagger},\tag{16}$$

$$\mathbf{A}\_{\oplus} = \mathbf{A}^{\dagger} \mathbf{A} \mathbf{A}^{\prime} \tag{17}$$

**Remark 3.5.** In Theorems 3.6 and 3.7, we will suppose that **A** ∈ CM *<sup>n</sup>* but

**A** ∉ EP *<sup>n</sup>* . Because, if **A** ∈ CM *<sup>n</sup>* and **A** ∈ EP *<sup>n</sup>* (in particular, **A** is Hermitian), then from Lemma 3.4 and the definitions of the Moore-Penrose and group inverses, it follows that **<sup>A</sup>**○# <sup>¼</sup> **<sup>A</sup>**○# <sup>¼</sup> **<sup>A</sup>**# <sup>¼</sup> **<sup>A</sup>**† .

**Theorem 3.6.** *Let* **A** ∈ CM *<sup>n</sup> and* rk**A**<sup>2</sup> <sup>¼</sup> rk**<sup>A</sup>** <sup>¼</sup> *s. Then, its right core inverse has the following determinantal representations*

$$a\_{ij}^{\otimes,r} = \frac{\sum\_{a \in I\_{\boldsymbol{\pi}} \left\langle j \right\rangle} \left| \left( \mathbf{A} \mathbf{A}^\* \right)\_j \left( \mathbf{u}\_{i.}^{(1)} \right) \right|\_a^a}{\sum\_{\beta \in I\_{\boldsymbol{\pi}, \boldsymbol{\pi}}} \left| \mathbf{A}^2 \right|\_\beta^\beta \sum\_{a \in I\_{\boldsymbol{\pi}, \boldsymbol{\pi}}} \left| \mathbf{A} \mathbf{A}^\* \right|\_a^a} = \tag{18}$$

$$\rho = \frac{\sum\_{\boldsymbol{\beta} \in J\_{\nu}} \left| \left( \mathbf{A}^{2} \right)\_{\boldsymbol{\cdot}} \left( \mathbf{u}\_{\cdot \boldsymbol{\cdot}}^{(2)} \right) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}}}{\sum\_{\boldsymbol{\beta} \in J\_{\nu}} \left| \mathbf{A}^{2} \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \sum\_{a \in I\_{\nu}} \left| \mathbf{A} \mathbf{A}^{\*} \right|\_{a}^{a}},\tag{19}$$

where

$$\begin{aligned} \mathbf{u}\_{i.}^{(1)} &= \left[ \sum\_{\boldsymbol{\beta} \in I\_{\boldsymbol{s}\boldsymbol{u}}\{\boldsymbol{\beta}\}} \left| \left(\mathbf{A}^{2}\right)\_{\boldsymbol{s}} (\widetilde{\mathbf{a}}\_{\boldsymbol{f}}) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \right] \in \mathbb{C}^{1 \times n}, \; f = \mathbf{1}, \ldots, n, \\\mathbf{u}\_{\boldsymbol{j}}^{(2)} &= \left[ \sum\_{\boldsymbol{a} \in I\_{\boldsymbol{s}\boldsymbol{s}}\{\boldsymbol{j}\}} \left| (\mathbf{A}\mathbf{A}^{\*})\_{\boldsymbol{j}.} (\widetilde{\mathbf{a}}\_{\boldsymbol{l}}) \right|\_{\boldsymbol{a}}^{\boldsymbol{a}} \right] \in \mathbb{C}^{n \times 1}, \; l = 1, \ldots, n. \end{aligned}$$

are the row and column vectors, respectively. Here **a**~*:<sup>f</sup>* and **a**~*l:* are the *f*th column and *l*th row of **A**~ ≔ **A**<sup>2</sup> **A**<sup>∗</sup> .

*Proof.* Taking into account (16), we have for #*A*,

$$a\_{ij}^{\bullet,r} = \sum\_{l=1}^{n} \sum\_{f=1}^{n} a\_{il}^{\bullet} a\_{lf} a\_{fj}^{\dagger}.\tag{20}$$

By substituting (14) and (15) in (20), we obtain

$$\begin{split} \mathbf{a}\_{ij}^{\boldsymbol{\theta},\boldsymbol{r}} &= \sum\_{l=1}^{n} \frac{\sum\_{f=1}^{n} \sum\_{\boldsymbol{\beta} \in I\_{\boldsymbol{s},\boldsymbol{u}}\{\boldsymbol{\beta}\}} \left| \left(\mathbf{A}^{2}\right)\_{\boldsymbol{\cdot},\boldsymbol{\mathsf{a}}} \left(\mathbf{a}\_{\boldsymbol{f}}\right)\right|\_{\boldsymbol{\beta}}^{\boldsymbol{\theta}} a\_{\boldsymbol{f}} \frac{\sum\_{a \in I\_{\boldsymbol{s},\boldsymbol{u}}\{\boldsymbol{j}\}} \left| \left(\mathbf{A}\mathbf{A}^{\*}\right)\_{\boldsymbol{j},\boldsymbol{\cdot}} \left(\mathbf{a}\_{\boldsymbol{l}}\right)\right|\_{a}^{a}}{\sum\_{a \in I\_{\boldsymbol{s},\boldsymbol{u}}} \left| \mathbf{A}\mathbf{A}^{\*}\right|\_{\boldsymbol{a}}^{a}} = \\ & \frac{\sum\_{l=1}^{n} \sum\_{l=1}^{n} \sum\_{\boldsymbol{\beta} \in I\_{\boldsymbol{s},\boldsymbol{u}}\{\boldsymbol{j}\}} \left| \left(\mathbf{A}^{2}\right)\_{\boldsymbol{j}} \left(\mathbf{e}\_{\boldsymbol{f}}\right)\right|\_{\boldsymbol{\beta}}^{\boldsymbol{\theta}} \tilde{a}\_{\boldsymbol{f}} \sum\_{a \in I\_{\boldsymbol{s},\boldsymbol{u}}\{\boldsymbol{j}\}} \left| \left(\mathbf{A}\mathbf{A}^{\*}\right)\_{\boldsymbol{j},\boldsymbol{\cdot}} \left(\mathbf{e}\_{\boldsymbol{l}}\right)\right|\_{a}^{a}}{\sum\_{\boldsymbol{\beta} \in I\_{\boldsymbol{s},\boldsymbol{u}}} \left| \mathbf{A}^{2}\right|\_{\boldsymbol{\beta}}^{\boldsymbol{\theta}} \sum\_{a \in I\_{\boldsymbol{s},\boldsymbol{u}}\left| \mathbf{A}\mathbf{A}^{\*}\right|\_{\boldsymbol{a}}^{a}}}, \end{split}$$

where **e***:<sup>l</sup>* and **e***<sup>l</sup>:* are the unit column and row vectors, respectively, such that all their components are 0, except the *l*th components which are 1; *a*~*lf* is the (*lf*)th element of the matrix **A**~ ≔ **A**<sup>2</sup> **A**<sup>∗</sup> .

Let

$$\mu\_{il}^{(1)} \coloneqq \sum\_{f=1}^{n} \sum\_{\beta \in I\_{\nu\mu}\{i\}} \left| \left(\mathbf{A}^2\right)\_{:i} \left(\mathbf{e}\_f\right) \right|\_{\beta}^{\beta} \bar{a}\_{\mathcal{I}l} = \sum\_{\beta \in I\_{\nu\nu}\{i\}} \left| \left(\mathbf{A}^2\right)\_{:i} \left(\bar{\mathbf{a}}\_{\mathcal{I}l}\right) \right|\_{\beta}^{\beta}, \quad i, l = 1, \dots, n.$$

Construct the matrix **<sup>U</sup>**<sup>1</sup> <sup>¼</sup> *<sup>u</sup>*ð Þ<sup>1</sup> *il* � �<sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>*. It follows that

$$\sum\_{l} \boldsymbol{u}\_{il}^{(1)} \sum\_{a \in I\_{l,n}\{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_.} (\mathbf{e}\_{l.}) \right|\_{a}^{a} = \sum\_{a \in I\_{l,n}\{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_.} (\mathbf{u}\_{l.}^{(1)}) \right|\_{a}^{a},$$

where **u**ð Þ<sup>1</sup> *<sup>i</sup>:* is the *i*th row of **U**1. So, we get (18). If we first consider

$$\mathbf{u}\_{\circlearrowleft}^{(2)} \coloneqq \sum\_{l} \vec{\mathbf{a}}\_{\circ l} \sum\_{a \in I\_{\iota a} \{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\cdot}} (\mathbf{e}\_{l\cdot}) \right|\_{a}^{a} = \sum\_{a \in I\_{\iota a} \{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\cdot}} (\mathbf{\bar{a}}\_{\circ l}) \right|\_{a}^{a}, \quad f, j = \mathbf{1}, \ldots, n.$$

and construct the matrix **<sup>U</sup>**<sup>2</sup> <sup>¼</sup> *<sup>u</sup>*ð Þ<sup>2</sup> *if* � � <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>*, then from

$$\sum\_{j=1}^{n} \sum\_{\beta \in J\_{\iota,\mathfrak{r}}\{i\}} |\left(\mathbf{A}^{2}\right)\_{\,i}\left(\mathbf{e}\_{\,\ell}\right)|\_{\beta}^{\beta} u\_{\,\tilde{\mathbf{y}}}^{(2)} = \sum\_{\beta \in J\_{\iota,\mathfrak{r}}\{i\}} \left|\left(\mathbf{A}^{2}\right)\_{\,i}\left(\mathbf{u}\_{\,\ell}^{(2)}\right)\right|\_{\beta}^{\beta},$$

it follows (19). □

Taking into account (17), the following theorem on the determinantal represen-

*<sup>n</sup> and* rk**A**<sup>2</sup> <sup>¼</sup> rk**<sup>A</sup>** <sup>¼</sup> *s. Then for its left core inverse*

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>v</sup>**ð Þ<sup>2</sup>

*β* P

, *l* ¼ 1, …, *n:*

.

�

<sup>5</sup><sup>∈</sup> <sup>1</sup>�*n*, *<sup>f</sup>* <sup>¼</sup> 1, …, *<sup>n</sup>*

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

� � � �

*: j*

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **<sup>A</sup>**<sup>2</sup> � � � � *α α* ,

� � � *β β*

P

P

3

� *β β*

3

<sup>5</sup><sup>∈</sup> *<sup>n</sup>*�<sup>1</sup>

� � � *α α*

**Definition 3.8.** A matrix **X** ∈ *<sup>n</sup>*�*<sup>n</sup>* is said to be the right core-EP inverse of

**XAX** <sup>¼</sup> **<sup>A</sup>**, *and* Cð Þ¼C **<sup>X</sup> <sup>X</sup>**<sup>∗</sup> ð Þ¼C **<sup>A</sup>***<sup>d</sup>* � �*:*

**Definition 3.9.** A matrix **X** ∈ *<sup>n</sup>*�*<sup>n</sup>* is said to be the left core-EP inverse of

**XAX** <sup>¼</sup> **<sup>A</sup>**, *and* Rð Þ¼R **<sup>X</sup> <sup>X</sup>**<sup>∗</sup> ð Þ¼R **<sup>A</sup>***<sup>d</sup>* � �*:*

**Remark 3.10.** Since <sup>C</sup> **<sup>A</sup>**<sup>∗</sup> ð Þ*<sup>d</sup>* � � ¼ R **<sup>A</sup>***<sup>d</sup>* � �, then the left core inverse **<sup>A</sup>**○† of **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* is similar to the ∗ core inverse introduced in [4], and the dual core-EP

Due to [4], we have the following representations the core-EP inverses of

**<sup>A</sup>**○† <sup>¼</sup> **<sup>A</sup>**f g 2,3,6*<sup>a</sup>* and <sup>C</sup> **<sup>A</sup>**○† � �⊆<sup>C</sup> **<sup>A</sup>***<sup>k</sup>* � �,

Thanks to [35], the following representations of the core-EP inverses will be

� �<sup>⊆</sup> <sup>R</sup> **<sup>A</sup>***<sup>k</sup>* � �*:*

**<sup>A</sup>**○† <sup>¼</sup> **<sup>A</sup>**f g 2,4,6*<sup>b</sup>* and <sup>R</sup> **<sup>A</sup>**○†

tation of the left core inverse can be proved similarly.

*Determinantal Representations of the Core Inverse and Its Generalizations*

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **<sup>A</sup>**<sup>2</sup> � �

�

<sup>P</sup> *<sup>α</sup> <sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

*β* ∈*Js*,*n*f g*i*

*α*∈*Is*,*n*f g*j*

*β* P

*<sup>j</sup>:* **<sup>v</sup>**ð Þ<sup>1</sup> *i:* � � � �

> *<sup>α</sup>*∈*Is*,*<sup>n</sup>* **<sup>A</sup>**<sup>2</sup> � � � � *α α* ¼

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>a</sup>***:<sup>f</sup>* � � � � �

> **A**<sup>2</sup> � � *j:* ð Þ **a***<sup>l</sup>:*

� � �

Here **a***:<sup>f</sup>* and **a***<sup>l</sup>:* are the *f*th column and *l*th row of **A** ≔ **A**<sup>∗</sup> **A**<sup>2</sup>

**3.2 Determinantal representations of the core-EP inverses**

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

� � � *α*

**Theorem 3.7.** *Let* **A** ∈ CM

*DOI: http://dx.doi.org/10.5772/intechopen.89341*

P

**v**ð Þ<sup>1</sup>

**v**ð Þ<sup>2</sup>

**A** ∈ *<sup>n</sup>*�*<sup>n</sup>* if it satisfies the conditions

**A** ∈ *<sup>n</sup>*�*<sup>n</sup>* if it satisfies the conditions

.

used for their determinantal representations.

It is denoted as **A**○†

It is denoted as **<sup>A</sup>**○†.

inverse introduced in [35].

