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In this paper, we obtain some new characterizations of the range symmetric matrices in the Minkowski Space M by using the Block representation of the matrices. These characterizations are used to establish some results on the partial ordering of the range symmetric matrices with respect to the Minkowski adjoint. Further, we establish some results regarding the partial ordering of m-projectors with respect to the Minkowski adjoint and manipulate them to characterize some sets of range symmetric elements in the Minkowski Space M. All the results obtained in this paper are an extension to the Minkowski space of those given by A. Hernandez,
*et al*. in [The star partial order and the eigenprojection at 0 on EP matrices, Applied Mathematics and Computation, 218: 10669-10678, 2012].

Let us denote by

Indefinite inner product is a scalar product defined by

where

is denoted by

The minkowski inverse of a matrix

[MI-1]:

[MI-2]:

[MI-3]:

[MI-4]:

However unlike the Moore-Penrose inverse of a matrix, the Minkowski inverse of a matrix does not exist always. In [

[RS-1]:

[RS-2]:

[RS-3]:

[RS-4]:

[RS-5]: their exist a G-unitary matrix

Partial orders on matrices has remained the topic of interest for many authors in the area of matrix theory and generalized inverse. Almost all authors who have worked on partial ordering of matrices have formulated the definition involving different kinds of generalized inverses and in particular the Moore-Penrose Inverse. Results involving partial orders on matrices in relation with their generalized inverse are scattered in the literature of the matrix theory and generalized inverses for instance see [

[PO-1]:

[PO-2]:

[PO-3]:

In any of the above cases we say

In this paper we obtain some characterizations of range symmetric matrices and utilize them to study the partial ordering of range symmetric matrices w.r.t the Min- kowski adjoint in Minkowski space and hence different characterizations of partial orders on range symmetric matrices are obtained. Finally we study the partial ordering on m-Projectors w.r.t the Minkowski adjoint. All the results obtained in this paper are an extension of those given in [

In this section we develop some properties of Range Symmetric matrices by utilizing the representation obtained in corollary in [

and

where

Theorem 1 Let

1.

2. If

Proof. We consider the decomposition of the matrix

From the statement (i) of the theorem, we get

This gives

If both the matrices

Theorem 2 Let

1.

2.

3.

Proof. (i)⇔(ii) Consider the representations of

Also

Therefore

From Equations (4) and (5) we have

Pre multiplying and post multiplying (6) by

From this equality, on using the fact that

(i)⇔(iii) From

Furthermore, doing some algebra we have,

Therefore the equality

Hence the equivalence follows.

Theorem 3 Let

1.

2.

3.

Proof. (i)⇔(ii) Since

(i)⇔(iii) Similarly

Theorem 4 Let

1.

2. There exists an invertible matrix

3. There exists an invertible matrix

Proof. (i)⇔(ii) Using [RS-4], there exists an invertible matrix

Now

(i)⇔(iii) From statement (ii) of the Theorem 3 and [RS-4], we have

Theorem 5 Let

1.

2. There exists an invertible matrix

3. There exists an invertible matrix

Proof. The proof follows on the same lines as in the above theorem, using the fact that two matrices

In this section some characterizations of predecessors of range symmetric matrices under the partial ordering w.r.t Minkowski adjoint. Using the equivalences of the defi- nition of Partial ordering w.r.t Minkowski adjoint that is [PO-1] and, [PO-2], it can

be easily verified that

Theorem 6 Let

1.

2. There exists

Proof. (i)⇔(ii) We consider the following block representation of

Then

and

Therefore the equality

However if

Example 1

Remark 1 If both the matrices

The next result gives some equivalent conditions for a matrix

Theorem 7 Let

1.

2.

3.

4.

5.

6.

Proof. (i)⇔(ii) From remark 1, we have

(ii)⇔(iii) For

(ii)⇔(iv) Using [PO-1] and substituting the representations of

On the same lines the equivalences (ii)⇔(v) and (iii)⇔(vi) follow by using the Remark 1 and statements [PO-1] and [PO-2].

The next result similar to Theorem 6 holds if we consider

Theorem 8 Let

1.

2. There exists

Proof. The proof follows on the same line as in Theorem 6

We again note that if

Theorem 9 Let

1.

2.

3.

4.

Proof. (i)⇔(ii) For

(i)⇒(iii) Since

(iii)⇒(i) Since

(i)⇔(iv) From Remark 1, we have

In the above results we have used the commutativity of

Theorem 10 Let

1.

2. There exists a G-unitary matrix

Proof. (i)⇒(ii) Consider the decomposition of

Follows at once by direct verification.

In this section we obtain some results on partial ordering of m-projectors w.r.t Minkowski adjoint. The following result from [

Lemma 1 Let

1.

2.

3.

4.

5.

6.

7.

8.

Lemma 2 Let

1. If

2.

3.

4. If

5.

6.

Proof. (i) Since

(ii)

(iii) From statement (ii) of Lemma 1 and the fact that

(iv) It is obvious from (ii) and (iii).

(v) Follows at once by using point (i) of the Lemma 2 and point (vi) of Lemma 1.

(vi) The statement follows at once on using the fact that

Lemma 3 Let

1. If

2. If

Proof. (i) The statement follows at once on using the [RS-3], [MI-3] and [MI-4].

(ii) If

Remark 2 Since

We generalize the function

Thus we have the following equations

and hence if R = 0, we get

Let us consider some sets with following notations:

and

Theorem 11 Let

Proof. The proof follows easily by utilizing Lemmas 1 and 3.

From the statement (i) of Lemma 3, it is obvious that

Remark 3 Let

The next result provides a characterization of the set

Theorem 12 Let

Proof. Let

The next result shows that the function

Theorem 13 Let

Proof. Let

and therefore

Also

Finally using (15) and (16) we get

However for the range symmetric matrices

Theorem 14 Let

Proof. The proof follows at once by using Theorem 13 and Remark 2.

Theorem 15 Let

Proof. Consider the decomposition of

Theorem 16 Let

Proof. Assume that

The second author was supported by UGC-BSR through grant No. F25-1/2014-15(BSR)/ 7-254/2009(BSR) (20.01.2015). This support is greatly appreciated.

Krishnaswamy, D. and Lone, M.S. (2016) Partial Ordering of Range Symmetric Matrices and M-Projectors with Respect to Minkowski Adjoint in Min- kowski Space. Advances in Linear Algebra & Matrix Theory, 6, 132-145. http://dx.doi.org/10.4236/alamt.2016.64013