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We introduce some new algebraic and topological properties of the Minkowski inverse

In this work, we consider matrices over the field of complex numbers as

The Moore-Penrose inverse is widely used in perturbation theory, singular systems, neural network problems, least-squares problems, optimization problems, and many other subjects [

The Moore-Penrose inverse can be explicitly expressed by the singular value decomposition (SVD) due to van Loan [

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 [

By letting

In [

Let

~-cancellation rule

Finally, a matrix

In this paper, some algebraic properties concerning the rank, range, existence, uniqueness, continuity, and reverse order law 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

Let

Let

(i)Since the following four matrix equations are satisfied:

(ii)Let

This means that

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

(iv) By using the matrix equations in (

(v) Since

(vi) Since

(vii) Equations (

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 [

Let

Since

The Minkowski inverse of a matrix

Let

Let

If

If

(i) Since

Let

Since

Now, since

By applying the four matrix equations in (

Let

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.

Let

For

Let

By using Theorem

Let

Suppose that

Let

Set

If

Let

The matrix

If only assumption (i) is violated but (ii) and (iii) hold of Theorem

Let

Let

Let

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

Let

Suppose that

Several attractive properties and conditions of the Minkowski inverse

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