**A** ∈ *<sup>n</sup>*�*<sup>n</sup>*

**7**

,

*<sup>i</sup>:* <sup>¼</sup> <sup>X</sup>

*: <sup>j</sup>* <sup>¼</sup> <sup>X</sup>

2 4

2 4

*ij* � �*, we have*

*a*#,*<sup>l</sup> ij* ¼

ð Þ¼ #*<sup>A</sup> <sup>a</sup>*#,*<sup>l</sup>*

where

*Determinantal Representations of the Core Inverse and Its Generalizations DOI: http://dx.doi.org/10.5772/intechopen.89341*

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 *<sup>n</sup> and* rk**A**<sup>2</sup> <sup>¼</sup> rk**<sup>A</sup>** <sup>¼</sup> *s. Then for its left core inverse* ð Þ¼ #*<sup>A</sup> <sup>a</sup>*#,*<sup>l</sup> ij* � �*, we have*

$$a\_{ij}^{\boldsymbol{\theta},l} = \frac{\sum\_{\boldsymbol{a}\in I\_{\boldsymbol{\nu}}\{j\}} \left| \left(\mathbf{A}^{2}\right)\_{j\cdot} \left(\mathbf{v}\_{i\cdot}^{(1)}\right) \right|\_{\boldsymbol{a}}^{\boldsymbol{a}}}{\sum\_{\boldsymbol{\beta}\in I\_{\boldsymbol{\nu},\boldsymbol{a}}} \left| \mathbf{A}^{\*}\mathbf{A}\right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \sum\_{\boldsymbol{a}\in I\_{\boldsymbol{\nu},\boldsymbol{a}}} \left| \mathbf{A}^{2}\right|\_{\boldsymbol{a}}^{\boldsymbol{a}}} = \frac{\sum\_{\boldsymbol{\beta}\in I\_{\boldsymbol{\nu},\boldsymbol{a}}\{i\}} \left| \left(\mathbf{A}^{\*}\mathbf{A}\right)\_{j} \left(\mathbf{v}\_{j}^{(2)}\right) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}}}{\sum\_{\boldsymbol{\beta}\in I\_{\boldsymbol{\nu},\boldsymbol{a}}} \left| \mathbf{A}^{\*}\mathbf{A}\right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \sum\_{\boldsymbol{a}\in I\_{\boldsymbol{\nu},\boldsymbol{a}}} \left| \mathbf{A}^{2}\right|\_{\boldsymbol{a}}^{\boldsymbol{a}}},$$

where

where

*Functional Calculus*

**u**ð Þ<sup>1</sup>

**u**ð Þ<sup>2</sup>

and *l*th row of **A**~ ≔ **A**<sup>2</sup>

*a*#,*<sup>r</sup> ij* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> l*¼1

P*<sup>n</sup> f*¼1 P*<sup>n</sup> l*¼1 P

Let

*u*ð Þ<sup>1</sup> *il* <sup>≔</sup> <sup>X</sup>*<sup>n</sup> f*¼1

where **u**ð Þ<sup>1</sup>

*u*ð Þ<sup>2</sup> *if* <sup>≔</sup><sup>X</sup> *l a*~*fl*

**6**

element of the matrix **A**~ ≔ **A**<sup>2</sup>

X *β* ∈*Js*,*n*f g*i*

Construct the matrix **<sup>U</sup>**<sup>1</sup> <sup>¼</sup> *<sup>u</sup>*ð Þ<sup>1</sup>

X *α*∈*Is*,*n*f g*j*

and construct the matrix **<sup>U</sup>**<sup>2</sup> <sup>¼</sup> *<sup>u</sup>*ð Þ<sup>2</sup>

X *β* ∈*Js*,*n*f g*i*

X*n f*¼1

X *α*∈*Is*,*n*f g*j*

> � � �

X *l u*ð Þ<sup>1</sup> *il*

*<sup>i</sup>:* <sup>¼</sup> <sup>X</sup>

*: <sup>j</sup>* <sup>¼</sup> <sup>X</sup>

**A**<sup>∗</sup> . *Proof.* Taking into account (16), we have for #*A*,

2 4 2 4

*β* ∈*Js*,*n*f g*i*

� � �

*a*#,*<sup>r</sup> ij* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> l*¼1

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>2</sup> � �

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>2</sup> � � � � *β β*

*: <sup>j</sup>* **e***:<sup>f</sup>* � � �

> � *β β*

*il* � �

ð Þ **e***<sup>l</sup>:*

� � � *α*

*if* � �

> � *β β u*ð Þ<sup>2</sup>

ð Þ **e***<sup>l</sup>:*

� � � *α*

*<sup>i</sup>:* is the *i*th row of **U**1. So, we get (18). If we first consider

*<sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α*∈*Is*,*n*f g*j*

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

*α*∈*Is*,*n*f g*j*

By substituting (14) and (15) in (20), we obtain

P

� �

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>2</sup> � � � � *β β* P

**A**<sup>∗</sup> .

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>j</sup>* **<sup>A</sup>**<sup>2</sup> � �

**A**<sup>2</sup> � � *:<sup>i</sup>* **e***:<sup>f</sup>* � � � � �

> � � �

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

**A**<sup>2</sup> � � *:<sup>i</sup>* **e***:<sup>f</sup>* � � � � �

P

P*<sup>n</sup> f*¼1 P

**A**<sup>2</sup> � � *:<sup>i</sup>* ~**a***:<sup>f</sup>* � � � � �

**AA**<sup>∗</sup> ð Þ*j:*

� *β β*

� � � *α α*

ð Þ **a**~*l:*

are the row and column vectors, respectively. Here **a**~*:<sup>f</sup>* and **a**~*l:* are the *f*th column

X*n f*¼1 *a*# *ilalf <sup>a</sup>*†

*:<sup>i</sup>* **a***:<sup>f</sup>* � � � � �

> � � � *β β a*~*fl* P

� *β β afl*

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

where **e***:<sup>l</sup>* and **e***<sup>l</sup>:* are the unit column and row vectors, respectively, such that all their components are 0, except the *l*th components which are 1; *a*~*lf* is the (*lf*)th

*<sup>a</sup>*~*fl* <sup>¼</sup> <sup>X</sup>

*β* ∈*Js*,*n*f g*i*

*<sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α*∈*Is*,*n*f g*j*

P

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

� � �

*α*

**A**<sup>2</sup> � � *:i* ð Þ **a**~*:<sup>l</sup>*

�

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>a</sup>**~*<sup>f</sup>:* � � �

**A**<sup>2</sup> � �

�

� �

� *β β*

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>u</sup>**ð Þ<sup>1</sup>

� � � �

� � � *α α*

*:<sup>i</sup>* **<sup>u</sup>**ð Þ<sup>2</sup> *:f* � � � �

� � � *β β* ,

*i:*

� � � *α α* ,

, *f*, *j* ¼ 1, …, *n:*

�

∈ *<sup>n</sup>*�*<sup>n</sup>*. It follows that

� �

*if* <sup>¼</sup> <sup>X</sup> *β* ∈*Js*,*n*f g*i*

it follows (19). □

∈ *<sup>n</sup>*�*<sup>n</sup>*, then from

3

3

<sup>5</sup><sup>∈</sup> *<sup>n</sup>*�<sup>1</sup>

<sup>5</sup><sup>∈</sup> <sup>1</sup>�*n*, *<sup>f</sup>* <sup>¼</sup> 1, …, *<sup>n</sup>*

, *l* ¼ 1, …, *n:*

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>a</sup>** <sup>∗</sup>

� �

*fj:* (20)

� � �

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

ð Þ **e***<sup>l</sup>:*

*l:*

*α*

,

, *i*, *l* ¼ 1, …, *n:*

� � � *α α* � � � *α*

¼

$$\begin{aligned} \mathbf{v}\_{i.}^{(1)} &= \left[ \sum\_{\boldsymbol{\beta} \in I\_{\boldsymbol{\beta},\boldsymbol{\alpha}}\{\boldsymbol{i}\}} \left| (\mathbf{A}^{\*}\mathbf{A})\_{\boldsymbol{i}} (\overline{\mathbf{a}}\_{\boldsymbol{f}}) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \right] \in \mathbb{C}^{1 \times n}, \; f = \mathbf{1}, ..., n, \\\mathbf{v}\_{\cdot \boldsymbol{j}}^{(2)} &= \left[ \sum\_{a \in I\_{\boldsymbol{\alpha},\boldsymbol{\alpha}}\{\boldsymbol{j}\}} \left| (\mathbf{A}^{2})\_{\boldsymbol{j}.} (\overline{\mathbf{a}}\_{\boldsymbol{l}}) \right|\_{a}^{a} \right] \in \mathbb{C}^{n \times 1}, \; l = 1, ..., n. \end{aligned}$$

Here **a***:<sup>f</sup>* and **a***<sup>l</sup>:* are the *f*th column and *l*th row of **A** ≔ **A**<sup>∗</sup> **A**<sup>2</sup> .

#### **3.2 Determinantal representations of the core-EP inverses**

Similar as in [4], we introduce two core-EP inverses. **Definition 3.8.** A matrix **X** ∈ *<sup>n</sup>*�*<sup>n</sup>* is said to be the right core-EP inverse of **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* if it satisfies the conditions

$$\mathbf{X} \mathbf{A} \mathbf{X} = \mathbf{A}, \text{and } \mathcal{C}(\mathbf{X}) = \mathcal{C}(\mathbf{X}^\*) = \mathcal{C} \left( \mathbf{A}^d \right).$$

It is denoted as **A**○† .

**Definition 3.9.** A matrix **X** ∈ *<sup>n</sup>*�*<sup>n</sup>* is said to be the left core-EP inverse of **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* if it satisfies the conditions

$$\mathbf{X} \mathbf{A} \mathbf{X} = \mathbf{A}, \text{and } \mathcal{R}(\mathbf{X}) = \mathcal{R}(\mathbf{X}^\*) = \mathcal{R}\left(\mathbf{A}^d\right).$$

It is denoted as **<sup>A</sup>**○†.

**Remark 3.10.** Since <sup>C</sup> **<sup>A</sup>**<sup>∗</sup> ð Þ*<sup>d</sup>* � � ¼ R **<sup>A</sup>***<sup>d</sup>* � �, then the left core inverse **<sup>A</sup>**○† of **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* 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** ∈ *<sup>n</sup>*�*<sup>n</sup>* ,

$$\mathbf{A}^{\copyright} = \mathbf{A}^{\{2,3,6a\}} \quad \text{and} \quad \mathcal{C}(\mathbf{A}^{\copyright}) \subseteq \mathcal{C}\left(\mathbf{A}^{k}\right),$$

$$\mathbf{A}\_{\copyright} = \mathbf{A}^{\{2,4,6b\}} \quad \text{and} \quad \mathcal{R}\left(\mathbf{A}\_{\copyright}\right) \subseteq \mathcal{R}\left(\mathbf{A}^{k}\right).$$

Thanks to [35], the following representations of the core-EP inverses will be used for their determinantal representations.

**Lemma 3.11.** *Let* **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup> and* Ind**<sup>A</sup>** <sup>¼</sup> *k. Then*

$$\mathbf{A}^{\oplus} = \mathbf{A}^{k} \left(\mathbf{A}^{k+1}\right)^{\dagger},\tag{21}$$

**Corollary 3.13.** *Let* **A** ∈ *<sup>n</sup>*�*<sup>n</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.89341*

*a*○# ,*<sup>r</sup> ij* ¼

*a*○# ,*<sup>l</sup> ij* ¼

to be the DMP inverse of **A** if it satisfies the conditions

.

*a<sup>d</sup>*,† *ij* ¼

¼

and has the following representation

**Theorem 3.15.** *Let* **A** ∈ *<sup>n</sup>*�*<sup>n</sup>*

**u**ð Þ<sup>1</sup>

**u**ð Þ<sup>2</sup>

*<sup>i</sup>:* <sup>¼</sup> <sup>X</sup> *β* ∈*Js* <sup>1</sup>,*n*f g*i*

*: <sup>j</sup>* <sup>¼</sup> <sup>X</sup>

2 4

2 4

**XAX** <sup>¼</sup> **<sup>X</sup>**, **XA** <sup>¼</sup> **<sup>A</sup>***<sup>d</sup>*

*ij* � � *has the following determinantal representations.*

P

P

P *β* ∈*Js*

P *β* ∈*Js*

*ij* � � *can be expressed as follows*

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **<sup>A</sup>**<sup>2</sup> **<sup>A</sup>**<sup>2</sup> � � <sup>∗</sup> � �

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **<sup>A</sup>**<sup>2</sup> **<sup>A</sup>**<sup>2</sup> � � <sup>∗</sup> � � �

**<sup>A</sup>**<sup>2</sup> � �

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>2</sup> � � <sup>∗</sup>

� � �

where **<sup>a</sup>**^*<sup>i</sup>:* is the *<sup>i</sup>*th row of **<sup>A</sup>**^ <sup>¼</sup> **A A**<sup>2</sup> � � <sup>∗</sup> and **<sup>a</sup>**�*: <sup>j</sup>* is the *<sup>j</sup>*th column of **<sup>A</sup>**� <sup>¼</sup> **<sup>A</sup>**<sup>2</sup> � � <sup>∗</sup>

The concept of the DMP inverse in complex matrices was introduced in [2] by S.

**Definition 3.14.** [2] Suppose **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* and Ind**<sup>A</sup>** <sup>¼</sup> *<sup>k</sup>*. A matrix **<sup>X</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* is said

Due to [2], if an arbitrary matrix satisfies the system of Eq. (27), then it is unique

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>u</sup>**ð Þ<sup>1</sup>

�

<sup>1</sup>,*n*f g*<sup>i</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

�

*:i* **a**~*:f* � � �

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

Here, **a**~*:<sup>f</sup>* and **a**^*<sup>l</sup>:* are the *f*th column and the *l*th row of **A**~ ≔ **A***<sup>k</sup>*þ<sup>1</sup>

*<sup>β</sup>* <sup>∈</sup>*Js*1,*<sup>n</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � � � � *β β* P

> <sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � � � � *β β* P

**<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

� �

> � � �

*α*∈*Is*,*n*f g*j*

� � � �

*:i* **u**ð Þ<sup>2</sup> *: j*

� � � �

� � � *β β*

ð Þ **a**~*<sup>l</sup>:*

� � � *α α* 3

<sup>5</sup><sup>∈</sup> *<sup>n</sup>*�<sup>1</sup>

3

**<sup>A</sup>***<sup>d</sup>*,† <sup>¼</sup> **<sup>A</sup>***<sup>d</sup>***AA**†

**A**, *and* **A***<sup>k</sup>*

**<sup>X</sup>** <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>*

*<sup>s</sup> ,* Ind**<sup>A</sup>** <sup>¼</sup> *k, and* rk **<sup>A</sup>***<sup>k</sup>* � � <sup>¼</sup> *<sup>s</sup>*1*. Then, its DMP inverse*

� � � *α α*

*α*

*i:*

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>β</sup>*

� � � *β β*

*β*

<sup>5</sup><sup>∈</sup> <sup>1</sup>�*<sup>n</sup>*, *<sup>f</sup>* <sup>¼</sup> 1, …, *<sup>n</sup>*,

, *l* ¼ 1, …, *n:*

**A**†

*:* (28)

*:* (27)

¼ (29)

, (30)

**A**<sup>∗</sup> .

� � �

� � � �

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>2</sup> � � <sup>∗</sup>

� �

P

P

**3.3 Determinantal representations of the DMP and MPD inverses**

P

*Determinantal Representations of the Core Inverse and Its Generalizations*

P

*ij* � � *and A*○# <sup>¼</sup> *<sup>a</sup>*○# ,*<sup>l</sup>*

Malik and N. Thome.

It is denoted as **A***<sup>d</sup>*,†

**<sup>A</sup>***<sup>d</sup>*,† <sup>¼</sup> *ad*,†

where

**9**

*a*○# ,*<sup>r</sup>*

*<sup>s</sup> ,* Ind**<sup>A</sup>** <sup>¼</sup> <sup>1</sup>*, and there exist A*○# *and A*○#*. Then A*○# <sup>¼</sup>

� � � �

*α*

*α*

� � � *β β* ,

,

**A**.

*j:* ð Þ **a**^*i:*

� � � *α α*

*:i* **a**�*: j*

� � � *β β*

**A**2

$$\mathbf{A}\_{\oplus} = \left(\mathbf{A}^{k+1}\right)^{\dagger} \mathbf{A}^{k}. \tag{22}$$

Moreover, if Ind**A** ¼ 1, then we have the following representations of the right and left core inverses

$$\mathbf{A}^{\otimes} = \mathbf{A} \left(\mathbf{A}^{2}\right)^{\dagger},\tag{23}$$

$$\mathbf{A}\_{\oplus} = \left(\mathbf{A}^{2}\right)^{\dagger}\mathbf{A}.\tag{24}$$

