Partial Ordering of Range Symmetric Matrices and M-Projectors with Respect to Minkowski Adjoint in Minkowski Space

In this paper, we obtain some new characterizations of the range symmetric matrices in the Minkowski Space  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  . 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].


Introduction and Preliminaries
Let us denote by .Where k I is the identity matrix of suitable order.r and s will denote the rank of the matrices R and S .Indefinite inner product is a scalar product defined by [ ] * , , , = = u v u Mv u Mv (1) where , denotes the conventional Hilbert Space inner product and M is a Hermitian matrix.This Hermitian matrix M is referred to as metric matrix.Minkowski Space  is an indefinite inner product space in which the metric matrix is denoted by G and is defined as G is called the Minkowski metric matrix.In case ( ) , then G is called the Minkowski metric tensor and is defined as For detailed study of indefinite linear algebra refer to [1].
However unlike the Moore-Penrose inverse of a matrix, the Minkowski inverse of a matrix does not exist always.In [2], Meenakshi showed that the Minkowski inverse of a matrix is called the Minkowski adjoint of the matrix R and 1 G and 2 G are the Minkowski metric matrices of suitable order m and n.A matrix  is said to be m-symmetric if = R R and is said to be G-unitary if and only if ~~n = = RR R R I .In [3], Meenakshi introduced the concept of range symmetric matrices in Minkowski Space and developed the Minkowski inverse of the range symmetric matrices and some equivalent conditions for a matrix to be range symmetric.A matrix In [4], the authors produced the necessary and sufficient conditions for the product of range symmetric matrices to be range symmetric and further showed that any block matrix in Minkowski space can be expressed as the product of range symmetric matrices.In [5] the authors studied the range symmetric matrices in relation with their Minkowski inverse and m-projectors.Summarizing the equivalent conditions for the definition of a range symmetric matrix form [3] [5] [6] the following equivalent conditions will be used in the forthcoming results: [RS-3]: [RS-5]: their exist a G-unitary matrix U such that 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 [PO-2]: [PO-3]: In any of the above cases we say R is predecessor of S or S is successor of R .We will use the notation ( ) to denote the set of all the matrices of index k.
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 Minkowski 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 [27] to the Minkowski space  .

Properties of Range Symmetric Matrices
In this section we develop some properties of Range Symmetric matrices by utilizing the representation obtained in corollary 2.6 in [5].Let ( )  be non-zero range symmetric matrices of rank r and s respectively.Then R and S , accord- ing to the above mentioned result, can be written as and where R U and S U are G-unitary and R D and S D are invertible matrices of order .r Theorem 1 Let ( )  be such that R is range symmetric.Then the fol-lowing statements are equivalent: 1.
= RS SR 2. If R is given by (2), then there exists Proof.We consider the decomposition of the matrix S , according to the size of blocks of R , as: From the statement (i) of the theorem, we get and hence the result follows.
If both the matrices R and S are range symmetric, then we have the following result for the commutativity.
Theorem 2 Let ( ) L M Then the following statements are equivalent:

( ) ( )
Proof.(i)⇔(ii) Consider the representations of R and S given by ( 2) and (3) respectively.With given From Equations ( 4) and ( 5) we have Pre multiplying and post multiplying (6) by S U and S U respectively and sub- stituting the matrix representation of

J D JD D J D J G K D JD D J D K
From this equality, on using the fact that R D and S D are nonsingular, we have ( ) ( ) , and ~0 R = J D K and hence the equivalence follows.
. Substituting the representations of S U and S U in the block representation of S given by (3) we have Furthermore, doing some algebra we have, 1. R is range symmetric.

Proof. (i)⇔(ii) Since
⊕ RR and ⊕ R R are m-symmetric idempotents, in fact m- projectors, on using [RS-3], we have R is range symmetric if and only if Hence the equivalence follows.
There exists an invertible matrix ( ) Proof.(i)⇔(ii) Using [RS-4], there exists an invertible matrix We partition E according to the blocks of R such that 1. R is range symmetric.2. There exists an invertible matrix ( ) There exists an invertible matrix ( ) Proof.The proof follows on the same lines as in the above theorem, using the fact that two matrices R and S are row equivalent if and only if ( ) ( )

Partial Ordering of Range Symmetric Matrices w.r.t Minkowski Adjoint
In this section some characterizations of predecessors of range symmetric matrices 2. There exists Proof.(i)⇔(ii) We consider the following block representation of R according to the block size of S as: , where 1 0 0 1

R S
Remark  3) and (7) of S and R respectively and Theorem 6.8.3.from [26], we have another equivalent condition for the partial ordering of range symmetric matrices w.r.t minkowski adjoint given by The next result gives some equivalent conditions for a matrix R to be range symmetric when S is range symmetric and S is the successor of R .
Theorem 7 Let ( )  such that S is a nonzero range symmetric matrix and ≤ R S , where S is given by (3) and R is given by (7).Then the following statements are equivalent: 1. R is range symmetric.

