Group Inverse of 2 × 2 Block Matrices over Minkowski Space

Necessary and sufficient conditions for the existence of the group inverse of the block matrix ~ ~ ~ 0 P P Q       in Minkowski Space are studied, where ~ ~ , P Q are both square and ( ) ( ) ~ ~ ≥ rank Q rank P . The representation of this group inverse and some related additive results are also given.


Introduction
Let F be a skew field and ( ) n n F ×  be the set of all matrices over F. For ( )  is said to be the group inverse of A, if , , AXA A XAX X AX XA = = = .
and is denoted by # X A = , and is unique by [1].
The generalized inverse of block matrix has important applications in statistical probability, mathematical programming, game theory, control theory etc. and for references see [2] [3] [4].The research on the existence and the representation of the group inverse for block matrices in Euclidean space has been done in wide range.For the literature of the group inverse of block matrix in Euclidean space, see [5]- [11].
In [12] the existence of anti-reflexive with respect to the generalized reflection antisymmetric matrix P and solution of the matrix equation AXB C = in Minkowski space  is given.In [13] necessary and sufficient condition for the existence of Re-nnd solution has been established of the matrix equation . In [14] partitioned matrix M in Minkowski space  was taken of the form to yield a formula for the inverse of M in terms of the Schur complement of D * .In this paper P * and P denote the conjugate transpose and Minkowski adjoint of a matrix P respectively.n I denotes the identity matrix of order n n × .Minkowski Space  is an indefinite inner product space in which the metric matrix associated with the indefinite inner product is denoted by G and is defined as G is called the Minkowski metric matrix.In case G is called the Minkowski metric tensor and is defined as , , , n Gu u u u − = − − [12].For any , the Minkowski adjoint of P denoted by P is defined as where * P is the usual Hermitian adjoint and G the Minkowski metric matrix of order n.We establish the necessary and sufficient condition for the existence and the representation of the group inverse of a block matrix , We also give a suf- ficient condition for ( ) to be similar to ( ) ~Q P .

Lemmas
Lemma 1.Let ( ) , then there are unitary matrices ( ) Proof.Since ( ) , = rank P r there are two unitary matrices ( ) , , , ( ) Then the group inverse of M exists in  if and only if the group inverse of P exists in  and ( ) , suppose group inverse of P exists in  and ( ) , and ( ) Then the group inverse of M exists in  if and only if the group inverse of P exists in  and ( ) ( ) Proof.The proof is same as Lemma 2. Lemma 4. Let ( ) then the following conclusions hold: 1) ( ) = , then by Lemma 1 we have ( ) ( ) where , , Q is invertible.By using Lemma 2 and 3 we get Then, Similarly we can prove 2) -5).

Main Results
Theorem 1.Let , The group inverse of M exists in  if and only if 21 0 0 0 Then ~P Q and ~Q P are similar.Proof.Suppose

( )
rank P r = , then by using Lemma 1, there are unitary matrices So ~P Q and ~Q P are similar.
in  .Conversely, suppose the group inverse of M exists in  , then it satisfies the fol- lowing conditions: 1)