**Theorem 3.12.** *Suppose* **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*n,* Ind**<sup>A</sup>** <sup>¼</sup> *k,* rk**A***<sup>k</sup>* <sup>¼</sup> *s, and there exist* **<sup>A</sup>**○† *and* **<sup>A</sup>**○†*. Then* **<sup>A</sup>**○† <sup>¼</sup> *<sup>a</sup>*○† ,*<sup>r</sup> ij* � � *and* **<sup>A</sup>**○† <sup>¼</sup> *<sup>a</sup>*○† , *<sup>l</sup> ij* � � *possess the determinantal representations, respectively,*

$$a\_{ij}^{\oplus,r} = \frac{\sum\_{a \in I, \iota \{j\}} \left| \left( \mathbf{A}^{k+1} \left( \mathbf{A}^{k+1} \right)^{\*} \right)\_{j.} \right|\_{a}^{a}}{\sum\_{a \in I\_{\iota,\iota}} \left| \mathbf{A}^{k+1} \left( \mathbf{A}^{k+1} \right)^{\*} \right|\_{a}^{a}},\tag{25}$$

$$a\_{ij}^{\mathfrak{D},l} = \frac{\sum\_{\boldsymbol{\beta} \in J\_{\boldsymbol{\varsigma},\boldsymbol{\imath}}\{i\}} \Big| \Big( \left( \mathbf{A}^{k+1} \right)^{\*} \mathbf{A}^{k+1} \Big)\_{\boldsymbol{\imath}} (\check{\mathbf{a}}\_{\cdot,j}) \Big|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} }{\sum\_{\boldsymbol{\beta} \in J\_{\boldsymbol{\imath},\boldsymbol{\imath}}\,\Big|} \Big( \mathbf{A}^{k+1} \Big)^{\*} \mathbf{A}^{k+1} \Big|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}}},\tag{26}$$

where **<sup>a</sup>**^*<sup>i</sup>:* is the *<sup>i</sup>*th row of **<sup>A</sup>**^ <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �<sup>∗</sup> and **a**�*: <sup>j</sup>* is the *j*th column of **<sup>A</sup>**� <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �<sup>∗</sup> **A***k* . *Proof.* Consider **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �† ¼ *a* ð Þ *k*þ1,† *ij* � � and **<sup>A</sup>***<sup>k</sup>* <sup>¼</sup> *<sup>a</sup>* ð Þ*k ij* � �. By (21), *a*○† ,*<sup>r</sup> ij* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> t*¼1 *a*ð Þ*<sup>k</sup> it <sup>a</sup>*ð Þ *<sup>k</sup>*þ1,† *tj :*

Taking into account (9) for the determinantal representation of **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �† , we get

$$a\_{ij}^{\oplus,r} = \sum\_{t=1}^{n} a\_{it}^{(k)} \frac{\sum\_{a \in I\_{\iota,n}\{j\}} \left| \left(\mathbf{A}^{k+1} \left(\mathbf{A}^{k+1}\right)^{\*}\right)\_{j.} \left(\mathbf{a}\_{\iota}^{(k+1,\*)}\right)\right|\_{a}^{a}}{\sum\_{a \in I\_{\iota,n}} \left| \mathbf{A}^{k+1} \left(\mathbf{A}^{k+1}\right)^{\*}\right|\_{a}^{a}},$$

where **a** ð Þ *k*þ1, ∗ *<sup>t</sup>:* is the *<sup>t</sup>*th row of **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �<sup>∗</sup> . Since P*<sup>n</sup> <sup>t</sup>*¼<sup>1</sup>*a*ð Þ*<sup>k</sup> it* **a** ð Þ *k*þ1, ∗ *<sup>t</sup>:* ¼ ^**a***<sup>i</sup>:*, then it follows (25).

The determinantal representation (26) can be obtained similarly by integrating (8) for the determinantal representation of **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �† 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 Core Inverse and Its Generalizations DOI: http://dx.doi.org/10.5772/intechopen.89341*

**Corollary 3.13.** *Let* **A** ∈ *<sup>n</sup>*�*<sup>n</sup> <sup>s</sup> ,* Ind**<sup>A</sup>** <sup>¼</sup> <sup>1</sup>*, and there exist A*○# *and A*○#*. Then A*○# <sup>¼</sup> *a*○# ,*<sup>r</sup> ij* � � *and A*○# <sup>¼</sup> *<sup>a</sup>*○# ,*<sup>l</sup> ij* � � *can be expressed as follows*

$$\begin{split} a\_{ij}^{\boldsymbol{\Phi},\boldsymbol{r}} &= \frac{\sum\_{\boldsymbol{a}\in I\_{\boldsymbol{\star},\boldsymbol{\star}}\{\boldsymbol{j}\}} \left| \left(\mathbf{A}^{2}\left(\mathbf{A}^{2}\right)^{\ast}\right)\_{\boldsymbol{j}} \left(\hat{\mathbf{a}}\_{\cdot}\right)\right|\_{\boldsymbol{a}}^{\boldsymbol{a}}}{\sum\_{\boldsymbol{a}\in I\_{\boldsymbol{\star},\boldsymbol{\star}}} \left|\mathbf{A}^{2}\left(\mathbf{A}^{2}\right)^{\ast}\right|\_{\boldsymbol{a}}^{\boldsymbol{a}}}, \end{split}$$

$$a\_{ij}^{\boldsymbol{\Phi},\boldsymbol{l}} = \frac{\sum\_{\boldsymbol{\beta}\in I\_{\boldsymbol{\star},\boldsymbol{\star}}\{\boldsymbol{i}\}} \left| \left(\left(\mathbf{A}^{2}\right)^{\ast}\mathbf{A}^{2}\right)\_{\boldsymbol{j}} \left(\mathbf{\dot{a}}\_{\cdot}\right)\right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}}}{\sum\_{\boldsymbol{\beta}\in I\_{\boldsymbol{\star},\boldsymbol{\star}}} \left|\left(\mathbf{A}^{2}\right)^{\ast}\mathbf{A}^{2}\right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}}},$$

where **<sup>a</sup>**^*<sup>i</sup>:* is the *<sup>i</sup>*th row of **<sup>A</sup>**^ <sup>¼</sup> **A A**<sup>2</sup> � � <sup>∗</sup> and **<sup>a</sup>**�*: <sup>j</sup>* is the *<sup>j</sup>*th column of **<sup>A</sup>**� <sup>¼</sup> **<sup>A</sup>**<sup>2</sup> � � <sup>∗</sup> **A**.

#### **3.3 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.

**Definition 3.14.** [2] Suppose **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* and Ind**<sup>A</sup>** <sup>¼</sup> *<sup>k</sup>*. A matrix **<sup>X</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* is said to be the DMP inverse of **A** if it satisfies the conditions

$$\mathbf{X} \mathbf{A} \mathbf{X} = \mathbf{X}, \mathbf{X} \mathbf{A} = \mathbf{A}^d \mathbf{A}, and \; \mathbf{A}^k \mathbf{X} = \mathbf{A}^k \mathbf{A}^\dagger. \tag{27}$$

It is denoted as **A***<sup>d</sup>*,† .

**Lemma 3.11.** *Let* **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup> and* Ind**<sup>A</sup>** <sup>¼</sup> *k. Then*

*and* **<sup>A</sup>**○† <sup>¼</sup> *<sup>a</sup>*○† , *<sup>l</sup>*

*a*○† ,*<sup>r</sup> ij* ¼

*a*○† ,*<sup>l</sup> ij* ¼

where **<sup>a</sup>**^*<sup>i</sup>:* is the *<sup>i</sup>*th row of **<sup>A</sup>**^ <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �<sup>∗</sup>

*a*ð Þ*<sup>k</sup> it*

*<sup>t</sup>:* is the *<sup>t</sup>*th row of **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �<sup>∗</sup>

(8) for the determinantal representation of **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �†

sions than they are obtained in Theorems 3.6 and 3.7.

P

*ij* � �

P

P

¼ *a*

*a*○† ,*<sup>r</sup> ij* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> t*¼1

and left core inverses

*Functional Calculus*

*Then* **<sup>A</sup>**○† <sup>¼</sup> *<sup>a</sup>*○† ,*<sup>r</sup>*

**<sup>A</sup>**� <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �<sup>∗</sup>

where **a**

follows (25).

**8**

**A***k* . *Proof.* Consider **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �†

> *a*○† ,*<sup>r</sup> ij* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> t*¼1

ð Þ *k*þ1, ∗

*ij* � � **<sup>A</sup>**○† <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �†

**<sup>A</sup>**○† <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �†

Moreover, if Ind**A** ¼ 1, then we have the following representations of the right

*<sup>A</sup>*○# <sup>¼</sup> **A A**<sup>2</sup> � �†

*<sup>A</sup>*○# <sup>¼</sup> **<sup>A</sup>**<sup>2</sup> � �†

**Theorem 3.12.** *Suppose* **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*n,* Ind**<sup>A</sup>** <sup>¼</sup> *k,* rk**A***<sup>k</sup>* <sup>¼</sup> *s, and there exist* **<sup>A</sup>**○† *and* **<sup>A</sup>**○†*.*

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � � � �<sup>∗</sup>

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �<sup>∗</sup>

� � �

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �<sup>∗</sup> � � �

**<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

� � �

and **<sup>A</sup>***<sup>k</sup>* <sup>¼</sup> *<sup>a</sup>*

*a*ð Þ*<sup>k</sup> it <sup>a</sup>*ð Þ *<sup>k</sup>*þ1,† *tj :*

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � � � �<sup>∗</sup>

The determinantal representation (26) can be obtained similarly by integrating

Taking into account the representations (23)-(24), we obtain the determinantal representations of the right and left core inverses that have more simpler expres-

Taking into account (9) for the determinantal representation of **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �†

� � �

P

� � � �

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �<sup>∗</sup>

� �

P

P

ð Þ *k*þ1,† *ij* � � **A***k*

, (21)

*:* (22)

, (23)

**A***:* (24)

*possess the determinantal representations, respectively,*

� � � �

*α*

*α*

� � � *β β*

and **a**�*: <sup>j</sup>* is the *j*th column of

. By (21),

ð Þ *k*þ1, ∗ *t:*

ð Þ *k*þ1, ∗

� � � �

*α*

*α*

,

*<sup>t</sup>:* ¼ ^**a***<sup>i</sup>:*, then it

in (22). □

, (25)

, (26)

, we get

*j:* ð Þ **a**^*<sup>i</sup>:*

� � � *α α*

*:i* **a**�*: j*

� � � *β β*

ð Þ*k ij* � �

*j:* **a**

*<sup>t</sup>*¼<sup>1</sup>*a*ð Þ*<sup>k</sup> it* **a**

� � � *α α*

� � �

*<sup>α</sup>*∈*Ir*,*<sup>m</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �<sup>∗</sup> � � �

. Since P*<sup>n</sup>*

**A***<sup>k</sup>*þ<sup>1</sup>

Due to [2], if an arbitrary matrix satisfies the system of Eq. (27), then it is unique and has the following representation

$$\mathbf{A}^{d,\dagger} = \mathbf{A}^d \mathbf{A} \mathbf{A}^\dagger. \tag{28}$$

**Theorem 3.15.** *Let* **A** ∈ *<sup>n</sup>*�*<sup>n</sup> <sup>s</sup> ,* Ind**<sup>A</sup>** <sup>¼</sup> *k, and* rk **<sup>A</sup>***<sup>k</sup>* � � <sup>¼</sup> *<sup>s</sup>*1*. Then, its DMP inverse* **<sup>A</sup>***<sup>d</sup>*,† <sup>¼</sup> *ad*,† *ij* � � *has the following determinantal representations.*

$$a\_{ij}^{d, \dagger} = \frac{\sum\_{a \in I\_{\iota u} \{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\iota}} \left( \mathbf{u}\_{i.}^{(1)} \right) \right|\_{a}^{a}}{\sum\_{\beta \in I\_{\iota\_{1}u}} \left| \mathbf{A}^{k+1} \right|\_{\beta}^{\beta} \sum\_{a \in I\_{\iota u}} \left| \mathbf{A} \mathbf{A}^\* \right|\_{a}^{a}} = \tag{29}$$

$$\hat{\boldsymbol{\beta}} = \frac{\sum\_{\boldsymbol{\beta} \in J\_{\boldsymbol{\tau}, \boldsymbol{u}}\{i\}} \Big| \left(\mathbf{A}^{k+1}\right)\_{\boldsymbol{\beta}} \left(\mathbf{u}\_{\boldsymbol{\beta}}^{(2)}\right) \Big|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}}}{\sum\_{\boldsymbol{\beta} \in J\_{\boldsymbol{\tau}, \boldsymbol{u}}} |\mathbf{A}^{k+1}|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \sum\_{\boldsymbol{\beta} \in J\_{\boldsymbol{\tau}, \boldsymbol{u}}} |\mathbf{A}\mathbf{A}^{\*}|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}}},\tag{30}$$

where

$$\begin{aligned} \mathbf{u}\_{i.}^{(1)} &= \left[ \sum\_{\beta \in I\_{\iota\_1, \iota} \{i\}} \left| \left( \mathbf{A}^{k+1} \right)\_{,i} (\mathbf{\bar{a}}\_{\mathcal{f}}) \right|\_{\beta}^{\beta} \right] \in \mathbb{C}^{1 \times n}, \ f = \mathbf{1}, \ldots, n, \\\ \mathbf{u}\_{.\boldsymbol{j}}^{(2)} &= \left[ \sum\_{a \in I\_{\iota \mathcal{F}} \{\boldsymbol{j}\}} \left| (\mathbf{A} \mathbf{A}^{\*})\_{\boldsymbol{j},} (\mathbf{\bar{a}}\_{\mathcal{l}}) \right|\_{a}^{a} \right] \in \mathbb{C}^{n \times 1}, \ l = \mathbf{1}, \ldots, n. \end{aligned}$$

Here, **a**~*:<sup>f</sup>* and **a**^*<sup>l</sup>:* are the *f*th column and the *l*th row of **A**~ ≔ **A***<sup>k</sup>*þ<sup>1</sup> **A**<sup>∗</sup> . *Proof.* Taking into account (28) for **A***<sup>d</sup>*,† , we get

$$a\_{ij}^{d, \dagger} = \sum\_{l=1}^{n} \sum\_{f=1}^{n} a\_{il}^{d} a\_{lj} a\_{fj}^{\dagger}. \tag{31}$$

The matrix **A**†,*<sup>d</sup>* is unique, and it can be represented as

*Determinantal Representations of the Core Inverse and Its Generalizations*

*ij* � � *has the following determinantal representations*

� � � �

� � � � *: j*

**<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>a</sup>**^*:<sup>f</sup>* � � � � �

Here, **a**^*<sup>l</sup>:* and **a**^*:<sup>f</sup>* are the *l*th row and the *f*th column of **A**^ ≔ **A**<sup>∗</sup> **A***<sup>k</sup>*þ<sup>1</sup>

*j:* ð Þ **a**^*<sup>l</sup>:*

� � � �

*α*

3

*α*

3

**A**1**A**† .

*. The matrix* **<sup>X</sup>** <sup>¼</sup> **<sup>A</sup>***<sup>c</sup>*,† *is the unique matrix that*

, *and* **XA** <sup>¼</sup> **<sup>A</sup>**†

**<sup>A</sup>***<sup>c</sup>*,† <sup>¼</sup> **<sup>Q</sup>***A***A***<sup>d</sup>***P***A:* (34)

*<sup>s</sup> ,* Ind**<sup>A</sup>** <sup>¼</sup> *m, and* rk **<sup>A</sup>***<sup>m</sup>* ð Þ¼ *<sup>s</sup>*1*. Then, the determinan-*

� � � *β β*

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � � � � *β β*

*ij* � � *can be expressed as*

*: j*

**A**1*:*

(35)

� *β β*

*Proof.* The proof is similar to the proof of Theorem 3.15. □

**Definition 3.18.** [6] Suppose **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* has the core-nilpotent decomposition **A** ¼ **A**<sup>1</sup> þ **A**2, where Ind**A**<sup>1</sup> ¼ Ind**A**, **A**<sup>2</sup> is nilpotent, and **A**1**A**<sup>2</sup> ¼ **A**2**A**<sup>1</sup> ¼ 0. The

Taking into account (34), it follows the next theorem about determinantal

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>v</sup>**ð Þ*<sup>l</sup>*

*β*

�

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

� �<sup>2</sup>

� � � �

P *β* ∈*Js*

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � � � � *α α* ¼

� � � *β β*

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>v</sup>**ð Þ<sup>1</sup>

*β* P *β* ∈*Is*

�

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

*: <sup>j</sup>* <sup>¼</sup> <sup>X</sup> *α*∈*Is* <sup>1</sup>,*n*f g*j*

*<sup>i</sup>:* <sup>¼</sup> <sup>X</sup>

CMP inverse of **A** is called the matrix **A***<sup>c</sup>*,† ≔ **A**†

representations of the quaternion CMP inverse.

