The usual star, left-star, right-star, plus order, minus order and Lowner ordering have been generalized to bimatrices. Also it is shown that all these orderings are partial orderings in bimatrices. The relationship between star partial order and minus partial order of bimatrices and their squares are examined.

1. International
OPEN ACCESS Journal
Of Modern Engineering Research (IJMER)
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.9| Sept. 2014 | 66|
On Some Partial Orderings for Bimatrices G. Ramesh1, N. Anbarasi2 1 Associate Professor of Mathematics, Govt Arts College (Auto), Kumbakonam 2Assistant Professor of Mathematics, Arasu Engineering College , Kumbakonam
I. Introduction And Preliminaries
Let ℂ푚푥푛 be the set of 푚푥푛 complex bimatrices. The symbols 퐴퐵 ∗,ℛ 퐴퐵 and 푟 퐴퐵 denote the conjugate transpose, range space and rank subtractivity of 퐴퐵 휖 ℂ푚푥푛 respectively. Further, 퐴퐵 † 휖 ℂ푛푥푚 stand for the Moore-Penrose inverse of 퐴퐵[10], that is the unique bimatrix satisfying the equations, 퐴퐵 퐴퐵 †퐴퐵=퐴퐵, 퐴퐵 † 퐴퐵 퐴퐵 †=퐴퐵 †,퐴퐵 퐴퐵 †= 퐴퐵 퐴퐵 † ∗ , 퐴퐵 †퐴퐵 = 퐴퐵 †퐴퐵 ∗ (1.1) and 퐼퐵푛 be the identity bimatrix of order 푛. Moreover, ℂ푛 퐸푃 , ℂ푛 퐻 and ℂ푛 ≥denote the subsets of ℂ푛푥푛 consisting of 퐸푃, Hermitian, and Hermitian non-negative definite bimatrices respectively. That is, ℂ푛 퐸푃= 퐴퐵휖 ℂ푛푥푛 ∶푅 퐴퐵 =푅 퐴퐵 ∗ ⟹푅 퐴1 =푅 퐴1∗ 푎푛푑 푅 퐴2 =푅 퐴2∗ ℂ푛 퐻= 퐴퐵휖 ℂ푛푥푛 ∶퐴퐵=퐴퐵 ∗⟹퐴1=퐴1∗ 푎푛푑 퐴2=퐴2∗ and ℂ푛 ≥= 퐴퐵휖 ℂ푛푥푛∶퐴퐵=퐿퐵 퐿퐵 ∗⟹퐴1=퐿1퐿1∗ ; 퐴2=퐿2퐿2 ∗ 푓표푟 푠표푚푒퐿퐵=퐿1∪퐿2∈ℂ푛푥푝 In this paper, the usual star, left-star, right-star, plus order, minus order and Lowner order have been generalized to bimatrices. Also it is shown that all these orderings are partial orderings in bimatrices. The relationship between star partial order and minus partial order of bimatrices and their squares are examined. Definition 1.1 The star ordering for bimatrices is defined by,
퐴퐵≤∗퐵퐵⟺ 퐴퐵 ∗퐴퐵=퐴퐵 ∗퐵퐵 that is, 퐴1∗퐴1=퐴1∗ 퐵1 ; 퐴2∗퐴2=퐴2∗퐵2 and 퐴퐵퐴퐵 ∗=퐵퐵퐴퐵 ∗ that is, 퐴1퐴1∗ =퐵1퐴1∗ ; 퐴2퐴2∗=퐵2퐴2∗
and can alternatively be specified as, 퐴퐵≤∗퐵퐵⟺ 퐴퐵 †퐴퐵=퐴퐵 †퐵퐵 that is, 퐴1†퐴1=퐴1†퐵1 ; 퐴2†퐴2=퐴2†퐵2 and 퐴퐵퐴퐵 †=퐵퐵퐴퐵 † that is, 퐴1퐴1†=퐵1퐴1† ; 퐴2퐴2†=퐵2퐴2† Example 1.2 Consider the bimatrices 퐴퐵= 1 10 0 ∪ 2 20 0
(1.2)
(1.3)
Abstract: The usual star, left-star, right-star, plus order, minus order and Lowner ordering have been generalized to bimatrices. Also it is shown that all these orderings are partial orderings in bimatrices. The relationship between star partial order and minus partial order of bimatrices and their squares are examined.
Keywords: Star partial order, left-star partial order, right-star partial order, plus order, minus order and lowner order.
AMS Classification: 15A09, 15A15, 15A57

