Partial Ordering of Range Symmetric Matrices and M-Projectors with Respect to Minkowski Adjoint in Minkowski Space ()
1. Introduction and Preliminaries
Let us denote by
the set of
matrices and when
we write
for
. The symbols
,
,
,
,
and
de- note the conjugate transpose, Minkowski adjoint, Minkowski inverse, Moore-Penrose inverse, range space and null space of a matrix
respectively.
denote the iden- tity matrix of order
. Further we denote by
the set of all m-projections. i.e.
. Also we use the convection according to which
and
. Where
is the identity matrix of suitable order. r and s will denote the rank of the matrices
and
.
Indefinite inner product is a scalar product defined by
(1)
where
denotes the conventional Hilbert Space inner product and
is a Hermitian matrix. This Hermitian matrix
is referred to as metric matrix. Min- kowski Space
is an indefinite inner product space in which the metric matrix
is denoted by
and is defined as
satisfying
and
.
is called the Minkowski metric matrix. In case
, then
is called the Minkowski metric tensor and is defined as
. For detailed study of indefinite linear algebra refer to [1] .
The minkowski inverse of a matrix
, introduced by Meenakshi in [2] , is the unique solution to the following four matrix equations:
[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 [2] , Meenakshi showed that the Minkowski inverse of a matrix
exists if and only if
, where
is called the Minkowski adjoint of the matrix
and
and
are the Minkowski metric matrices of suitable order m and n. A matrix
is said to be m-symmetric if
and is said to be G-unitary if and only if
. 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
is said to be range symmetric if and only if
. 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 con- ditions will be used in the forthcoming results:
[RS-1]:
is range symmetric.
[RS-2]:
.
[RS-3]:
.
[RS-4]:
.
[RS-5]: their exist a G-unitary matrix
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 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 [7] - [19] . Partial ordering on matrices has a wide range of applications in different fields which include electrical networks, statistics, generalized inverses etc. see [20] [21] [22] [23] . Different kinds of partial orders on matrices have been studied which include Star partial ordering
introduced by Drazin [24] , minus partial order
introduced by Hartwig [25] , Sharp partial
order introduced by Mitra [19] , followed by left star ordering
and right star ordering
. In [26] , Punithavalli introduced the partial ordering on matrices in Minkowski space w.r.t the Minkowski adjoint. She studied the partial ordering, left partial ordering and right partial ordering w.r.t the Minkowski adjoint on Range symmetric matrices. She also established some equivalent conditions for the reverse order law to hold in relation to the partial ordering w.r.t Minkowski adjoint. Form ( [26] , page 79), we have for any two matrices
,
is said to be below
under the partial order w.r.t Minkowski adjoint, denoted by
, if one of the following equivalent condition is satisfied:
[PO-1]:
and
.
[PO-2]:
and
.
[PO-3]:
and
.
In any of the above cases we say
is predecessor of
or
is successor of
. 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 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 [27] to the Minkowski space
.
2. Properties of Range Symmetric Matrices
In this section we develop some properties of Range Symmetric matrices by utilizing the representation obtained in corollary
in [5] . Let
be non-zero range symmetric matrices of rank
and
respectively. Then
and
, accord- ing to the above mentioned result, can be written as
(2)
and
(3)
where
and
are G-unitary and
and
are invertible matrices of order ![]()
Theorem 1 Let
be such that
is range symmetric. Then the fol- lowing statements are equivalent:
1. ![]()
2. If
is given by (2), then there exists
and
such that
with
.
Proof. We consider the decomposition of the matrix
, according to the size of blocks of
, as:
.
From the statement (i) of the theorem, we get
.
This gives
,
and
and hence the result follows.
If both the matrices
and
are range symmetric, then we have the following result for the commutativity.
Theorem 2 Let
be range symmetric matrices. If
Then the following statements are equivalent:
1.
.
2.
.
3.
.
Proof. (i)⇔(ii) Consider the representations of
and
given by (2) and (3) res- pectively. With given
, we have
(4)
Also
![]()
Therefore
(5)
From Equations (4) and (5) we have
(6)
Pre multiplying and post multiplying (6) by
and
respectively and sub- stituting the matrix representation of
and
we get
![]()
From this equality, on using the fact that
and
are nonsingular, we have
,
, and
and hence the equivalence follows.
(i)⇔(iii) From
, using the fact that
is G-unitary, we have
and hence
. Substituting the re- presentations of
and
in the block representation of
given by (3) we have
![]()
Furthermore, doing some algebra we have,
![]()
Therefore the equality
, on using the fact that
,
and
are nonsingular, gives
![]()
Hence the equivalence follows.