*tal representations of its CMP inverse* **<sup>A</sup>***<sup>c</sup>*,† <sup>¼</sup> *<sup>a</sup><sup>c</sup>*,†

*ac*,† *ij* ¼

**Theorem 3.20.** *Let* **A** ∈ *<sup>n</sup>*�*<sup>n</sup>*

**Lemma 3.19.** [6] *Let* **A** ∈ *<sup>n</sup>*�*<sup>n</sup>*

Moreover,

**11**

*satisfies the following system of equations:*

2 4

*β* ∈*Js*,*n*f g*i*

**3.4 Determinantal representations of the CMP inverse**

**XAX** <sup>¼</sup> **<sup>X</sup>**, **AXA** <sup>¼</sup> **<sup>A</sup>**1, **AX** <sup>¼</sup> **<sup>A</sup>**1**A**†

P

P

2 4

**Theorem 3.17.** *Let* **A** ∈ *<sup>n</sup>*�*<sup>n</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.89341*

P

**v**ð Þ<sup>1</sup>

**v**ð Þ<sup>2</sup>

P

**<sup>A</sup>**†,*<sup>d</sup>* <sup>¼</sup> *<sup>a</sup>*†,*<sup>d</sup>*

*a*†,*<sup>d</sup> ij* ¼

where

**<sup>A</sup>**†,*<sup>d</sup>* <sup>¼</sup> **<sup>A</sup>**†

**AA***<sup>d</sup>*

P *α*∈*Is*

*α*∈*Is*

<sup>5</sup><sup>∈</sup> *<sup>n</sup>*�<sup>1</sup>

*<sup>s</sup> ,* Ind**<sup>A</sup>** <sup>¼</sup> *<sup>k</sup>*, *and* rk**A***<sup>k</sup>* <sup>¼</sup> *<sup>s</sup>*1*. Then, its MPD inverse*

<sup>1</sup>,*n*f g*<sup>j</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

<sup>P</sup> *<sup>α</sup>*

, *l* ¼ 1, …, *n*

� � �

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup> β* P

<sup>5</sup><sup>∈</sup> <sup>1</sup>�*<sup>n</sup>*, *<sup>l</sup>* <sup>¼</sup> 1, …, *<sup>n</sup>:*

*:* (33)

� � �

*<sup>j</sup>:* **<sup>v</sup>**ð Þ<sup>2</sup> *i:*

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � � � � *α α* ,

.

� � � �

*α*

By substituting (12) and (9) for the determinantal representations of **A***<sup>d</sup>* and **A**† in (31), we get

$$\begin{split} &a\_{ij}^{d,\uparrow} = \\ &\sum\_{l=1}^{n} \sum\_{f=1}^{n} \frac{\sum\_{\beta \in J\_{\eta,\boldsymbol{x}}\{i\}} \left| \left(\mathbf{A}^{k+1}\right)\_{\boldsymbol{i}} \left(\mathbf{a}^{(k)}\_{l}\right) \right|\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{\boldsymbol{x},\boldsymbol{a}}} \left| \mathbf{A}^{k+1}\right|\_{\beta}^{\beta}} a\_{lj} \frac{\sum\_{a \in I\_{\boldsymbol{x}}\{j\}} \left| \left(\mathbf{A}\mathbf{A}^{\*}\right)\_{\boldsymbol{j}} \left(\mathbf{a}^{\*}\_{l}\right) \right|\_{a}^{a}}{\sum\_{a \in I\_{\boldsymbol{x},\boldsymbol{a}}} \left| \mathbf{A}\mathbf{A}^{\*}\right|\_{a}^{a}} = \\ &\sum\_{l=1}^{n} \sum\_{f=1}^{n} \frac{\sum\_{\beta \in I\_{\boldsymbol{x},\boldsymbol{a}}\{i\}} \left| \left(\mathbf{A}^{k+1}\right)\_{\boldsymbol{i}} \left(\mathbf{e}\_{l}\right)\_{\boldsymbol{j}} \right|\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{\boldsymbol{x},\boldsymbol{a}}} \left| \mathbf{A}^{k+1}\right|\_{\beta}^{\beta}} \tilde{d}\_{l} \tilde{\mu} \frac{\sum\_{a \in I\_{\boldsymbol{x},\boldsymbol{a}}\{j\}} \left(\left(\mathbf{A}\mathbf{A}^{\*}\right)\_{\boldsymbol{j}} \left(\mathbf{e}\_{l}\right)\_{\boldsymbol{a}} \right)\_{a}^{a}}{\sum\_{a \in I\_{\boldsymbol{x},\boldsymbol{a}}} \left| \mathbf{A}\mathbf{A}^{\*}\right|\_{\boldsymbol{a}}},\end{split} \tag{32}$$

where **e***:<sup>l</sup>* and **e***<sup>l</sup>:* are the *l*th unit column and row vectors, and *a*~*lf* is the ð Þ *lf* th element of the matrix **<sup>A</sup>**<sup>~</sup> <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> **A**<sup>∗</sup> . If we put

$$\mu\_{\boldsymbol{\tilde{y}}^{\boldsymbol{\varepsilon}}}^{(1)} := \sum\_{l=1}^{n} \sum\_{\boldsymbol{\beta} \in J\_{\boldsymbol{\tau}\_{1}, \boldsymbol{\pi}}\{i\}} \left| \left(\mathbf{A}^{k+1}\right)\_{:,i} (\mathbf{e}\_{I}) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \tilde{a}\_{\boldsymbol{\mathcal{Y}}} = \sum\_{\boldsymbol{\beta} \in J\_{\boldsymbol{\tau}\_{1}, \boldsymbol{\pi}}\{i\}} \left| \left(\mathbf{A}^{k+1}\right)\_{:,i} (\mathbf{\bar{a}}\_{\boldsymbol{\mathcal{Y}}}) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}},$$

as the *f*th component of the row vector **u**ð Þ<sup>1</sup> *<sup>i</sup>:* <sup>¼</sup> *<sup>u</sup>*ð Þ<sup>1</sup> *<sup>i</sup>*<sup>1</sup> , …, *<sup>u</sup>*ð Þ<sup>1</sup> *in* h i, then from

$$\sum\_{j=1}^{n} u\_{\circ j}^{(1)} \sum\_{a \in I\_{\iota \pi} \{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\cdot}} (\mathbf{e}\_{\circ \cdot}) \right|\_{a}^{a} = \sum\_{a \in I\_{\iota \pi} \{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\cdot}} (\mathbf{u}\_{\iota \cdot}^{(1)}) \right|\_{a}^{a},$$

it follows (29). If we initially obtain

$$\mu\_{lj}^{(2)} \coloneqq \sum\_{f=1}^{n} \tilde{a}\_{lf} \sum\_{a \in I\_{\iota\pi}\{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\iota}} (\mathbf{e}\_{\mathcal{f}.}) \right|\_{a}^{a} = \sum\_{a \in I\_{\iota\pi}\{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\iota}} (\tilde{\mathbf{a}}\_{l.}) \right|\_{a}^{a},$$

as the *l*th component of the column vector **u**ð Þ<sup>2</sup> *: <sup>j</sup>* <sup>¼</sup> *<sup>u</sup>*ð Þ<sup>2</sup> <sup>1</sup> *<sup>j</sup>* , …, *<sup>u</sup>*ð Þ<sup>2</sup> *nj* h i, then from

$$\sum\_{l=1}^{n} \sum\_{\substack{\beta \in J\_{\mathbf{1},n}\{i\}}} \left| \left(\mathbf{A}^{k+1}\right)\_{\,:,i} (\mathbf{e}\_{\,l}) \right|\_{\beta}^{\beta} u\_{l\bar{j}}^{(2)} = \sum\_{\beta \in J\_{\mathbf{1},n}\{i\}} \left| \left(\mathbf{A}^{k+1}\right)\_{\,:,i} \left(\mathbf{u}\_{\cdot \bar{j}}^{(2)}\right) \right|\_{\beta}^{\beta},$$

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 **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* and Ind**<sup>A</sup>** <sup>¼</sup> *<sup>k</sup>*. A matrix **<sup>X</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* is said to be the MPD inverse of **A** if it satisfies the conditions

$$\mathbf{X} \mathbf{A} \mathbf{X} = \mathbf{X}, \mathbf{A} \mathbf{X} = \mathbf{A} \mathbf{A}^d, \text{and } \mathbf{X} \mathbf{A}^k = \mathbf{A}^\dagger \mathbf{A}^k.$$

It is denoted as **A**†,*<sup>d</sup>* . *Determinantal Representations of the Core Inverse and Its Generalizations DOI: http://dx.doi.org/10.5772/intechopen.89341*

The matrix **A**†,*<sup>d</sup>* is unique, and it can be represented as

$$\mathbf{A}^{\dagger,d} = \mathbf{A}^{\dagger} \mathbf{A} \mathbf{A}^{d}. \tag{33}$$

**Theorem 3.17.** *Let* **A** ∈ *<sup>n</sup>*�*<sup>n</sup> <sup>s</sup> ,* Ind**<sup>A</sup>** <sup>¼</sup> *<sup>k</sup>*, *and* rk**A***<sup>k</sup>* <sup>¼</sup> *<sup>s</sup>*1*. Then, its MPD inverse* **<sup>A</sup>**†,*<sup>d</sup>* <sup>¼</sup> *<sup>a</sup>*†,*<sup>d</sup> ij* � � *has the following determinantal representations*

$$a\_{ij}^{\uparrow,d} = \frac{\sum\_{\beta \in I\_{\forall i}} \left| \left( \mathbf{A}^\* \mathbf{A} \right)\_{\cdot} \left( \mathbf{v}\_{\cdot j}^{(1)} \right) \right|\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{\forall i}} |\mathbf{A}^\* \mathbf{A}|\_{\beta}^{\beta} \sum\_{\beta \in I\_{\forall i}} \left| \mathbf{A}^{k+1} \right|\_{a}^{a}} = \frac{\sum\_{a \in I\_{1,\forall}} \left| \left( \mathbf{A}^{k+1} \right)\_{j\_{\cdot}} \left( \mathbf{v}\_{i.}^{(2)} \right) \right|\_{a}^{a}}{\sum\_{a \in I\_{1,\forall}} |\mathbf{A}^\* \mathbf{A}|\_{\beta}^{\beta} \sum\_{a \in I\_{\forall}} \left| \mathbf{A}^{k+1} \right|\_{a}^{a}},$$

where

*Proof.* Taking into account (28) for **A***<sup>d</sup>*,†

<sup>1</sup>,*n*f g*<sup>i</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

�

P *β* ∈*Js*

> P *β* ∈*Js*

in (31), we get

*Functional Calculus*

*ad*,† *ij* ¼

X*n l*¼1

X*n f*¼1

X*n l*¼1

X*n f*¼1

*u*ð Þ<sup>1</sup> *if* <sup>≔</sup> <sup>X</sup>*<sup>n</sup> l*¼1

P *β* ∈*Js*

element of the matrix **<sup>A</sup>**<sup>~</sup> <sup>¼</sup> **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup>

X*n f*¼1

*u*ð Þ<sup>2</sup> *lj* <sup>≔</sup> <sup>X</sup>*<sup>n</sup> f*¼1 *a*~*lf*

X*n l*¼1

It is denoted as **A**†,*<sup>d</sup>*

**10**

X *<sup>β</sup>* <sup>∈</sup>*Js*1,*n*f g*<sup>i</sup>*

logical to consider the following definition.

be the MPD inverse of **A** if it satisfies the conditions

.

*u*ð Þ<sup>1</sup> *if*

it follows (29). If we initially obtain

P *β* ∈*Js* *ad*,† *ij* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> l*¼1

*:i* **a** ð Þ*k :l*

> *:i* ð Þ **e***:<sup>l</sup>*

**<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>e</sup>***<sup>f</sup>:* � � �

� � � �

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � � � � *β β*

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � � � � *β β*

> � � �

<sup>1</sup>,*n*f g*<sup>i</sup>* **<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

� � �

X *β* ∈*Js* <sup>1</sup>,*n*f g*i*

as the *f*th component of the row vector **u**ð Þ<sup>1</sup>

X *α*∈*Is*,*n*f g*j*

� �

X *α*∈*Is*,*n*f g*j*

as the *l*th component of the column vector **u**ð Þ<sup>2</sup>

� � �

**<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

� �

> *:i* ð Þ **e***:<sup>l</sup>*

, we get

*fj:* (31)

*f:*

*α*

*α*

**<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

� �

*<sup>i</sup>*<sup>1</sup> , …, *<sup>u</sup>*ð Þ<sup>1</sup> *in* h i

> � � �

**<sup>A</sup>***<sup>k</sup>*þ<sup>1</sup> � �

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>u</sup>**ð Þ<sup>1</sup>

� � � �

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

<sup>1</sup> *<sup>j</sup>* , …, *<sup>u</sup>*ð Þ<sup>2</sup> *nj* h i

> *:i* **u**ð Þ<sup>2</sup> *: j*

� � � �

**A***k :*

*:i* **a**~*:f* � � �

*i:*

� � � *α*

� � *α*

,

� � � *β β* ,

, then from

� � � *α α* ,

ð Þ **a**~*<sup>l</sup>:*

� � � *β β* ,

� � � *α α* ,

, then from

¼

(32)

X*n f*¼1 *ad ilalf <sup>a</sup>*†

By substituting (12) and (9) for the determinantal representations of **A***<sup>d</sup>* and **A**†

� � � *β β*

� � � *β β*

**A**<sup>∗</sup> . If we put

*:i* ð Þ **e***:<sup>l</sup>*

*alf* P

*a*~*lf* P

where **e***:<sup>l</sup>* and **e***<sup>l</sup>:* are the *l*th unit column and row vectors, and *a*~*lf* is the ð Þ *lf* th

� � � *β β*

� � � *α*

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>e</sup>***<sup>f</sup>:* � � �

> � � � *β β u*ð Þ<sup>2</sup>

*<sup>a</sup>*~*lf* <sup>¼</sup> <sup>X</sup> *β* ∈*Js* <sup>1</sup>,*n*f g*i*

*<sup>i</sup>:* <sup>¼</sup> *<sup>u</sup>*ð Þ<sup>1</sup>

�

*: <sup>j</sup>* <sup>¼</sup> *<sup>u</sup>*ð Þ<sup>2</sup>

�

*<sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α*∈*Is*,*n*f g*j*

> � � � *α*

*lj* <sup>¼</sup> <sup>X</sup> *β* ∈*Js* <sup>1</sup>,*n*f g*i*

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

**Definition 3.16.** Suppose **<sup>A</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* and Ind**<sup>A</sup>** <sup>¼</sup> *<sup>k</sup>*. A matrix **<sup>X</sup>** <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* is said to

**XAX** <sup>¼</sup> **<sup>X</sup>**, **AX** <sup>¼</sup> **AA***<sup>d</sup>*, *and* **XA***<sup>k</sup>* <sup>¼</sup> **<sup>A</sup>**†

*<sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α*∈*Is*,*n*f g*j*

*<sup>α</sup>* <sup>∈</sup>*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*j:* **<sup>a</sup>** <sup>∗</sup>

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>e</sup>***<sup>f</sup>:* � � ��

�

� � � �

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

$$\begin{aligned} \mathbf{v}\_{\boldsymbol{j}}^{(1)} &= \left[ \sum\_{a \in I\_{l,n}\{\boldsymbol{j}\}} \left| \left( \mathbf{A}^{k+1} \right)\_{\boldsymbol{j}.} (\hat{\mathbf{a}}\_{l}) \right|\_{a}^{a} \right] \in \mathbb{C}^{n \times 1}, \ l = 1, \ldots, n \\\\ \mathbf{v}\_{\boldsymbol{i}}^{(2)} &= \left[ \sum\_{\boldsymbol{\beta} \in I\_{l,n}\{\boldsymbol{i}\}} \left| (\mathbf{A}^{\*} \mathbf{A})\_{\boldsymbol{i}} (\hat{\mathbf{a}}\_{\boldsymbol{\beta}}) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \right] \in \mathbb{C}^{1 \times n}, \ l = 1, \ldots, n. \end{aligned}$$

Here, **a**^*<sup>l</sup>:* and **a**^*:<sup>f</sup>* are the *l*th row and the *f*th column of **A**^ ≔ **A**<sup>∗</sup> **A***<sup>k</sup>*þ<sup>1</sup> . *Proof.* The proof is similar to the proof of Theorem 3.15. □

#### **3.4 Determinantal representations of the CMP inverse**

**Definition 3.18.** [6] Suppose **A** ∈ *<sup>n</sup>*�*<sup>n</sup>* has the core-nilpotent decomposition **A** ¼ **A**<sup>1</sup> þ **A**2, where Ind**A**<sup>1</sup> ¼ Ind**A**, **A**<sup>2</sup> is nilpotent, and **A**1**A**<sup>2</sup> ¼ **A**2**A**<sup>1</sup> ¼ 0. The CMP inverse of **A** is called the matrix **A***<sup>c</sup>*,† ≔ **A**† **A**1**A**† .

