Some New Algebraic and Topological Properties of the Minkowski Inverse in the Minkowski Space

We introduce some new algebraic and topological properties of the Minkowski inverse A ⊕ of an arbitrary matrix A ∈ M m,n (including singular and rectangular) in a Minkowski space μ. Furthermore, we show that the Minkowski inverse A ⊕ in a Minkowski space and the Moore-Penrose inverse A + 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 A ⊕ is also derived.

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 = diag( 1 , 2 , . . . , ) ∈ , 1 ≥ 2 ≥ ⋅ ⋅ ⋅ ≥ > 0, and 2 ( = 1, 2, . . . , ) 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 2 The Scientific World Journal [12][13][14][15][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].
In [21,22], the Minkowski inner product on C is defined by ( , V) = [ , V], 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 ∈ C , 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 −1 = and ( ∼ ) = ( ). More generally, if ∈ , , then the Minkowski conjugate transpose of is defined by ∼ = 1 * 2 (where 1 and 2 are the Minkowski metric matrices of orders × and × , resp.), and it satisfies the following algebraic properties as in the following result.
Then, the following one given: 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.

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: (iv) ⊕ and ⊕ are idempotents (i.e., ⊕ and ⊕ are projectors on ( ) and ( ⊕ ), resp.), Proof. (i) Since the following four matrix equations are satisfied: then, by (6), we get the result.
The Scientific World Journal 3 (ii) Let 1 and 2 be two Minkowski metric tensors such that ⊕ 1 and ⊕ 2 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).
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. 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).

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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 ‖ ‖ 02 = (tr( ∼ )) 1/2 , then the following example shows that ∼ and ∼ are, in general, not invertible matrices and also ‖ ‖ 02 ̸ = ‖ ‖ 2 .
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,

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.
For ‖ ⊕ ‖ 02 ̸ = 0, 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.
If ∈ , and 0 is the largest eigenvalue of ∼ , then ‖ ‖ 02 = √ 0 in the Minkowski space . Here, if is thesymmetric projector, then it is easy to show that ⊕ = . Since the eigenvalues of a projector are only 0 and 1, then we have ‖ ‖ 02 = ‖ ⊕ ‖ 02 = 1, and by applying Corollary 12, we get the following result.

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.