Characterizations of the Generalized MPCEP Inverse of Rectangular Matrices

Abstract

In this paper, we introduce a new generalized inverse, called the G-MPCEP inverse of a complex matrix. We investigate some characterizations, representations, and properties of this new inverse. Cramer’s rule for the solution of a singular equation is also presented. Moreover, the determinantal representations for the G-MPCEP inverse are studied. Finally, the G-MPCEP inverse being used in solving appropriate systems of linear equations is established.

1. Introduction

Throughout this paper, we denote the set of all complex matrices by . And the set of matrices with rank by . For , the symbols , , , and stand for the conjugate transpose, the rank, the null space, and the range space of , respectively. Moreover, will refer to the identity matrix. Let , the smallest positive integer for which is called the index of and is denoted by . represents the projector on the subspace along the subspace . For , stands for the orthogonal projection onto . The symbol represents the subset of all complex matrices with index 1.

Next, let us review the definitions of some generalized inverses. For , the Moore-Penrose inverse of is the unique matrix satisfying the following four Penrose equations [1]:

The Moore-Penrose inverse can be used to represent orthogonal projectors onto and onto , respectively. A matrix that satisfies the equality is called an inner inverse or -inverse of , and a matrix that satisfies the equality is called an outer inverse or -inverse of .

The core-EP inverse is an outer inverse which was presented in [2] for an arbitrary square matrix. For a square matrix and , there exists the core-EP inverse as the unique matrix for which the next equalities hold,

Recall that, by [3, 4], the core-EP inverse can be expressed as .

The -inverse of , denoted by [5], is the unique matrix satisfying where [5, 6].

In the paper [7], two new generalized inverses have emerged by combining the Moore-Penrose inverse and the core-EP inverse, which are the MPCEP inverse (MP-Core-EP) and the dual MPCEP inverse (CEPMP) , respectively [7]. Precisely, the MPCEP inverse of presents a unique solution to the matrix system [7],

Notice that

The dual MPCEP inverse of presents a unique solution to the matrix system [7],

And

And in [8], representations and properties for the MPCEP inverse were introduced.

Definition 1 (see [9]). Let , be a subspace of of dimension , and a subspace of of dimension . Then, has a inverse such that and if and only if in which case, is unique and denoted by

Denote by , the set of all such that , , and satisfy the abovementioned conditions, and exists.

Lemma 2 (see [9]). Let . Then, (1),(2)

Lemma 3 (see [3]). Let . Then, the system of matrix equations, possesses the unique solution .

Definition 4 (see [3]). Let . The generalized core-EP (or g-core-EP) inverse of the given matrix is defined as

Lemma 5 (see [3]). Let . Then, (1) is a projector onto along ,(2) is the orthogonal projector onto (3)

Lemma 6 (Urquhart formula) (see [10, 11]). If , , then, where is fixed, but the arbitrary element of

The main structure of this paper is as follows. In Section 2, we introduce the G-MPCEP inverse. Then, we give some representations and characterizations of the G-MPCEP inverse. In Section 3, Cramer’s rule for the solution of a singular equation is presented. In Section 4, the determinantal representations of the G-MPCEP inverse are developed. In Section 5, we give the application of the G-MPCEP inverse in solving linear equations.

2. Definition, Characterizations, and Representations of the G-MPCEP Inverse

Theorem 7. Let , be a subspace of of dimension , and a subspace of of dimension . Suppose that exists. Then, the system of matrix equations, has the unique solution

Proof. We can easily show that (10) holds for If different matrices and satisfy the system of matrix Equation (10), then, which further implies Thus, the solution to the system (10) is unique.

Definition 8. Let , be a subspace of of dimension , and a subspace of of dimension . Suppose that exists. The generalized MP-Core-EP (or G-MPCEP) inverse of is defined as

Corollary 9. Various important special cases of the defined G-MPCEP inverse, which recover some previously studied outer inverses, are given as follows: (1)For , and the G-MPCEP inverse becomes the MPCEP inverse. Indeed,(2)If then .

Let , , in the following conclusions satisfy the assumption of Theorem 7.

Theorem 10. Let . Then,

Proof. Since we have

Theorem 11. Let . Then,

Proof. Since , then the G-MPCEP inverse can be expressed by

Theorem 12. Let . Then, (1) is a projector onto along ,(2) is a projector onto along ,(3) is the orthogonal projector onto ,(4)

Proof. (1)Since is an outer inverse of , we have which is a projector. Notice thatAlso, we obtain (2)Sincewe get By Lemma 2, we have
On the other hand, (3)By (2) and Lemma 5, we can obtain which is the orthogonal projector onto .(4)This part is clear by

Theorem 13. Let . Then, is the -inverse of .

Proof. From the definition of the -inverse, we can get and On the other hand, from Theorem 12, we have

Theorem 14. Let . Then,

Proof. Using , we obtain, on the basis of the Urquhart formula [10, 11] (see Lemma 6),

Theorem 15. Let . Then is the unique solution to the constrained matrix equation

Proof. By Theorem 12, notice that (30) holds for . Let (30) be satisfied for different . Because we see and . Then, the inclusions and yield . So, i.e., is the unique solution to the constrained system (30).