**Lemma 3.19.** [6] *Let* **A** ∈ *<sup>n</sup>*�*<sup>n</sup> . The matrix* **<sup>X</sup>** <sup>¼</sup> **<sup>A</sup>***<sup>c</sup>*,† *is the unique matrix that satisfies the following system of equations:*

$$\mathbf{X} \mathbf{A} \mathbf{X} = \mathbf{X}, \mathbf{A} \mathbf{X} \mathbf{A} = \mathbf{A}\_1, \mathbf{A} \mathbf{X} = \mathbf{A}\_1 \mathbf{A}^\dagger, \text{and } \mathbf{X} \mathbf{A} = \mathbf{A}^\dagger \mathbf{A}\_1.$$

Moreover,

$$\mathbf{A}^{c,\dagger} = \mathbf{Q}\_A \mathbf{A}^d \mathbf{P}\_A. \tag{34}$$

Taking into account (34), it follows the next theorem about determinantal representations of the quaternion CMP inverse.

**Theorem 3.20.** *Let* **A** ∈ *<sup>n</sup>*�*<sup>n</sup> <sup>s</sup> ,* Ind**<sup>A</sup>** <sup>¼</sup> *m, and* rk **<sup>A</sup>***<sup>m</sup>* ð Þ¼ *<sup>s</sup>*1*. Then, the determinantal representations of its CMP inverse* **<sup>A</sup>***<sup>c</sup>*,† <sup>¼</sup> *<sup>a</sup><sup>c</sup>*,† *ij* � � *can be expressed as*

$$a\_{ij}^{\epsilon,\dagger} = \frac{\sum\_{\beta \in I\_{\iota,\pi}} \left| \left( \mathbf{A}^\* \mathbf{A} \right)\_{\cdot} \left( \mathbf{v}\_{\cdot j}^{(l)} \right) \right|\_{\beta}^{\beta}}{\left( \sum\_{\beta \in I\_{\iota,\pi}} \left| \mathbf{A}^\* \mathbf{A} \right|\_{\beta}^{\beta} \right)^2 \sum\_{\beta \in I\_{\iota\_1,\pi}} \left| \mathbf{A}^{m+1} \right|\_{\beta}^{\beta}} \tag{35}$$

*Functional Calculus*

$$a\_{ij}^{\varepsilon,\dagger} = \frac{\sum\_{a \in I\_{\pi}\{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_j \left( \mathbf{w}\_i^{(l)} \right) \right|\_a^a}{\left( \sum\_{a \in I\_{\pi}\mathbf{a}} |\mathbf{A} \mathbf{A}^\*|\_a^a \right)^2 \sum\_{\beta \in I\_{\nu\_1 \nu}} \left| \mathbf{A}^{m+1} \right|\_\beta^\beta} \tag{36}$$

where **e***:<sup>t</sup>* is the *t*th unit column vector, **e***<sup>k</sup>:* is the *k*th row vector, and *a*^*tk* is the

� *α <sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α*∈*Is*1,*n*f g*j*

as the *t*th component of a column vector **u***:<sup>l</sup>* ¼ *u*1*l*, …, *unl* ½ �. Then from

� *β*

� �

*β* P *α*∈*Is*

� �

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

� �<sup>2</sup>

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

ð Þ **e***:<sup>t</sup>*

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

**A**<sup>2</sup> � � *j:* ð Þ **e***<sup>k</sup>:*

� � �

� �

� *β β v* ð Þ1 *tj* <sup>¼</sup> <sup>X</sup> *β* ∈*Js*,*n*f g*i*

*<sup>β</sup> utl* <sup>¼</sup> <sup>X</sup>

ð Þ **u***:<sup>l</sup>*

Construct the matrix **<sup>U</sup>** <sup>¼</sup> ð Þ *utl* <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>*, where *utl* is given by (42), and denote

ð Þ **e***:<sup>t</sup>*

*β*

ð Þ **e***<sup>k</sup>:*

*: <sup>j</sup>* given by (37). If we initially put

� *β*

ð Þ **e***:<sup>t</sup>*

� � � *α*

*<sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α*∈*Is*,*n*f g*j*

� � � *α*

*<sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α* ∈*Is*,*n*f g*j*

*: <sup>j</sup>* ¼ *v*

*<sup>β</sup>u*^*tk* <sup>¼</sup> <sup>X</sup>

�

*β* ∈*Js*,*n*f g*i*

*<sup>i</sup>:* <sup>¼</sup> *<sup>w</sup>*ð Þ<sup>1</sup>

� *β βu*^*tk*<sup>P</sup>

> P *α*∈*Is*

� *β β* P

<sup>P</sup> *<sup>α</sup>*

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � � � � *α α* P

*β* ∈*Js*,*n*f g*i*

ð Þ **e***:<sup>t</sup>*

**A***<sup>m</sup>*þ<sup>1</sup> � � *l:* ð Þ **a**^*t:*

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

� � �

� �

�

ð Þ **u***:<sup>l</sup>*

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

*<sup>β</sup>* <sup>¼</sup> **AA**<sup>∗</sup> j j*<sup>α</sup>*

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

ð Þ1 *nj* h i, then from

*: j*

� � � *β β* ,

ð Þ **u**^*:<sup>k</sup>*

� *β β*

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>v</sup>**ð Þ<sup>1</sup>

� � � �

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

*<sup>i</sup>*<sup>1</sup> , …, *<sup>w</sup>*ð Þ<sup>1</sup>

*<sup>j</sup>:* **<sup>w</sup>**ð Þ<sup>1</sup> *i:* � � � �

� �

*in* h i, then from

� � � *α α* ,

�

**A**<sup>2</sup> � �

�

� � �

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � � � � *α α*

> � � �

ð Þ1 <sup>1</sup> *<sup>j</sup>* , …, *v* � *β β*,

ð Þ **a**€*<sup>l</sup>:*

� � � *α*

*α :*

*<sup>α</sup>*, we have

ð Þ **e***<sup>k</sup>:*

ð Þ **u**^*<sup>t</sup>:*

� � � *α α* � � � *α α*

*:*

� �

� *α*

*<sup>α</sup>* (42)

�

.

**A***<sup>m</sup>*þ<sup>1</sup> � � *l:* ð Þ **e***<sup>k</sup>:*

*Determinantal Representations of the Core Inverse and Its Generalizations*

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

� �

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

*<sup>β</sup>* <sup>∈</sup>*Jr*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

�

� � �

P

*<sup>u</sup>*^*tk* <sup>X</sup> *α*∈*Is*,*n*f g*j*

is the *t*th component of a column vector **v**ð Þ<sup>1</sup>

�

X *β* ∈*Js*,*n*f g*i*

as the *k*th component of the row vector **w**ð Þ<sup>1</sup>

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

�

� �

**<sup>U</sup>**^ <sup>≔</sup> **UAA**<sup>∗</sup> . Then, taking into account that **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

�

� �

�

ð Þ *tk* th element of **<sup>A</sup>**^ <sup>¼</sup> **<sup>A</sup>**<sup>∗</sup> **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.89341*

*<sup>a</sup>*^*tk* <sup>X</sup> *α*∈*Is* <sup>1</sup>,*n*f g*j*

> X *β* ∈*Js*,*n*f g*i*

> > P

�

X *t*

Denote

we have

*a<sup>c</sup>*,† *ij* <sup>¼</sup> <sup>X</sup> *l*

*a<sup>c</sup>*,† *ij* ¼

If we put that

*v* ð Þ1 *tj* <sup>≔</sup><sup>X</sup> *k*

> X *t*

it follows (35) with **v**ð Þ<sup>1</sup>

*w*ð Þ<sup>1</sup> *ik* <sup>≔</sup><sup>X</sup> *t*

> X *k*

**13**

*w*ð Þ<sup>1</sup> *ik* <sup>X</sup> *α*∈*Is*,*n*f g*j*

X *β* ∈*Js*,*n*f g*i*

P *t* P *k* P

*utl* <sup>≔</sup><sup>X</sup> *k*

for all *l* ¼ 1, 2, where

$$\mathbf{v}\_{\cdot j}^{(1)} = \left[ \sum\_{a \in I\_{\iota, n} \{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\cdot}} (\hat{\mathbf{u}}\_{\iota}) \right|\_{a}^{a} \right] \in \mathbb{C}^{n \times 1}, t = 1, \ldots, n,\tag{37}$$

$$\mathbf{w}\_{i.}^{(1)} = \left[ \sum\_{\boldsymbol{\beta} \in I\_{\boldsymbol{\Lambda}} \{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{:i} (\hat{\mathbf{u}}\_k) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \right] \in \mathbb{C}^{1 \times n}, k = \mathbf{1}, \ldots, n,\tag{38}$$

$$\mathbf{v}\_{\cdot j}^{(2)} = \left[ \sum\_{a \in I\_{\wedge \pi} \{j\}} \left| (\mathbf{A}^\* \mathbf{A})\_{j\_{\cdot}} (\tilde{\mathbf{g}}\_{\cdot \cdot}) \right|\_{a}^{a} \right] \in \mathbb{C}^{n \times 1}, t = 1, \ldots, n,\tag{39}$$

$$\mathbf{w}\_{i.}^{(2)} = \left[ \sum\_{\boldsymbol{\beta} \in I\_{\boldsymbol{\lambda}^\*} \{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{.i} (\tilde{\mathbf{g}}\_k) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \right] \in \mathbb{C}^{1 \times n}, k = 1, \ldots, n. \tag{40}$$

Here, **<sup>u</sup>**^*<sup>t</sup>:* is the *<sup>t</sup>*th row and **<sup>u</sup>**^*:<sup>k</sup>* is the *<sup>k</sup>*th column of **<sup>U</sup>**^ <sup>≔</sup> **UAA**<sup>∗</sup> , **<sup>g</sup>**~*<sup>t</sup>:* is the *<sup>t</sup>*th row and **<sup>g</sup>**~*:<sup>k</sup>* is the *<sup>k</sup>*th column of **<sup>G</sup>**<sup>~</sup> <sup>≔</sup> **<sup>A</sup>**<sup>∗</sup> **AG**, and the matrices **<sup>U</sup>** <sup>¼</sup> *uij* � �<sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* and **<sup>G</sup>** <sup>¼</sup> *gij* � � <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* are such that

$$\mathfrak{u}\_{\vec{\eta}} = \sum\_{a \in I\_{\mathfrak{s}\_1 \mathfrak{s}} \{\vec{j}\}} \left| \left( \mathbf{A}^{m+1} \right)\_{\vec{j}.} (\mathring{\mathbf{a}}\_{\vec{\iota}}) \right|\_{a}^{a}, \quad \mathbf{g}\_{\vec{\eta}} = \sum\_{\beta \in I\_{\mathfrak{s}\_1 \mathfrak{s}} \{\vec{i}\}} \left| \left( \mathbf{A}^{m+1} \right)\_{\vec{\iota}} (\mathring{\mathbf{a}}\_{\cdot \mathbf{j}}) \right|\_{\beta}^{\beta}.$$

where **a**^*<sup>i</sup>:* is the *i*th row of **A**^ ≔ **A**<sup>∗</sup> **A***<sup>m</sup>*þ<sup>1</sup> and **a**~*: <sup>j</sup>* is the *j*th column of **A**~ ≔ **A***<sup>m</sup>*þ<sup>1</sup> **A**<sup>∗</sup> . *Proof.* Taking into account (34), we get

$$a\_{ij}^{c, \dagger} = \sum\_{l=1}^{n} \sum\_{k=1}^{n} q\_{il}^{A} a\_{lk}^{d} p\_{kj}^{A},\tag{41}$$

where **<sup>Q</sup>***<sup>A</sup>* <sup>¼</sup> *<sup>q</sup><sup>A</sup> il* � �, **<sup>A</sup>***<sup>d</sup>* <sup>¼</sup> *<sup>a</sup><sup>d</sup> il* � �, and **<sup>P</sup>***<sup>A</sup>* <sup>¼</sup> *<sup>p</sup><sup>A</sup> il* � �.

a. Taking into account the expressions (13), (10), and (11) for the determinantal representations of **A***<sup>d</sup>* , **Q***A*, and **P***A*, respectively, we have

$$a\_{\vec{\eta}}^{c,\dagger} = \sum\_{l} \sum\_{t} \frac{\sum\_{\beta \in I\_{l\nu}\{\vec{\iota}\}} \left| (\mathbf{A}^\* \mathbf{A})\_{:,l} (\dot{\mathbf{a}}\_t) \right|\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{l\nu}} |\mathbf{A}^\* \mathbf{A}|\_{\beta}^{\beta}} \frac{\sum\_{a \in I\_{l\_1\nu}\{l\}} \left| \left( \mathbf{A}^{m+1} \right)\_{l} \left( \mathbf{a}\_t^{(m)} \right) \right|\_{a}^{a}}{\sum\_{a \in I\_{l\_1\nu}} |\mathbf{A}^{m+1}|\_{a}^{a}} \frac{\left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\bar{j}}} \right| \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\bar{j}}} \right|\_{a}^{a}}{\sum\_{a \in I\_{l\_{\bar{\eta}}}} |\mathbf{A} \mathbf{A}^\*|\_{\bar{\eta}}^{a}},$$

where **a**\_*:<sup>t</sup>* is the *t*th column of **A**<sup>∗</sup> **A**, **a**€*<sup>l</sup>:* is the *l*th row of **AA**<sup>∗</sup> , and **a** ð Þ *m <sup>t</sup>:* is the *t*th row of **A***<sup>m</sup>*. So, it is clear that

$$a\_{\vec{\eta}}^{c,\uparrow} = \sum\_{l} \sum\_{t} \sum\_{k} \frac{\sum\_{\beta \in I\_{us}[\boldsymbol{\ell}]} \left| (\mathbf{A}^{\ast} \mathbf{A})\_{\boldsymbol{j}} (\mathbf{e}\_{l}) \right|\_{\beta}^{\beta} \cdot \left\|\_{\mathbf{f}} \boldsymbol{\tilde{a}}\_{\text{il}} \sum\_{a \in I\_{1s}[\boldsymbol{\ell}]} \left| (\mathbf{A}^{m+1})\_{\boldsymbol{l}} (\mathbf{e}\_{k}) \right|\_{a}^{a} \sum\_{a \in I\_{1s}[\boldsymbol{\ell}]} \left| (\mathbf{A} \mathbf{A}^{\ast})\_{\boldsymbol{j}} (\mathbf{\bar{a}}\_{\text{l}}) \right|\_{a}^{a}}{\sum\_{a \in I\_{1s}} \left| \mathbf{A} \mathbf{A}^{\ast} \right|\_{a}^{a}},$$

*Determinantal Representations of the Core Inverse and Its Generalizations DOI: http://dx.doi.org/10.5772/intechopen.89341*

where **e***:<sup>t</sup>* is the *t*th unit column vector, **e***<sup>k</sup>:* is the *k*th row vector, and *a*^*tk* is the ð Þ *tk* th element of **<sup>A</sup>**^ <sup>¼</sup> **<sup>A</sup>**<sup>∗</sup> **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> .