Theorem 3 Let
be such that
exists. Then the following state- ments are equivalent:
1.
is range symmetric.
2.
.
3.
.
Proof. (i)⇔(ii) Since
and
are m-symmetric idempotents, in fact m- projectors, on using [RS-3], we have
is range symmetric if and only if
. Also from [MI-1] and [MI-2] we have
and
. Therefore
. Hence the equivalence follows.
(i)⇔(iii) Similarly
and
are idempotents such that
and
. Again using [RS-3], the result fol- lows.
Theorem 4 Let
be a non zero matrix. Then the following statements are equivalent:
1.
is range symmetric.
2. There exists an invertible matrix
and
such that
with
.
3. There exists an invertible matrix
and
such that
with ![]()
Proof. (i)⇔(ii) Using [RS-4], there exists an invertible matrix
such that
. We partition
according to the blocks of
such that
![]()
Now
, gives
, using the fact that
is in- vertible and
is G-unitary.
(i)⇔(iii) From statement (ii) of the Theorem 3 and [RS-4], we have
, the equivalence follows on the same lines as above
Theorem 5 Let
be a nonzero matrix. Then the following statements are equivalent:
1.
is range symmetric.
2. There exists an invertible matrix
and
such that
with
.
3. There exists an invertible matrix
and
such that
with
.
Proof. The proof follows on the same lines as in the above theorem, using the fact that two matrices
and
are row equivalent if and only if
and utilizing the statement (iii) of Theorem 3 and [RS-2].
3. Partial Ordering of Range Symmetric Matrices w.r.t Minkowski Adjoint
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
and
are m-symmetric.
Theorem 6 Let
such that
is a nonzero range symmetric matrix. Then the following statements are equivalent:
1.
.
2. There exists
such that
(7)
Proof. (i)⇔(ii) We consider the following block representation of
according to the block size of
as:
![]()
Then
![]()
and
![]()
Therefore the equality
gives
and
and
Also computing
and
and using the equality
, we get
and
. Thus
and
i.e.,
.
However if
is range symmetric and
, then
need not be range sym- metric e.g. consider the matrices
Example 1
![]()
Remark 1 If both the matrices
are range symmetric and
, then using the statements [PO-1], [PO-2] and [RS-3], it can be easily observed that
. Using the representations (3) and (7) of
and
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
and
. Furthermore,
is range symmetric, we have
.
The next result gives some equivalent conditions for a matrix
to be range symmetric when
is range symmetric and
is the successor of
.
Theorem 7 Let
such that
is a nonzero range symmetric matrix and
, where
is given by (3) and
is given by (7). Then the following statements are equivalent:
1.
is range symmetric.
2.
.
3.
.
4.
.
5.
.
6.
is range symmetric.
Proof. (i)⇔(ii) From remark 1, we have
. Now using the facts that
;
being invertible and
is G-unitary and substituting the repre- sentations of
and
from (3) and (7) respectively in the above equality and doing some simple algebra leads to ![]()
(ii)⇔(iii) For
,
. Again using Remark 1 and substituting the respective representations of
and
, the equivalence follows.
(ii)⇔(iv) Using [PO-1] and substituting the representations of
,
and
, the equivalence follows after some computation.
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
to be range symmetric and decompose
in terms of representation for ![]()
Theorem 8 Let
such that
is a nonzero range symmetric matrix. Then the following statements are equivalent:
1.
.
2. There exists
such that
(8)
Proof. The proof follows on the same line as in Theorem 6
We again note that if
and
is range symmetric, then
need not be range symmetric. Consider Example 1. In the following result we establish some equivalent conditions for
when
is range symmetric and
.
Theorem 9 Let
be given by (2) and (8) respectively such that
is a nonzero range symmetric matrix and
. Then the following statements are equivalent:
1.
is range symmetric.
2.
is range symmetric.
3.
.
4.
.
Proof. (i)⇔(ii) For
, since
is nonsingular and
is G- unitary, direct verification shows that
. Therefore
and
.
being range sym- metric, by [RS-3] we have
. This gives
and the equi- valence follows.
(i)⇒(iii) Since
and
and
are range symmetric, using the observation mentioned in Remark 1 i.e.,
, we have
, the equivalence follows.
(iii)⇒(i) Since
and
is range symmetric, again by the same fact that
and
commute, using (iii) i.e.,
, we get
is range symmetric.
(i)⇔(iv) From Remark 1, we have
. This gives
. Now using the fact that
is range symmetric the equivalence follows.
In the above results we have used the commutativity of
and
and
and
. However if we assume the conditions given in the above theorem with an additional assumption that
, then the conditions obtained by interchanging
and
are also equivalent.