Corollary 16. If and , the MPCEP inverse of is the unique matrix satisfying

Theorem 17. Let . The following statements are mutually equivalent for . (1)(2)(3)(4)(5)(6)

Proof.  
(1)⇒(2): On the basis of , one can verify The rest of the proof is evident by Theorem 7.
(2)⇒(3): Using and , we obtain (3)⇒(4): By , we get The assumption implies (4)⇒(5): Since , , we have Because , we see that (5)⇒(6): For the assumptions, we have (6)⇒(1): Note that

Theorem 18. Let and let such that . Then the following statements are equivalent. (1)(2)(3)(4) for arbitrary

Proof. (1)⇒(2): The equalities imply (2)⇒(3): The assumption yields and . Applying the condition , we get (3)⇒(1): Since , and , we obtain By , we have (1)⇒(4): Recall that all solutions of the equation are obtained as a sum of a particular solution of and the general solution of the homogeneous equation . According to [9], the general solution of is given by for arbitrary . In the same way, the general solution of is given by for arbitrary .
(4)⇒(1): If and , for arbitrary , then

Using the full-rank decomposition of , we develop the next expression for the G-MPCEP inverse .

Theorem 19. Let . If has the full-rank decomposition and , then

Proof. The proof follows from in conjunction with Theorem 10.

Theorem 20. Let . Suppose that satisfies and Assume that is full rank factorization of and is invertible. Then,

Proof. By [12], we observe that An application of Theorem 10 leads to which was our goal.

Corollary 21. Let , , satisfy the assumption of Theorem 7. Suppose has the full-rank decomposition . If and is the full-rank decomposition of , then

The G-MPCEP inverse can be further characterized under the assumptions and for some and . The notation will be used in order to simplify presentations.

Theorem 22. Let , , and . Then the next assertions are equivalent. (1) just coincides with , defined by(2), ,(3)

Proof. By [9], The equality implies, for By [9], By the equality , we get, for (1)⇒(2): Applying , we conclude (2)⇒(1): Notice that and give (1)⇒(3): Because , it follows that (3)⇒(1): From the assumptions and , we get

3. Cramer’s Rule for the Solution of a Singular Equation

Theorem 23. Let with , . Suppose that and are full-column rank matrices such that Then the bordered matrix is nonsingular and

Proof. Since we obtain By we can obtain Let we have Thus, is nonsingular and

Using the relationship between the G-MPCEP inverse and a nonsingular bordered matrix, we give Cramer’s rule for solving a singular linear equation, denotes the matrix obtained by replacing th column of with , where is the th column of .

Theorem 24. Let and . If then the restricted matrix equation is consistent and it has the unique solution

Proof. Since then Clearly, is a solution of . also satisfies the restricted condition because Finally, we show the uniqueness of . If also satisfies (65), we can get , then

Theorem 25. Let . Suppose and having full column rank such that If then the unique solution of the singular linear equation (65) is given by

Proof. Since and , we have It follows from (69) that the solution of satisfies By Theorem 23, the coefficient matrix of (70) is nonsingular. Using (60) and (69), we can obtain Therefore, and (68) follows from the classical Cramer’s rule [1].

4. The Determinantal Representation of the G-MPCEP Inverse

The following notations are used in order to express determinantal representations of the generalized inverses.

Let and be subsets with Suppose stands for a submatrix of with rows and columns indexed by and . Then, and denote a principal minor. Assume that represents strictly increasing sequences of integers chosen from {1,…,}. For fixed and , put

Denote by and the th column and the th row of , respectively. The notations and stand for matrices arising from stating the row vector instead of its th row and the column vector instead of its th column.

Lemma 26. [13] Let . Then the Moore-Penrose inverse possesses the determinantal representations

Lemma 27. [14] Let be of rank , be a subspace of of dimension , and be a subspace of of dimension . In addition, suppose that satisfies and If exists, then Furthermore,

Lemma 28. [15] Let and be the same as in Lemma 27. Write . Suppose that the generalized inverse of exists. Then, can be represented as follows:

Theorem 29. Let , and and be the same as in Lemma 27. Write . Then its G-MPCEP inverse has the following determinantal representations where

Proof. According to Definition 8 for , we obtain By substituting (74) and (79) for the determinantal representations of and in (83), we get where is the unit th column vector, is the unit th row vector, and is the th element of the matrix . If we put as the th component of the row vector , then from get (80). If we initially obtain for the th component of the column vector , then by get (81).

5. Application

In this section, we will give an application of the G-MPCEP inverse in solving linear equations.

Theorem 30. Let and . Then, the equation is consistent and its general solution is for arbitrary .

Proof. For given by (90), we get and so is a solution to (89).
If is a solution to 5.1, we have Thus, i.e., the solution is of the form (90).

Theorem 31. Let and . Then, the general solution of is given by for arbitrary

Proof. Notice that of the form (95) is a solution to (94): Let be a solution to (94). Then, by , we deduce that has the form (95):

Theorem 32. Let and . Then, the general solution to is given by for arbitrary

Proof. If is represented by (99), then Hence, is a solution of (98).
On the other hand, assume that is a solution to (98). Using one can conclude that Thus, the solution to (98) possesses the form (99). Since we observe the identities , which confirm the second identity in (99).

Data Availability

There is no data in this work.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by the Guangxi Natural Science Foundation (no. 2018 GXNSFDA281023), the National Natural Science Foundation of China (no. 12061015), and the Special Fund for Science and Technological Bases and Talents of Guangxi (no. GUIKE AD21220024).