Denote

*ac*,† *ij* ¼

*: <sup>j</sup>* <sup>¼</sup> <sup>X</sup>

*<sup>i</sup>:* <sup>¼</sup> <sup>X</sup>

*: <sup>j</sup>* <sup>¼</sup> <sup>X</sup>

*<sup>i</sup>:* <sup>¼</sup> <sup>X</sup>

2 4

2 4

2 4

2 4

*α*∈*Is*,*n*f g*j*

*β* ∈*Js*,*n*f g*i*

*α*∈*Is*,*n*f g*j*

*β* ∈*Js*,*n*f g*i*

**A***<sup>m</sup>*þ<sup>1</sup> � � *j:* ð Þ **a**^*<sup>i</sup>:*

> *ac*,† *ij* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> l*¼1

*il*

� *β β*

ð Þ **e***:<sup>t</sup>*

*<sup>β</sup>* <sup>∈</sup> *Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

� *β <sup>β</sup> a*^*tk* P *α*∈ *Is*

> *β* P *α* ∈*Is*

� �

P *α*∈ *Is*

where **a**\_*:<sup>t</sup>* is the *t*th column of **A**<sup>∗</sup> **A**, **a**€*<sup>l</sup>:* is the *l*th row of **AA**<sup>∗</sup> , and **a**

� � �

*Proof.* Taking into account (34), we get

� �, **<sup>A</sup>***<sup>d</sup>* <sup>¼</sup> *<sup>a</sup><sup>d</sup>*

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>a</sup>**\_ ð Þ*:<sup>t</sup>* �

*<sup>β</sup>* <sup>∈</sup> *Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

P

�

� �

*β*

*il*

representations of **A***<sup>d</sup>*

P

row of **A***<sup>m</sup>*. So, it is clear that

P

P

for all *l* ¼ 1, 2, where

*Functional Calculus*

**v**ð Þ<sup>1</sup>

**w**ð Þ<sup>1</sup>

**v**ð Þ<sup>2</sup>

**w**ð Þ<sup>2</sup>

∈ *<sup>n</sup>*�*<sup>n</sup>* are such that

where **<sup>Q</sup>***<sup>A</sup>* <sup>¼</sup> *<sup>q</sup><sup>A</sup>*

*uij* <sup>¼</sup> <sup>X</sup> *α*∈*Is* <sup>1</sup>,*n*f g*j*

*gij* � �

*a<sup>c</sup>*,† *ij* <sup>¼</sup> <sup>X</sup> *l* X *t*

*ac*,† *ij* <sup>¼</sup> <sup>X</sup> *l* X *t* X *k*

**12**

P

P

� � �

�

� �

and **<sup>g</sup>**~*:<sup>k</sup>* is the *<sup>k</sup>*th column of **<sup>G</sup>**<sup>~</sup> <sup>≔</sup> **<sup>A</sup>**<sup>∗</sup> **AG**, and the matrices **<sup>U</sup>** <sup>¼</sup> *uij*

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*j:* **<sup>w</sup>**ð Þ*<sup>l</sup>*

*α*

ð Þ **u**^*t:*

ð Þ **u**^*:<sup>k</sup>*

� � � *α α*

� *β β*

� � � *α α*

> � *β β*

, *gij* <sup>¼</sup> <sup>X</sup> *β* ∈*Js* <sup>1</sup>,*n*f g*i*

> *il* � �.

a. Taking into account the expressions (13), (10), and (11) for the determinantal

, **Q***A*, and **P***A*, respectively, we have

<sup>1</sup>,*n*f g*<sup>l</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � �

�

*α*∈ *Is*

*<sup>l</sup>:* **a** ð Þ *m t:* � � � �

<sup>P</sup> *<sup>α</sup>*

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � � � � *α α*

<sup>1</sup>,*n*f g*<sup>l</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � �

�

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � � � � *α α* *l:* ð Þ **e***<sup>k</sup>:*

� *α α* P

� �

� � � *α*

P

*<sup>α</sup>*<sup>∈</sup> *Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*<sup>∈</sup> *Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

ð Þ *m*

*<sup>α</sup>*<sup>∈</sup> *Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

� � �

*α*∈ *Is*,*<sup>n</sup>*

� � �

Here, **<sup>u</sup>**^*<sup>t</sup>:* is the *<sup>t</sup>*th row and **<sup>u</sup>**^*:<sup>k</sup>* is the *<sup>k</sup>*th column of **<sup>U</sup>**^ <sup>≔</sup> **UAA**<sup>∗</sup> , **<sup>g</sup>**~*<sup>t</sup>:* is the *<sup>t</sup>*th row

3

3

3

3

<sup>5</sup><sup>∈</sup> *<sup>n</sup>*�<sup>1</sup>

<sup>5</sup><sup>∈</sup> *<sup>n</sup>*�<sup>1</sup>

�

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

� �<sup>2</sup>

**AA**<sup>∗</sup> ð Þ*j:*

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

� �

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *<sup>j</sup>:* **<sup>g</sup>**~*<sup>t</sup>:* � � �

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>g</sup>**~*:<sup>k</sup>* � � � � �

> � � � *α α*

where **a**^*<sup>i</sup>:* is the *i*th row of **A**^ ≔ **A**<sup>∗</sup> **A***<sup>m</sup>*þ<sup>1</sup> and **a**~*: <sup>j</sup>* is the *j*th column of **A**~ ≔ **A***<sup>m</sup>*þ<sup>1</sup>

X*n k*¼1 *qA il a<sup>d</sup> lkpA*

� �, and **<sup>P</sup>***<sup>A</sup>* <sup>¼</sup> *<sup>p</sup><sup>A</sup>*

� � � �

P

*i:*

*<sup>β</sup>* <sup>∈</sup>*Js*1,*<sup>n</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � � � � *β β*

� � � *α α*

(36)

, *t* ¼ 1, …, *n*, (37)

, *t* ¼ 1, …, *n*, (39)

� �<sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>* and **<sup>G</sup>** <sup>¼</sup>

**A**<sup>∗</sup> .

ð Þ **a**€*<sup>l</sup>:*

*α*

*<sup>t</sup>:* is the *t*th

<sup>P</sup> *<sup>α</sup>*

**AA**<sup>∗</sup> j j*<sup>α</sup> α* ð Þ €**a***<sup>l</sup>:*

� � � *α*

,

� � � *α*

,

<sup>5</sup><sup>∈</sup> <sup>1</sup>�*n*, *<sup>k</sup>* <sup>¼</sup> 1, …, *<sup>n</sup>*, (38)

<sup>5</sup><sup>∈</sup> <sup>1</sup>�*<sup>n</sup>*, *<sup>k</sup>* <sup>¼</sup> 1, …, *<sup>n</sup>:* (40)

**A***<sup>m</sup>*þ<sup>1</sup> � �

*:<sup>i</sup>* ~**a***: <sup>j</sup>* � � � � �

*kj*, (41)

� *β β* ,

$$u\_{tl} \coloneqq \sum\_{k} \hat{a}\_{tk} \sum\_{a \in I\_{t\_1 \pi} \{j\}} \left| \left( \mathbf{A}^{m+1} \right)\_{l.} (\mathbf{e}\_{k.}) \right|\_{a}^{a} = \sum\_{a \in I\_{t\_1 \pi} \{j\}} \left| \left( \mathbf{A}^{m+1} \right)\_{l.} (\hat{\mathbf{a}}\_{t.}) \right|\_{a}^{a} \tag{42}$$

as the *t*th component of a column vector **u***:<sup>l</sup>* ¼ *u*1*l*, …, *unl* ½ �. Then from

$$\sum\_{t} \sum\_{\beta \in I\_{\iota,\mathfrak{r}}\{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{\,\iota} (\mathbf{e}\_{\,\iota}) \right|\_{\beta}^{\beta} u\_{\mathrm{tl}} = \sum\_{\beta \in I\_{\iota,\mathfrak{r}}\{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{\,\iota} (\mathbf{u}\_{\,\iota}) \right|\_{\beta}^{\beta},$$

we have

$$a\_{\vec{\boldsymbol{y}}}^{\epsilon,\uparrow} = \sum\_{l} \frac{\sum\_{\beta \in I\_{\iota,\boldsymbol{u}}\{\boldsymbol{i}\}} \left| (\mathbf{A}^{\ast}\mathbf{A})\_{\boldsymbol{i}}(\mathbf{u}\_{\boldsymbol{l}}) \right|\_{\beta}^{\beta} \sum\_{a \in I\_{\iota,\boldsymbol{u}}\{\boldsymbol{j}\}} \left| (\mathbf{A}\mathbf{A}^{\ast})\_{\boldsymbol{j}}(\mathbf{\bar{a}}\_{\boldsymbol{l}}) \right|\_{a}^{a}}{\sum\_{\beta \in I\_{\iota,\boldsymbol{u}}} |\mathbf{A}^{\ast}\mathbf{A}|\_{\beta}^{\beta} \sum\_{a \in I\_{\iota,\boldsymbol{u}}} \left| \mathbf{A}^{m+1} \right|\_{a}^{a} \sum\_{a \in I\_{\iota,\boldsymbol{u}}} |\mathbf{A}\mathbf{A}^{\ast}|\_{a}^{a}}.$$

Construct the matrix **<sup>U</sup>** <sup>¼</sup> ð Þ *utl* <sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>*, where *utl* is given by (42), and denote **<sup>U</sup>**^ <sup>≔</sup> **UAA**<sup>∗</sup> . Then, taking into account that **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup> <sup>β</sup>* <sup>¼</sup> **AA**<sup>∗</sup> j j*<sup>α</sup> <sup>α</sup>*, we have

$$a\_{ij}^{c, \dagger} = \frac{\sum\_{\iota} \sum\_{k} \sum\_{\beta \in J\_{\iota\pi} \{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{\,\iota} (\mathbf{e}\_{\mathbf{1}})\_{\,\beta}^{\beta} \hat{u}\_{\imath k} \sum\_{a \in I\_{\iota\pi} \{j\}} \right| (\mathbf{A} \mathbf{A}^\*)\_{\,\iota} (\mathbf{e}\_{\mathbf{k}, \cdot}) \Big|\_{a}^{a}}{\left(\sum\_{\beta \in J\_{\iota\pi}} |\mathbf{A}^\* \mathbf{A}|\_{\beta}^{\beta}\right)^2 \sum\_{a \in I\_{\iota\_1 \imath}} \left| \mathbf{A}^{m+1} \right|\_{a}^{a}}.$$

If we put that

$$\left| \boldsymbol{\upsilon}\_{ij}^{(1)} := \sum\_{k} \hat{\boldsymbol{u}}\_{tk} \sum\_{a \in I\_{\iota\iota^{\sharp}} \{j\}} \left| (\mathbf{A} \mathbf{A}^{\*} \,)\_{j\cdot} (\mathbf{e}\_{k\cdot}) \right|\_{a}^{a} = \sum\_{a \in I\_{\iota\iota^{\sharp}} \{j\}} \left| (\mathbf{A} \mathbf{A}^{\*} \,)\_{j\cdot} (\hat{\mathbf{u}}\_{\iota\cdot}) \right|\_{a}^{a}$$

is the *t*th component of a column vector **v**ð Þ<sup>1</sup> *: <sup>j</sup>* ¼ *v* ð Þ1 <sup>1</sup> *<sup>j</sup>* , …, *v* ð Þ1 *nj* h i, then from

$$\sum\_{\mathfrak{t}} \sum\_{\boldsymbol{\beta} \in J\_{\boldsymbol{\tau}, \boldsymbol{\mathsf{e}}}\{\boldsymbol{i}\}} \left| (\mathbf{A}^\* \mathbf{A})\_{\boldsymbol{i}} (\mathbf{e}\_{\boldsymbol{\mathcal{I}}}) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \boldsymbol{v}\_{\boldsymbol{\mathcal{I}}}^{(1)} = \sum\_{\boldsymbol{\beta} \in J\_{\boldsymbol{\tau}, \boldsymbol{\mathsf{e}}}\{\boldsymbol{i}\}} \left| (\mathbf{A}^\* \mathbf{A})\_{\boldsymbol{i}} \left(\mathbf{v}\_{\boldsymbol{\cdot} \boldsymbol{j}}^{(1)}\right) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}},$$

it follows (35) with **v**ð Þ<sup>1</sup> *: <sup>j</sup>* given by (37). If we initially put

$$\mathcal{w}\_{ik}^{(1)} := \sum\_{\mathfrak{l}} \sum\_{\beta \in I\_{\iota,\mathfrak{r}}\{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{\,i} (\mathbf{e}\_{\iota}) \right|\_{\beta}^{\beta} \hat{\mu}\_{tk} = \sum\_{\beta \in I\_{\iota,\mathfrak{r}}\{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{\,i} (\hat{\mathbf{u}}\_k) \right|\_{\beta}^{\beta}$$

as the *k*th component of the row vector **w**ð Þ<sup>1</sup> *<sup>i</sup>:* <sup>¼</sup> *<sup>w</sup>*ð Þ<sup>1</sup> *<sup>i</sup>*<sup>1</sup> , …, *<sup>w</sup>*ð Þ<sup>1</sup> *in* h i, then from

$$\sum\_{k} w\_{ik}^{(1)} \sum\_{a \in I\_{\iota, \mathfrak{r}}\{j\}} \left| \left(\mathbf{A}^{2}\right)\_{j.} (\mathbf{e}\_{k.}) \right|\_{a}^{a} = \sum\_{a \in I\_{\iota, \mathfrak{r}}\{j\}} \left| \left(\mathbf{A}^{2}\right)\_{j.} \left(\mathbf{w}\_{i.}^{(1)}\right) \right|\_{a}^{a},$$


as the *t*th component of a column vector **v**ð Þ<sup>2</sup>

�

X *β* ∈*Js*,*n*f g*i*

as the *k*th component of a row vector **w**ð Þ<sup>2</sup>

� � � �

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

� �

*Determinantal Representations of the Core Inverse and Its Generalizations*

ð Þ **e***:<sup>t</sup>*

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

**A** ¼

2 6 4

2 6 4

� �

� *β β v* ð Þ2 *tj* <sup>¼</sup> <sup>X</sup> *β* ∈*Js*,*n*f g*i*

*: <sup>j</sup>* given by (39).

ð Þ **e***:<sup>t</sup>*

ð Þ **e***<sup>k</sup>:*

� � � *α*

� *β*

X *t*

Thus, we have (35) with **v**ð Þ<sup>2</sup>

If, now, we denote

*w*ð Þ<sup>2</sup> *ik* <sup>≔</sup><sup>X</sup> *t*

X *k*

**4. An example**

Since

2 6 4

**AA**<sup>∗</sup> <sup>¼</sup>

Since

**15**

then by (25),

Given the matrix

*w*ð Þ<sup>2</sup> *ik* <sup>X</sup> *α*∈*Is*,*n*f g*j*

So, finally, we have (36) with **w**ð Þ<sup>2</sup>

4 2**i** 2**i** �2**i** 3 �1 �2**i** �1 3

3 7 <sup>5</sup>, **<sup>A</sup>**<sup>2</sup> <sup>¼</sup>

find **<sup>A</sup>**○† and **<sup>A</sup>**○† by (25) and (26), respectively.

*a*○† ,*<sup>r</sup>* <sup>11</sup> ¼

**<sup>A</sup>**^ <sup>¼</sup> **<sup>A</sup>**<sup>2</sup> **<sup>A</sup>**<sup>3</sup> � � <sup>∗</sup> <sup>¼</sup> <sup>16</sup>

P

X *β* ∈*Js*,*n*f g*i*

*DOI: http://dx.doi.org/10.5772/intechopen.89341*

*: <sup>j</sup>* ¼ *v*

*<sup>β</sup> <sup>g</sup>*~*tk* <sup>¼</sup> <sup>X</sup>

*<sup>i</sup>:* <sup>¼</sup> *<sup>w</sup>*ð Þ<sup>2</sup>

*<sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α*∈*Is*,*n*f g*j*

*<sup>i</sup>:* given by (40).