S S
= JD D J .   3) and ( 7) respectively in the above equality and doing some simple algebra leads to substituting the respective representations of ⊕ R and S , the equivalence follows.
(ii)⇔(iv) Using [PO-1] and substituting the representations of R , R and S , the equivalence follows after some computation.
The next result similar to Theorem 6 holds if we consider R to be range symmetric and decompose S in terms of representation for R Theorem 8 Let ( )  such that R is a nonzero range symmetric matrix.Then the following statements are equivalent: 2. There exists ( ) Proof.The proof follows on the same line as in Theorem 6 We again note that if ≤ R S and R is range symmetric, then S need not be range symmetric.Consider Example 1.In the following result we establish some equivalent conditions for S when R is range symmetric and ≤ R S .
Theorem 9 Let ( )  be given by ( 2) and (8) respectively such that R is a nonzero range symmetric matrix and ≤ R S .Then the following statements are equivalent: 1. S is range symmetric.2. M is range symmetric.

I S S U U M M
. S being range sym- (i)⇔(iv) From Remark 1, we have Now using the fact that S is range symmetric the equivalence follows.
In the above results we have used the commutativity of R and ⊕ S and ⊕ R and S .However if we assume the conditions given in the above theorem with an additional assumption that = RS SR , then the conditions obtained by interchanging R and S are also equivalent.
Theorem 10 Let ( )  be range symmetric such that R is a non zero matrix.Then the following statements are equivalent: 2. There exists a G-unitary matrix Proof.(i)⇒(ii) Consider the decomposition of S given by (3) i.e., 0 0 0 U .Since S is range symmetric, therefore by Theorem 6, there exists Using Theorem 7, we have J is range symmetric.We consider the following block representation of J as 0 0 0 () ⇒ () Follows at once by direct verification.

Partial Ordering of M-Projectors
In this section we obtain some results on partial ordering of m-projectors w.r.t Minkowski adjoint.The following result from [5], with two more obvious conditions, will be used extensively in the forthcoming results.
Lemma 1 Let R ∈  be range symmetric, then 1.

( ) ( )
if and only if R is nonsingular.( ) 4. If R is nonzero singular matrix then R and R P  are incomparable under the partial ordering w.r.t Minkowski adjoint.

5.
S R (iii) From statement (ii) of Lemma 1 and the fact that ( ) and hence by point (iv) of Lemma 1 R is invertible.Again by the same argument i.e, point (iv) of Lemma 1 converse holds.
(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 0

E F S P P G H
, where the decomposition is done according to the blocks of ( ) transpose, Minkowski adjoint, Minkowski inverse, Moore-Penrose inverse, range space and null space of a matrix A respectively.n I denote the iden- tity matrix of order n n × .Further we denote by mp n  the set of all m-projections.i.e. we use the convection according to which w


J DTherefore the equality = RS SR , on using the fact that R D , S D and r be such that ⊕ R exists.Then the following statements are equivalent:


be a non zero matrix.Then the following statements are equivalent:1.R is range symmetric.2. There exists an invertible matrix ( )


(i)⇔(iii) From statement (ii) of the Theorem 3 and [RS-4], we have ( ) ( ) be a nonzero matrix.Then the following statements are equivalent: and utilizing the statement (iii) of Theorem 3 and [RS-2].


under the partial ordering w.r.t Minkowski adjoint.Using the equivalences of the definition of Partial ordering w.r.t Minkowski adjoint that is [PO-1] and, [PO-2], it can be easily verified that ~, such that S is a nonzero range symmetric matrix.Then the following statements are equivalent: statements [PO-1], [PO-2] and [RS-3], it can be easily observed that ⊕ ⊕ = R S SR .Using the representations (

6 .
J is range symmetric.Proof.(i)⇔(ii) From remark 1, we have ⊕ ⊕ = S R RS .Now using the facts that D being invertible and S U is G-unitary and substituting the repre- sentations of S and R from ( ⇒(iii) Since ≤ R S ~ and R and S are range symmetric, using the observation mentioned in Remark 1 i.e., ⊕ ⊕ = R S SR , we have S , the equivalence follows.(iii)⇒(i)Since ≤ R S ~ and R is range symmetric, again by the same fact that ⊕ Rand S commute, using (iii) i.e., we get S is range symmetric.

1 .
If R is range symmetric, then R P  is m-symmetric and hence range symmetric2.If R P  is range symmetric, then so is R Proof.(i)The statement follows at once on using the [RS-3], [MI-3] and [MIblock representation of R , where the partition is done according to the blocks of R that M is nonsingular and the result follows.Remark 2 Since R P  is a m-projector[5], we have ( ) ., we take P  as a function of R , then converse is obvious.
[19]articular 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[7]-[19].Partial partial ordering of matrices have formulated the definition involving different kinds of generalized inverses and