#### Abstract

We introduce some new algebraic and topological properties of the Minkowski inverse of an arbitrary matrix (including singular and rectangular) in a Minkowski space . Furthermore, we show that the Minkowski inverse in a Minkowski space and the Moore-Penrose inverse in a Hilbert space are different in many properties such as the existence, continuity, norm, and SVD. New conditions of the Minkowski inverse are also given. These conditions are related to the existence, continuity, and reverse order law. Finally, a new representation of the Minkowski inverse is also derived.

#### 1. Introduction and Preliminaries

In this work, we consider matrices over the field of complex numbers as and real numbers as . The set of -by- complex matrices is denoted by . For simplicity, we write instead of or , and when , we write instead of . The notations , , ,, , , , , , , , and stand for the transpose, conjugate transpose, -symmetric, rank, range, null space, trace, determinant, Frobenius norm, Moore-Penrose inverse, Minkowski inverse, and set of all eigenvalues of a matrix , respectively.

The Moore-Penrose inverse is widely used in perturbation theory, singular systems, neural network problems, least-squares problems, optimization problems, and many other subjects [1–8]. The Moore-Penrose inverse of an arbitrary matrix is defined to be the unique solution of the following four matrix equations [3, 4, 8–10]: and it is often denoted by . Note that if we designate any matrix that satisfying the th matrix equation () in (1) is called the -inverse and denoted by .

The Moore-Penrose inverse can be explicitly expressed by the singular value decomposition (SVD) due to van Loan [11]. For any matrix with , there exist unitary matrices and satisfying and such that where , , and () are the nonzero eigenvalues of . Then, the Moore-Penrose inverse can be represented as

Some algebraic properties concerning the null space, range, rank, continuity, and some representations of some types of the generalized inverses of a given matrix over complex and real fields are widely studied by many researchers [12–16]. The Minkowski inverse of an arbitrary matrix is one of the important generalized inverses for solving matrix equations in the Minkowski space with respect to the generalized reflection antisymmetric matrix [17]. Some methods such as iterative, Borel summable, Euler-Knopp summable, Newton-Raphson, and Tikhonov’s methods are used for representation and computation of the Minkowski inverse in the Minkowski space [18, 19].

By letting be the space of complex -tuples, we will index the components of a complex vector in from to ; that is, . In addition to that, let be the Minkowski metric tensor defined by . Clearly, the Minkowski metric matrix is defined by [18, 20] and and .

In [21, 22], the Minkowski inner product on is defined by , where denotes the conventional Hilbert (unitary) space inner product. The space with the Minkowski inner product is called a Minkowski space and is denoted by . For any square matrix and vectors and , we have where is called the Minkowski conjugate transpose of in the Minkowski space . Naturally, the matrix is called -symmetric in the Minkowski space if . Now, it is easy to show that is -symmetric if and only if is Hermitian if and only if is Hermitian. Also, it is easy to verify that and . More generally, if , then the Minkowski conjugate transpose of is defined by (where and are the Minkowski metric matrices of orders and , resp.), and it satisfies the following algebraic properties as in the following result.

Lemma 1. *Let . Then, the following one given:*(i)* is unique,*(ii)*,*(iii)*,*(iv)*~-cancellation rule ,*(v)*,*(vi)*,*(vii)*.*

Finally, a matrix is said to be a range symmetric in unitary space (or equivalently is said to be EP) if . For further properties of EP matrices, one may refer to [3, 4, 10, 11].

In this paper, some algebraic properties concerning the rank, range, existence, uniqueness, continuity, and reverse order law of the Minkowski inverse are introduced. The relationships between and are also discussed. Furthermore, a new representation of related to the full-rank factorization of the matrix is derived, and new conditions for the existence and continuity of are also given.

#### 2. Some Algebraic Properties of the Minkowski Inverse

In this section, we derive some attractive algebraic properties and the reverse order law property of the Minkowski inverse in a Minkowski space.

The Minkowski inverse of an arbitrary matrix (including singular and rectangular), analogous to the Moore-Penrose inverse, is defined as follows.

*Definition 2. *Let be any matrix in the Minkowski space . Then, the Minkowski inverse of is the matrix which satisfies the following four matrix equations:

Theorem 3. *Let be any matrix in the Minkowski space . Then, the Minkowski inverse satisfies the following properties:*(i)*,*(ii)* is a unique matrix,*(iii)*,*(iv)* and are idempotents (i.e., and are projectors on and , resp.),*(v)* and are invertible matrices, where ,*(vi)*,*(vii)*.*

*Proof. *(i)Since the following four matrix equations are satisfied:
then, by (6), we get the result.

(ii)Let and be two Minkowski metric tensors such that and are two Minkowski inverses of a matrix ; then, by using Lemma 1 and Theorem 3(i), we have

This means that is a unique matrix.

(iii) It follows by applying the four matrix equations in (6).

(iv) By using the matrix equations in (6), we have and .

(v) Since is an idempotent matrix, then eigenvalues of are or . That is, or . So, for all , we have (i.e., is an invertible matrix). Similarly, we can prove that is also an invertible matrix.

(vi) Since , then, from ~-cancellation, we have
Now, by using the four matrix equations in (6), Theorem 3, and Lemma 1, we have
Consequently, (9), (10), and (11) show that .

(vii) Equations (9) and (10) show that . Now, by applying Theorem 3(vi), we have ; then, the equality holds.

The reverse order law property for the Moore-Penrose inverse of the product of two matrices is investigated by many researchers; one may refer to [23]. Analogous to Greville’s conditions that were stated in [6], we reached the following result.