200 �*ii i* �*i* �*i* �*i*

4 00 2 � 2*i* 0 0 �2 � 2*i* 0 0

then rk**<sup>A</sup>** <sup>¼</sup> 2 and rk**A**<sup>2</sup> <sup>¼</sup> rk**A**<sup>3</sup> <sup>¼</sup> 1, and *<sup>k</sup>* <sup>¼</sup> Ind**<sup>A</sup>** <sup>¼</sup> 2 and *<sup>r</sup>*<sup>1</sup> <sup>¼</sup> 1. So, we shall

2 6 4

*<sup>α</sup>*∈*I*1,3f g<sup>1</sup> **<sup>A</sup>**<sup>3</sup> **<sup>A</sup>**<sup>3</sup> � � <sup>∗</sup> � �

*<sup>α</sup>*∈*I*1,3 **<sup>A</sup>**<sup>3</sup> **<sup>A</sup>**<sup>3</sup> � � <sup>∗</sup> � � �

� � �

P

ð Þ2 <sup>1</sup> *<sup>j</sup>* , …, *v*

�

*β* ∈*Js*,*n*f g*i*

�

3 7 5*:*

> 3 7 <sup>5</sup>, **<sup>A</sup>**<sup>3</sup> <sup>¼</sup>

2 1 þ **i** �1 þ **i** 1 � **i** 1 **i** �1 � **i i** 1

> 1*:* ð Þ **a**^1*:*

� � � *α α* � � � *α α*

¼ 1 4 *:*

2 6 4

> 3 7 5,

*<sup>i</sup>*<sup>1</sup> , …, *<sup>w</sup>*ð Þ<sup>2</sup> *in* h i, then

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>w</sup>**ð Þ<sup>2</sup>

� � � �

*i:*

� � � *α α :*

8 00 4 � 4*i* 0 0 �4 � 4*i* 0 0 3 7 5,

ð Þ2 *nj* h i, then

*: j*

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>g</sup>**~*:<sup>k</sup>* � � � � �

� � � *β β :*

> � *β β*

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>v</sup>**ð Þ<sup>2</sup>

� � � �

$$a\_{ij}^{\star,\dagger} = \sum\_{k} \sum\_{t} \frac{\sum\_{\boldsymbol{\beta} \in I\_{\iota\boldsymbol{\mu}}(\boldsymbol{i})} \left| (\mathbf{A}^{\star}\mathbf{A})\_{\boldsymbol{j}} (\dot{\mathbf{a}}\_{t}) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \frac{\sum\_{\boldsymbol{\beta} \in I\_{\iota\boldsymbol{\mu}}(\boldsymbol{t})} \left| \left(\mathbf{A}^{\boldsymbol{m}+1}\right)\_{\iota} \left(\mathbf{a}\_{k}^{(m)}\right) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \frac{\sum\_{a \in I\_{\iota\boldsymbol{\mu}}(\boldsymbol{j})} \left| (\mathbf{A}\mathbf{A}^{\star})\_{\boldsymbol{j}} (\ddot{\mathbf{a}}\_{k}) \right|\_{a}^{a}}{\sum\_{\boldsymbol{\beta} \in I\_{\iota\boldsymbol{\mu}}} \left| \mathbf{A}\mathbf{A}^{\star} \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}}}.$$

Therefore,

$$\begin{split} a\_{ij}^{\epsilon,\dagger} &= \sum\_{l} \sum\_{k} \sum\_{\iota} \frac{\sum\_{\beta \in I\_{\iota\iota}\{\iota\}} \left( (\mathbf{A}^{\ast} \mathbf{A})\_{\iota} (\dot{\mathbf{a}}\_{\iota}) \right)\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{\iota\iota}} |\mathbf{A}^{\ast} \mathbf{A}|\_{\beta}^{\beta}} \times \\\\ &\frac{\sum\_{\beta \in I\_{\iota\iota}\{\iota\}} \left| \left( \mathbf{A}^{m+1} \right)\_{\iota} (\mathbf{e}\_{k}) \right|\_{\beta}^{\beta}}{\sum\_{\beta \in I\_{\iota\iota}} |\mathbf{A}^{m+1}|\_{\beta}^{\beta}} \, \ddot{a}\_{kl} \, \frac{\sum\_{a \in I\_{\iota\iota}\{\iota\}} \left| (\mathbf{A} \mathbf{A}^{\ast})\_{\iota} (\mathbf{e}\_{l}) \right|\_{a}^{a}}{\sum\_{a \in I\_{\iota\iota}} |\mathbf{A} \mathbf{A}^{\ast}|\_{a}^{a}} .\end{split}$$

where **e***:<sup>k</sup>* is the *k*th unit column vector, **e***<sup>l</sup>:* is the *l*th unit row vector, and *a*~*kl* is the ð Þ *kl* th element of **<sup>A</sup>**<sup>~</sup> <sup>¼</sup> **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> **A**<sup>∗</sup> .

If we denote

$$\mathbf{g}\_{\mathcal{U}} \coloneqq \sum\_{l} \sum\_{\boldsymbol{\beta} \in J\_{l\_{1}^{n}}\{\boldsymbol{t}\}} \left| \left( \mathbf{A}^{m+1} \right)\_{\boldsymbol{t}} (\mathbf{e}\_{k}) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \tilde{a}\_{\mathcal{U}} = \sum\_{\boldsymbol{\beta} \in J\_{l\_{1}^{n}}\{\boldsymbol{t}\}} \left| \left( \mathbf{A}^{m+1} \right)\_{\boldsymbol{t}} (\tilde{\mathbf{a}}\_{\mathcal{I}}) \right|\_{\boldsymbol{\beta}}^{\boldsymbol{\beta}} \tag{43}$$

as the *<sup>l</sup>*th component of a row vector **<sup>g</sup>***<sup>t</sup>:* <sup>¼</sup> *gt*1, …, *gtn* � �, then

$$\sum\_{l} \mathbf{g}\_{tl} \sum\_{a \in I\_{\rm ul} \{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\cdot}} (\mathbf{e}\_{l\cdot}) \right|\_{a}^{a} = \sum\_{a \in I\_{\rm ul} \{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\cdot}} (\mathbf{g}\_{t\cdot}) \right|\_{a}^{a}.$$

From this, it follows that

$$a\_{ij}^{\epsilon,\dagger} = \sum\_{t} \frac{\sum\_{\beta \in I\_{\iota,\boldsymbol{\pi}}\{i\}} \left| (\mathbf{A}^{\*}\mathbf{A})\_{,i} (\dot{\mathbf{a}}\_{\boldsymbol{\pi}}) \right|\_{\beta}^{\beta} \sum\_{a \in I\_{\iota,\boldsymbol{\pi}}\{j\}} \left| (\mathbf{A}\mathbf{A}^{\*})\_{j} \left(\mathbf{g}\_{t}\right) \right|\_{a}^{a}}{\sum\_{\beta \in I\_{\iota,\boldsymbol{\pi}}} |\mathbf{A}^{\*}\mathbf{A}|\_{\beta}^{\beta} \sum\_{a \in I\_{\iota,\boldsymbol{\pi}}} \left| \mathbf{A}^{m+1} \right|\_{a}^{a} \sum\_{a \in I\_{\iota,\boldsymbol{\pi}}} |\mathbf{A}\mathbf{A}^{\*}|\_{\boldsymbol{\pi}}}.$$

Construct the matrix **<sup>G</sup>** <sup>¼</sup> *gtl* � �<sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>*, where *gtl* is given by (43). Denote **G**~ ≔ **A**<sup>∗</sup> **AG**. Then,

$$a\_{ij}^{c, \dagger} = \frac{\sum\_{\iota} \sum\_{k} \sum\_{\beta \in I\_{\iota \iota} \{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{\iota} (\mathbf{e}\_{\iota})\_{\beta}^{\beta} \tilde{g}\_{ik} \sum\_{a \in I\_{\iota \iota} \{j\}} \right| (\mathbf{A} \mathbf{A}^\*)\_{j} (\mathbf{e}\_{k \cdot}) \Big|\_{a}^{a}}{\sum\_{\beta \in I\_{\iota \iota}} \left( |\mathbf{A}^\* \mathbf{A}|\_{\beta}^{\beta} \right)^2 \sum\_{a \in I\_{\iota \iota} \cup} |\mathbf{A}^{m+1}|\_{a}^{a}}.$$

If we denote

$$\boldsymbol{\nu}\_{tj}^{(2)} \coloneqq \sum\_{k} \tilde{\mathbf{g}}\_{tk} \sum\_{a \in I\_{\boldsymbol{\nu}\boldsymbol{x}}\{j\}} \left| (\mathbf{A} \mathbf{A}^{\*})\_{j\cdot} (\mathbf{e}\_{k\cdot}) \right|\_{a}^{a} = \sum\_{a \in I\_{\boldsymbol{\nu}\boldsymbol{x}}\{j\}} \left| (\mathbf{A} \mathbf{A}^{\*})\_{j\cdot} (\tilde{\mathbf{g}}\_{t\cdot}) \right|\_{a}^{a}$$

*Determinantal Representations of the Core Inverse and Its Generalizations DOI: http://dx.doi.org/10.5772/intechopen.89341*

as the *t*th component of a column vector **v**ð Þ<sup>2</sup> *: <sup>j</sup>* ¼ *v* ð Þ2 <sup>1</sup> *<sup>j</sup>* , …, *v* ð Þ2 *nj* h i, then

$$\sum\_{t} \sum\_{\beta \in J\_{\iota,n}\{i\}} |(\mathbf{A}^\* \mathbf{A})\_{\cdot i} (\mathbf{e}\_t)|\_{\beta}^{\beta} v\_{\eta}^{(2)} = \sum\_{\beta \in J\_{\iota,n}\{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{\cdot i} (\mathbf{v}\_{\cdot j}^{(2)}) \right|\_{\beta}^{\beta}.$$

Thus, we have (35) with **v**ð Þ<sup>2</sup> *: <sup>j</sup>* given by (39). If, now, we denote

$$w\_{ik}^{(2)} := \sum\_{t} \sum\_{\beta \in J\_{\iota,\mathbf{x}}\{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{,i} (\mathbf{e}\_{\mathbf{z}}) \right|\_{\beta}^{\beta} \tilde{g}\_{tk} = \sum\_{\beta \in J\_{\iota,\mathbf{x}}\{i\}} \left| (\mathbf{A}^\* \mathbf{A})\_{,i} (\tilde{g}\_{\mathbf{z}}) \right|\_{\beta}^{\beta}$$

as the *k*th component of a row vector **w**ð Þ<sup>2</sup> *<sup>i</sup>:* <sup>¼</sup> *<sup>w</sup>*ð Þ<sup>2</sup> *<sup>i</sup>*<sup>1</sup> , …, *<sup>w</sup>*ð Þ<sup>2</sup> *in* h i, then

$$\sum\_{k} w\_{ik}^{(2)} \sum\_{a \in I\_{\iota\mu} \{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\iota}} (\mathbf{e}\_{k.}) \right|\_{a}^{a} = \sum\_{a \in I\_{\iota\mu} \{j\}} \left| (\mathbf{A} \mathbf{A}^\*)\_{j\_{\iota}} \left(\mathbf{w}\_{i.}^{(2)}\right) \right|\_{a}^{a}.$$

So, finally, we have (36) with **w**ð Þ<sup>2</sup> *<sup>i</sup>:* given by (40).

### **4. An example**

it follows (36) with **w**ð Þ<sup>1</sup>

P

Therefore,

*Functional Calculus*

P

*ac*,† *ij* <sup>¼</sup> <sup>X</sup> *l*

P *β* ∈*Js*

the ð Þ *kl* th element of **<sup>A</sup>**<sup>~</sup> <sup>¼</sup> **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup>

X *l gtl*

From this, it follows that

Construct the matrix **G** ¼ *gtl*

P *t* P *k* P

*a<sup>c</sup>*,† *ij* <sup>¼</sup> <sup>X</sup> *t*

**G**~ ≔ **A**<sup>∗</sup> **AG**. Then,

*ac*,† *ij* ¼

If we denote

**14**

*v* ð Þ2 *tj* <sup>≔</sup><sup>X</sup> *k g*~*tk*

*gtl* <sup>≔</sup><sup>X</sup> *l*

If we denote

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>a</sup>**\_ ð Þ*:<sup>t</sup>* �

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

X *k*

<sup>1</sup>,*n*f g*<sup>t</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � �

�

P *β* ∈*Js*

X *β* ∈*Js* <sup>1</sup>,*n*f g*t*

�

X *α*∈*Is*,*n*f g*j*

P

P

� �

*β*

X *t*

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � � � � *β β*

**A***<sup>m</sup>*þ<sup>1</sup> � � *:t* ð Þ **e***:<sup>k</sup>*

as the *l*th component of a row vector **g***<sup>t</sup>:* ¼ *gt*1, …, *gtn*

� � �

� �

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>a</sup>**\_ ð Þ*:<sup>t</sup>* �

*<sup>β</sup>* <sup>∈</sup>*Jr*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>*

�

� � �

P

X *α* ∈*Is*,*n*f g*j*

P

*:t* ð Þ **e***:<sup>k</sup>*

**A**<sup>∗</sup> .

� �

P

� *β β*

> � *β β*

ð Þ **e***<sup>l</sup>:*

� �

*β* P *α*∈*Is*

� �

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

� � � *α*

*a*~*kl*

where **e***:<sup>k</sup>* is the *k*th unit column vector, **e***<sup>l</sup>:* is the *l*th unit row vector, and *a*~*kl* is

*ac*,† *ij* <sup>¼</sup> <sup>X</sup> *k* X *t*

*<sup>i</sup>:* given by (38).