Theorem 10 Let
be range symmetric such that
is a non zero matrix. Then the following statements are equivalent:
1.
.
2. There exists a G-unitary matrix
,
and
such that
and ![]()
Proof. (i)⇒(ii) Consider the decomposition of
given by (3) i.e.,
. Since
is range symmetric, therefore by Theorem 6, there exists
such that
with
. Using Theorem 7, we have
is range symmetric. We consider the following block representation of
as
, where
is G-unitary and
is invertible. Since
, by Theorem 8, we can find
such that
. Thus
is nonsingular when
. Taking
, we have
, where
is G-unitary.
Follows at once by direct verification.
4. 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
be range symmetric, then
1.
.
2.
is idempotent.
3.
.
4.
if and only if
is nonsingular.
5.
then
.
6.
.
7.
is invertible then
and if
, then
.
8.
has index atmost one.
Lemma 2 Let
. Then
1. If
, then
.
2.
.
3.
is invertible.
4. If
is nonzero singular matrix then
and
are incomparable under the partial ordering w.r.t Minkowski adjoint.
5.
, then
.
6.
.
Proof. (i) Since
. This gives
.
(ii)
, then
. Conversely if
,
and
and hence
.
(iii) From statement (ii) of Lemma 1 and the fact that
, if
, then
and hence by point (iv) of Lemma 1
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
.
Lemma 3 Let
. Then
1. If
is range symmetric, then
is m-symmetric and hence range symmetric
2. If
is range symmetric, then so is ![]()
Proof. (i) The statement follows at once on using the [RS-3], [MI-3] and [MI-4].
(ii) If
, the the result is trivial. Let
such that
, then by point (v) of Lemma 1 we have
. Also using point (ii) of Lemma 1 we get
i.e.,
. Thus we have
. Consider the block re- presentation of
, where the partition is done according to the blocks of
such that
. Using [MI-1] and [MI-2], we get
. Therefore
. This shows that
is nonsingular and the result follows.
Remark 2 Since
is a m-projector [5] , we have
. If we write
i.e., we take
as a function of
, then
. Thus
. However
in general. Consider the decomposition
, we have
i.e,
, which is the fundamental represen- tation of a m-projector. Hence we conclude that
if and only if
is a m- projector.
We generalize the function
by defining it as:
(9)
Thus we have the following equations
(10)
and hence if R = 0, we get
(11)
Let us consider some sets with following notations:
(12)
(13)
and
(14)
Theorem 11 Let
and
be the sets defined in (12), (13) and (14) respectively. Then,
and
.
Proof. The proof follows easily by utilizing Lemmas 1 and 3.
From the statement (i) of Lemma 3, it is obvious that
. However the reverse inclusion does not hold in general. Consider the matrix
. If possible suppose there exist a matrix
such that
, then by Lemma 3, we have
, which is absurd, since
but
. There- fore
.
Remark 3 Let
, then by using Lemma 1
. If
, then by Remark 2, we have
and if
, then by Theorem 11, we have
.
The next result provides a characterization of the set
![]()
Theorem 12 Let
be range symmetric given by (2), then
![]()
Proof. Let
. Then
. Therefore for
, we have
such that
i.e.,
and the result follows.
The next result shows that the function
, when restricted to the set
is mo- notonically decreasing w.r.t the partial ordering w.r.t Minkowski adjoint.
Theorem 13 Let
, such that
. Then ![]()
Proof. Let
, such that
. Since
is range symmetric we have
![]()
and therefore
(15)
Also
(16)
Finally using (15) and (16) we get
and
. Hence
.
However for the range symmetric matrices
and
, we have
but
. Thus we have the following result.
Theorem 14 Let
, such that
. Then
.
Proof. The proof follows at once by using Theorem 13 and Remark 2.
Theorem 15 Let
be range symmetric and
be such that
. Then
.
Proof. Consider the decomposition of
as given in (2). Then from Theorem 8 we get
and hence
. Thus if we assume
, then from Theorem 6, we get
with
. There- fore
. Hence
Conversely if we assume that
, then
is range symmetric and finally from Theorem 13, the result follows.
Theorem 16 Let
has the representation as given in point
of Lemma 1 and
. Then
if and only if there exists
such that ![]()
Proof. Assume that
. Then from Theorem 13 we have
. Let
, where the blocks are partitioned according to the blocks of
. Using the above mentioned equality we have
,
and
and hence
Also
. Taking
, where the decomposition is done according to the blocks of
. Then from the equation
we get
,
,
,
and
and therefore
with
. Clearly
has index one. The converse is obvious.
Acknowledgements
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.