Theorem 4. *Let and be two matrices in the Minkowski space such that the Minkowski inverses , , and exist. Then, if and only if and .*

*Proof. *Since is a projector on as in Theorem 3(iv), then
Now, by Definition 2 and Theorem 3, we have
Taking the Minkowski conjugate transpose of the two sides of (13), we have
Multiplying the right side and the left side of (15) by and , respectively, we have
Since , then we have
Also, multiplying the right side and the left side of (14) by and , respectively, and applying Theorem 3 and Definition 2 for , we have
Since is a projector on , we have
Equations (17) and (19) imply that satisfies the first, third, and fourth equations in (6). Finally, by taking the Minkowski conjugate transpose of the two sides of the first and the second equations in (6) for matrices and and by using Theorem 3(vi), we have
This equation shows that
Consequently, satisfies the second equation in (6).

#### 3. Existence of the Minkowski Inverse

The Minkowski inverse of a matrix exists if and only if [12]. In this section, we give some equivalent conditions for the existence and derive a new representation of the Minkowski inverse. If is a matrix of full row rank (column rank), then and are invertible matrices of orders and , respectively, in a Hilbert (Euclidian) space. Here, in a Minkowski space, if we define , then the following example shows that and are, in general, not invertible matrices and also .

*Example 5. *Let . Then, , and
Note that (i.e., ), and hence is not invertible. Also, , and , which are not equal.

Lemma 6. *Let and be two matrices. Then, the following are considered.*(i)*If , then .*(ii)*If , then .*

*Proof. *(i) Since , then is a left invertible; thus, there exists a matrix such that . Hence, , which implies that .Similarly, we can prove (ii).

Theorem 7. *Let be a rank factorization of rank . Then, and exist if and only if .*

*Proof. *Since , then and are of orders and , respectively, and . Hence,
where , , and are the Minkowski metric matrices of orders , , and , respectively. Since and are matrices of full ranks (i.e., ), then, by Lemma 6, we have . Similarly, we can prove that .

Now, since and are square matrices of order , then they are invertible if and only if they are of rank .

By applying the four matrix equations in (6), we can get a new representation of the Minkowski inverse as shown in the following result.

Theorem 8. *Let be a rank factorization of rank . Then,
*

#### 4. Some Topological Properties of the Minkowski Inverse

In this section, we establish some attractive topological properties and new conditions for the continuity of the Minkowski inverse in a Minkowski space.

It is known that, in normed algebra of bounded linear operators, the map of linear invertible operators associated with its usual inverse is continuous. The following example shows that this property is not valid in the Minkowski space.

*Example 9. *Let be a sequence of matrices for ; then, , , and . Note that does not exist. That is, for the map , we have

For , the following results are very important for finding the new conditions for the continuity of the Minkowski inverse of rectangular matrices in a Minkowski space.

Lemma 10. *Let . Then, for any ,
*

* Proof. *By using Theorem 3((iv) and (vii)), then, for any , we have . Thus, , and then we get the result.

Lemma 11. *Let and such that and . Then,
*

* Proof. *Suppose that and are the basis of . Then, the set is a subset of . Now, suppose that , for ; then, , and we have
Now, by using Lemma 10, we have , which is impossible, and thus . As is linearly independent, it follows that (for all ). Consequently, .

Corollary 12. *Let and such that ,, and . Then,
*

*Proof. *Set , and since , then, by using Lemma 11, we have which implies that
Since , then we also have which implies that
Now, the result follows by using (30) and (31).

If and is the largest eigenvalue of , then in the Minkowski space . Here, if is the -symmetric projector, then it is easy to show that . Since the eigenvalues of a projector are only and , then we have , and by applying Corollary 12, we get the following result.

Corollary 13. *Let and be two -symmetric projectors such that . Then,
*

Theorem 14. *The matrix can be written by using SVD as in the form with , , and is a diagonal if and only if the following conditions hold:*(i)* are nonnegative real numbers,*(ii)* is diagonalizable,*(iii)*.*

If only assumption (i) is violated but (ii) and (iii) hold of Theorem 14, then we can still get singular value decomposition (SVD). But in the Minkowski space, each of the assumptions can fail even if the other two hold. This is illustrated by the following three counterexamples [18].

*Example 15. *Let . Then, , and . Hence, which are not real numbers.

*Example 16. *Let . Then, , and . Hence, and cannot be diagonalized.

*Example 17. *Let . Then, , and . Hence, .

The following result gives the equivalent conditions for the continuity to be held for the Minkowski inverse of any rectangular matrix.

Theorem 18. *Let be a sequence of matrices such that . Then, if and only if .*

*Proof. *Suppose that , and set such that and . Then,
which means that there exists such that, for any ,
Since and are -symmetric projectors, then, by Corollary 13, we have
Conversely, suppose that satisfies the SVD conditions as in Theorem 14; then, , where is a diagonal matrix and and are unitary matrices. Suppose also that and for any . Now, set
Then,
Since , then, for and , we have
But , and then, by Lemma 11, we have
On the other hand,
Now, by using (39) and (40), we get that , which means that exists. Also by using the Schur complement, we have
where and . Now, if and , then, by Definition 2, we can see that
is the Minkowski inverse of . Since , then . Therefore, , , and . Now, by using the fact that the map which transforms an invertible matrix to its inverse is continuous, consequently, we find that
which completes the proof of Theorem 18.

#### 5. Conclusion

Several attractive properties and conditions of the Minkowski inverse in the Minkowski space are presented. In our opinion, it is worth extending these properties and establishing some necessary and sufficient conditions for the reverse order rule of the weighted Minkowski inverse in the Minkowski space of two and multiple matrix products.

#### Acknowledgment

The authors express their sincere thanks to the referees for the careful reading of the paper and several helpful suggestions.