� *β β* P *β* ∈*Js*

b. By using the determinantal representation (12) for **A***<sup>d</sup>* in (41), we have

<sup>1</sup>,*n*f g*<sup>t</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � �

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � � � � *β β*

�

*<sup>β</sup>* <sup>∈</sup>*Js*,*n*f g*<sup>i</sup>* **<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:<sup>i</sup>* **<sup>a</sup>**\_ ð Þ*:<sup>t</sup>* � �*<sup>β</sup>*

*<sup>β</sup>* <sup>∈</sup>*Js*,*<sup>n</sup>* **<sup>A</sup>**<sup>∗</sup> j j **<sup>A</sup>** *<sup>β</sup>*

P

*<sup>a</sup>*~*kl* <sup>¼</sup> <sup>X</sup> *β* ∈*Js* <sup>1</sup>,*n*f g*t*

> *<sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α* ∈*Is*,*n*f g*j*

> > � *β β* P

ð Þ **e***:<sup>t</sup>*

*β* � �<sup>2</sup>

ð Þ **e***<sup>k</sup>:*

� � � *α*

� *β <sup>β</sup>g*~*tk* P

P

*<sup>α</sup>* <sup>¼</sup> <sup>X</sup> *α*∈*Is*,*n*f g*j*

<sup>1</sup>,*<sup>n</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � � � � *α α* P

*β*

P

*:<sup>t</sup>* **a** ð Þ *m :k* � � � �

*β*

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

� � � �

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

> **A***<sup>m</sup>*þ<sup>1</sup> � � *:t* ð Þ **a**~*:<sup>l</sup>*

� �

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>g</sup>***<sup>t</sup>:* � � �

*<sup>α</sup>*∈*Is*,*n*f g*<sup>j</sup>* <sup>ð</sup>**AA**<sup>∗</sup> j j*<sup>j</sup>:* **<sup>g</sup>***<sup>t</sup>:*

*<sup>α</sup>* <sup>∈</sup>*Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

� � �

�

� �, then

� �

� �<sup>∈</sup> *<sup>n</sup>*�*<sup>n</sup>*, where *gtl* is given by (43). Denote

*<sup>α</sup>*∈*Is*1,*<sup>n</sup>* **<sup>A</sup>***<sup>m</sup>*þ<sup>1</sup> � � � � *α α*

> � �

� � � *β β*

P

ð Þ **e***<sup>l</sup>:*

*α*

� � � *α*

� *β*

� � � *α α :*

� �Þ*<sup>α</sup> α*

*α :*

ð Þ **e***<sup>k</sup>:*

� � � *α α* � � � *α α*

*:*

*<sup>α</sup>*∈*Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

**AA**<sup>∗</sup> ð Þ*<sup>j</sup>:* **<sup>g</sup>**~*<sup>t</sup>:* � � �

*<sup>β</sup>* (43)

*:*

*<sup>α</sup>*<sup>∈</sup> *Is*,*n*f g*<sup>j</sup>* **AA**<sup>∗</sup> ð Þ*<sup>j</sup>:*

<sup>P</sup> *<sup>α</sup> <sup>α</sup>*<sup>∈</sup> *Is*,*<sup>n</sup>* **AA**<sup>∗</sup> j j*<sup>α</sup>*

� � �

ð Þ **a**€*<sup>k</sup>:*

*α*

� � � *α*

*:*

Given the matrix

$$\mathbf{A} = \begin{bmatrix} 2 & \mathbf{0} & \mathbf{0} \\ -i & i & i \\ -i & -i & -i \end{bmatrix}.$$

Since

$$\mathbf{A}\mathbf{A}^\* = \begin{bmatrix} 4 & 2\mathbf{i} & 2\mathbf{i} \\ -2\mathbf{i} & 3 & -1 \\ -2\mathbf{i} & -1 & 3 \end{bmatrix}, \quad \mathbf{A}^2 = \begin{bmatrix} 4 & 0 & 0 \\ 2 - 2\mathbf{i} & 0 & 0 \\ -2 - 2\mathbf{i} & 0 & 0 \end{bmatrix}, \quad \mathbf{A}^3 = \begin{bmatrix} 8 & 0 & 0 \\ 4 - 4\mathbf{i} & 0 & 0 \\ -4 - 4\mathbf{i} & 0 & 0 \end{bmatrix},$$

then rk**<sup>A</sup>** <sup>¼</sup> 2 and rk**A**<sup>2</sup> <sup>¼</sup> rk**A**<sup>3</sup> <sup>¼</sup> 1, and *<sup>k</sup>* <sup>¼</sup> Ind**<sup>A</sup>** <sup>¼</sup> 2 and *<sup>r</sup>*<sup>1</sup> <sup>¼</sup> 1. So, we shall find **<sup>A</sup>**○† and **<sup>A</sup>**○† by (25) and (26), respectively.

Since

$$
\hat{\mathbf{A}} = \mathbf{A}^2 \left( \mathbf{A}^3 \right)^\* = 16 \begin{bmatrix} 2 & 1+\mathbf{i} & -\mathbf{1}+\mathbf{i} \\ 1-\mathbf{i} & \mathbf{1} & \mathbf{i} \\ -\mathbf{1}-\mathbf{i} & \mathbf{i} & \mathbf{1} \end{bmatrix},
$$

then by (25),

$$\mathbf{a}\_{11}^{\oplus,r} = \frac{\sum\_{a \in I\_{13}\{1\}} \left| \left(\mathbf{A}^3(\mathbf{A}^3)^\*\right)\_{1.}(\hat{\mathbf{a}}\_{1.}) \right|\_{a}^{a}}{\sum\_{a \in I\_{13}} \left| \mathbf{A}^3(\mathbf{A}^3)^\* \right|\_{a}^{a}} = \frac{1}{4} \text{.} $$

By similarly continuing, we get

$$\mathbf{A}^{\otimes} = \frac{1}{8} \begin{bmatrix} 2 & \mathbf{1} + \mathbf{i} & -\mathbf{1} + \mathbf{i} \\\\ \mathbf{1} - \mathbf{i} & \mathbf{1} & \mathbf{i} \\\\ -\mathbf{1} - \mathbf{i} & \mathbf{i} & \mathbf{1} \end{bmatrix}.$$

**<sup>G</sup>**<sup>~</sup> <sup>¼</sup> **<sup>A</sup>**<sup>∗</sup> **AA**<sup>~</sup> <sup>¼</sup> <sup>16</sup>

*Determinantal Representations of the Core Inverse and Its Generalizations*

� *β <sup>β</sup>* ¼ *det*

*<sup>α</sup>*∈*I*2,3f g<sup>1</sup> **AA**<sup>∗</sup> ð Þ1*:* **<sup>w</sup>**ð Þ<sup>2</sup>

*α*

�2**i** 3 � �

�

*<sup>α</sup>* <sup>∈</sup>*I*2,3 **AA**<sup>∗</sup> j j*<sup>α</sup>*

<sup>4608</sup> *det* 384 192**<sup>i</sup>**

**<sup>A</sup>***<sup>c</sup>*,† <sup>¼</sup> <sup>1</sup> 12

inverses, the DMP, MPD, and CMP inverses are derived.

� �<sup>2</sup>

� � � �

P

2 6 4

**<sup>A</sup>**<sup>∗</sup> ð Þ **<sup>A</sup>** *:*<sup>1</sup> **<sup>g</sup>**~*:*<sup>1</sup> � � � � �

Furthermore, by (40),

<sup>11</sup> <sup>¼</sup> <sup>X</sup>

So, by (36), we get

*ac*,† <sup>11</sup> ¼

*β* ∈*J*2,3f g1

By similar calculations, we get

*DOI: http://dx.doi.org/10.5772/intechopen.89341*

<sup>1</sup>*:* <sup>¼</sup> ½ � 384, 96**i**, 96**<sup>i</sup>** , **<sup>w</sup>**ð Þ<sup>2</sup>

P

P

<sup>¼</sup> <sup>1</sup>

By similarly continuing, we derive

*w*ð Þ<sup>2</sup>

**w**ð Þ<sup>2</sup>

**5. Conclusions**

**17**

6 3**i** 3**i** �2**i** 1 1 �2**i** 1 1

6 0 �2**i** 2 � �

<sup>2</sup>*:* ¼ �½ � <sup>192</sup>**i**, 96, 96 , **<sup>w</sup>**ð Þ<sup>2</sup>

1*:*

*<sup>β</sup>* <sup>∈</sup>*J*1,3 **<sup>A</sup>**<sup>3</sup> � � � � *β β*

� � � �

4 2**i** 2**i** �2**i** 1 1 �2**i** 1 1

In this chapter, we get the direct method to find the core inverse and its generalizations that are based on their determinantal representations. New determinantal

representations of the right and left core inverses, the right and left core-EP

� � � *α α*

<sup>þ</sup> *det* 384 192**<sup>i</sup>** �2**i** 3

> 3 7 5*:*

3 7 5*:*

þ *det*

� � � �

6 0 �2**i** 2

<sup>3</sup>*:* ¼ �½ � 192**i**, 96, 06 *:*

¼ 1 3 *:* ¼ 24*:*

2 6 4

By analogy, due to (26), we have

$$\mathbf{A}\_{\oplus} = \frac{1}{2} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}.$$

The DMP inverse **A***<sup>d</sup>*,† can be found by Theorem 3.15. Since

$$
\tilde{\mathbf{A}} = \mathbf{A}^3 \mathbf{A}^\* = 4 \begin{bmatrix} 4 & 2\mathbf{i} & 2\mathbf{i} \\ 2 - 2\mathbf{i} & \mathbf{1} + \mathbf{i} & \mathbf{1} + \mathbf{i} \\ -2 - 2\mathbf{i} & \mathbf{1} - \mathbf{i} & \mathbf{1} - \mathbf{i} \end{bmatrix}.
$$

and rk **<sup>A</sup>**<sup>3</sup> � � <sup>¼</sup> 1, then

$$\mathbf{u}\_1^{(1)} = \mathbf{\tilde{a}}\_1, \ \mathbf{u}\_2^{(1)} = \mathbf{\tilde{a}}\_2, \ \mathbf{u}\_3^{(1)} = \mathbf{\tilde{a}}\_3.$$

Furthermore, by (29),

$$a\_{11}^{d,\dagger} = \frac{\sum\_{a \in I\_{23}\{1\}} \left| (\mathbf{A}\mathbf{A}^\*)\_{1} \left(\mathbf{u}\_{1}^{(1)}\right) \right|\_{a}^{a}}{\sum\_{\beta \in I\_{13}} \left| \mathbf{A}^3 \right|\_{\beta}^{\beta} \sum\_{a \in I\_{23}} \left| \mathbf{A}\mathbf{A}^\* \right|\_{a}^{a}} = \frac{1}{192} \left( \det \begin{bmatrix} \mathbf{16} & \mathbf{8i} \\ -2\mathbf{i} & \mathbf{3} \end{bmatrix} + \det \begin{bmatrix} \mathbf{16} & \mathbf{8i} \\ -2\mathbf{i} & \mathbf{3} \end{bmatrix} \right) = \frac{1}{3}.$$

By similarly continuing, we get

$$\mathbf{A}^{d,\dagger} = \frac{1}{12} \begin{bmatrix} 4 & 2\mathbf{i} & 2\mathbf{i} \\ 2 - 2\mathbf{i} & \mathbf{1} + \mathbf{i} & \mathbf{1} + \mathbf{i} \\ -2 - 2\mathbf{i} & \mathbf{1} - \mathbf{i} & \mathbf{1} - \mathbf{i} \end{bmatrix}.$$

Similarly by Theorem 3.17, we get

$$\mathbf{A}^{\dagger,d} = \frac{1}{4} \begin{bmatrix} 2 & 0 & 0 \\ -\mathbf{i} & 0 & 0 \\ -\mathbf{i} & 0 & 0 \end{bmatrix}.$$

Finally, by theorem, we find the CMP inverse **<sup>A</sup>***<sup>c</sup>*,† <sup>¼</sup> *<sup>a</sup><sup>c</sup>*,† *ij* � �. Since rk**A**<sup>3</sup> <sup>¼</sup> 1, then **<sup>G</sup>** <sup>¼</sup> **<sup>A</sup>**<sup>~</sup> and

*Determinantal Representations of the Core Inverse and Its Generalizations DOI: http://dx.doi.org/10.5772/intechopen.89341*

$$
\ddot{\mathbf{G}} = \mathbf{A}^\* \mathbf{A} \ddot{\mathbf{A}} = \mathbf{16} \begin{bmatrix} 6 & 3\mathbf{i} & 3\mathbf{i} \\ -2\mathbf{i} & \mathbf{1} & \mathbf{1} \\ -2\mathbf{i} & \mathbf{1} & \mathbf{1} \end{bmatrix}.
$$

Furthermore, by (40),

By similarly continuing, we get

*Functional Calculus*

By analogy, due to (26), we have

**<sup>A</sup>**○† <sup>¼</sup> <sup>1</sup> 8

**<sup>A</sup>**○† <sup>¼</sup> <sup>1</sup> 2

The DMP inverse **A***<sup>d</sup>*,† can be found by Theorem 3.15. Since

**<sup>A</sup>**<sup>∗</sup> <sup>¼</sup> <sup>4</sup>

<sup>1</sup> <sup>¼</sup> **<sup>a</sup>**~1*:*, **<sup>u</sup>**ð Þ<sup>1</sup>

� � � *α α*

*α*

**<sup>A</sup>**†,*<sup>d</sup>* <sup>¼</sup> <sup>1</sup> 4

Finally, by theorem, we find the CMP inverse **<sup>A</sup>***<sup>c</sup>*,† <sup>¼</sup> *<sup>a</sup><sup>c</sup>*,†

2 6 4

<sup>¼</sup> <sup>1</sup> <sup>192</sup> *det*

1*:*

*<sup>α</sup>*∈*I*2,3 **AA**<sup>∗</sup> j j*<sup>α</sup>*

**<sup>A</sup>***<sup>d</sup>*,† <sup>¼</sup> <sup>1</sup> 12

**<sup>A</sup>**<sup>~</sup> <sup>¼</sup> **<sup>A</sup>**<sup>3</sup>

**u**ð Þ<sup>1</sup>

� � � �

*<sup>α</sup>*∈*I*2,3f g<sup>1</sup> **AA**<sup>∗</sup> ð Þ1*:* **<sup>u</sup>**ð Þ<sup>1</sup>

�

By similarly continuing, we get

Similarly by Theorem 3.17, we get

and rk **<sup>A</sup>**<sup>3</sup> � � <sup>¼</sup> 1, then

Furthermore, by (29),

*<sup>β</sup>* <sup>∈</sup>*J*1,3 **<sup>A</sup>**<sup>3</sup> � � � � *β β* P

P

P

then **<sup>G</sup>** <sup>¼</sup> **<sup>A</sup>**<sup>~</sup> and

**16**

*a<sup>d</sup>*,† <sup>11</sup> ¼ 2 1 þ **i** �1 þ **i**

1 � **i** 1 **i** �1 � **i i** 1

> 100 000 000

4 2**i** 2**i**

þ *det*

� � � �

*ij* � �

. Since rk**A**<sup>3</sup> <sup>¼</sup> 1,

16 8**i** �2**i** 3

¼ 1 3 *:*

2 � 2**i** 1 þ **i** 1 þ **i** �2 � 2**i** 1 � **i** 1 � **i**

<sup>3</sup> ¼ **a**~3*::*

16 8**i** �2**i** 3 � �

<sup>2</sup> <sup>¼</sup> **<sup>a</sup>**~2*:*, **<sup>u</sup>**ð Þ<sup>1</sup>

4 2**i** 2**i**

2 � 2**i** 1 þ **i** 1 þ **i** �2 � 2**i** 1 � **i** 1 � **i**

3 7 5*:*

$$w\_{11}^{(2)} = \sum\_{\beta \in I\_{2,3}\{1\}} \left| (\mathbf{A}^\* \mathbf{A})\_{:1} (\tilde{\mathbf{g}}\_{:1}) \right|\_{\beta}^{\beta} = \left( \det \begin{bmatrix} \mathbf{6} & \mathbf{0} \\ -2\mathbf{i} & 2 \end{bmatrix} + \det \begin{bmatrix} \mathbf{6} & \mathbf{0} \\ -2\mathbf{i} & 2 \end{bmatrix} \right) = 24.7$$

By similar calculations, we get

**w**ð Þ<sup>2</sup> *:* <sup>¼</sup> ½ � 384, 96**i**, 96**<sup>i</sup>** , **<sup>w</sup>**ð Þ<sup>2</sup> *:* ¼ �½ � <sup>192</sup>**i**, 96, 96 , **<sup>w</sup>**ð Þ<sup>2</sup> *:* ¼ �½ � 192**i**, 96, 06 *:*

So, by (36), we get

$$\begin{split} a\_{11}^{c,\uparrow} &= \frac{\sum\_{a \in I\_{2,3}} \left| (\mathbf{A} \mathbf{A}^\*)\_{1} \left( \mathbf{w}\_{1}^{(2)} \right) \right|\_{a}^{a}}{\left( \sum\_{a \in I\_{2,3}} |\mathbf{A} \mathbf{A}^\*|\_{a}^{a} \right)^2 \sum\_{\beta \in I\_{1,3}} \left| \mathbf{A}^3 \right|\_{\beta}^{\beta}} \\ &= \frac{1}{4608} \left( \det \begin{bmatrix} 384 & 192 \mathbf{i} \\ -2 \mathbf{i} & 3 \end{bmatrix} + \det \begin{bmatrix} 384 & 192 \mathbf{i} \\ -2 \mathbf{i} & 3 \end{bmatrix} \right) = \frac{1}{3} .\end{split}$$

By similarly continuing, we derive

$$\mathbf{A}^{c,\dagger} = \frac{1}{12} \begin{bmatrix} 4 & 2\mathbf{i} & 2\mathbf{i} \\ -2\mathbf{i} & 1 & 1 \\ -2\mathbf{i} & 1 & 1 \end{bmatrix}.$$

#### **5. Conclusions**

In this chapter, we get the direct method to find the core inverse and its generalizations that are based on their determinantal representations. New determinantal representations of the right and left core inverses, the right and left core-EP inverses, the DMP, MPD, and CMP inverses are derived.

*Functional Calculus*