2. On Some Partial Orderings for Bimatrices
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.9| Sept. 2014 | 67|
and 퐵퐵= 1 11 −1 ∪ 2 22 −2 ⟹퐴퐵≤∗퐵퐵 Definition 1.3 The left - star ordering for bimatrices is defined by,
퐴퐵∗≤퐵퐵⟺퐴퐵 ∗퐴퐵=퐴퐵 ∗퐵퐵 that is, 퐴1∗ 퐴1=퐴1∗ 퐵1 ; 퐴2∗퐴2=퐴2∗퐵2 and ℛ 퐴퐵 ⊆ℛ 퐵퐵 that is, ℛ 퐴1 ⊆ℛ 퐵1 ; ℛ 퐴2 ⊆ℛ 퐵2 Definition 1.4
The right - star ordering for bimatrices is defined by,
퐴퐵≤∗퐵퐵⟺퐴퐵퐴퐵 ∗=퐵퐵퐴퐵 ∗ that is, 퐴1퐴1∗ =퐵1퐴1∗ ; 퐴2퐴2∗=퐵2퐴2∗ and ℛ 퐴퐵 ∗ ⊆ℛ 퐵퐵∗ that is, ℛ 퐴1∗ ⊆ℛ 퐵1∗ ; ℛ 퐴2∗ ⊆ℛ 퐵2∗ Definition 1.5 The plus - order for bimatrices is defined as, 퐴퐵⋖퐵퐵 whenever 퐴퐵 †퐴퐵=퐴퐵 †퐵퐵 and 퐴퐵퐴퐵 †=퐵퐵퐴퐵 †, for some reflexive generalized inverse 퐴퐵 † of 퐴퐵. ( satisfying both 퐴퐵퐴퐵 †퐴퐵=퐴퐵 and 퐴퐵 †퐴퐵퐴퐵 †=퐴퐵 †). Definition 1.6 The minus (rank Subtractivity) ordering is defined for bimatrices as, 퐴퐵≤− 퐵퐵⟺ r(퐵퐵−퐴퐵)=r 퐵퐵 −r 퐴퐵 that is, r(퐵1−퐴1)= r 퐵1 − r 퐴1 and r(퐵2−퐴2)= r 퐵2 − r 퐴2 (1.6) or as, 퐴퐵≤− 퐵퐵⟺ 퐴퐵퐵퐵 †퐵퐵=퐴퐵, 퐵퐵퐵퐵 †퐴퐵= 퐴퐵 and 퐴퐵퐵퐵 †퐴퐵=퐴퐵 (1.7) Note 1.7 i) It can be shown that, 퐴퐵⋖퐵퐵⟺ r(퐵퐵−퐴퐵)=r 퐵퐵 − r 퐴퐵 and so the plus order is equivalent to rank subtractivity. ii) From (1.4) and (1.5) it is seen that 퐴퐵≤∗퐵퐵⟺ 퐴퐵 ∗∗≤ 퐵퐵∗ Definition 1.8 The Lowner partial ordering denoted by ≤퐿, for Which 퐴퐵,퐵퐵 휖 ℂ푛푋푛 is defined by 퐴퐵≤퐿퐵퐵⟺퐵퐵−퐴퐵 휖 ℂ푛 ≥. Result 1.9 Show that,the relation ∗≤ is a partial ordering. Proof (1) 퐴퐵∗≤퐴퐵⇒퐴1∗≤퐴1 and 퐴2∗≤퐴2 holds trivially. (2) If 퐴퐵 ∗퐴퐵= 퐴퐵 ∗퐵퐵 and ℛ 퐵퐵 ⊆ ℛ 퐴퐵 then 퐴퐵=퐴1∪퐴2 = 퐴1†∗ 퐴1∗ 퐴1∪ 퐴2†∗ 퐴2∗퐴2 = 퐴1†∗ 퐴1∗ 퐵1∪ 퐴2†∗ 퐴2∗퐵2 = 퐴1 퐴1† ∗ 퐵1∪ 퐴2 퐴2† ∗ 퐵2 =퐵1∪퐵2
(1.4)
(1.5)

3. On Some Partial Orderings for Bimatrices
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.9| Sept. 2014 | 68|
=퐴퐵=퐵퐵 (3) If 퐴퐵 ∗퐴퐵=퐴퐵 ∗퐵퐵 and 퐵퐵∗ 퐵퐵=퐵퐵∗ 퐶퐵 hold along with ℛ 퐴퐵 ⊆ℛ 퐵퐵 and ℛ 퐵퐵 ⊆ℛ 퐶퐵 ,then 퐴퐵 ∗퐴퐵=퐴퐵 ∗퐵퐵 =퐴1 ∗퐵1∪퐴2 ∗퐵2 =퐴1 ∗퐵1†∗ 퐵1∗ 퐵1∪퐴2 ∗퐵2†∗ 퐵2∗ 퐵2 =퐴1 ∗퐵1†∗ 퐵1∗ 퐶1∪퐴2 ∗퐵2†∗ 퐵2∗ 퐶2 = 퐵1 퐵1† 퐴1 ∗ 퐶1∪ 퐵2 퐵2† 퐴2 ∗ 퐶2 =퐴1 ∗퐶1∪퐴2 ∗퐶2 퐴퐵 ∗ 퐴퐵 =퐴퐵 ∗ 퐶퐵 푎푛푑 ℛ 퐴퐵 ⊆ ℛ 퐶퐵 Similarly, it can be verified that all the orderings are partial orderings. Lemma 1.10 [1] Let 퐴,퐵∈ℂ푚푥푛 and let 푎=푟 퐴 <푟 퐵 =푏. Then 퐴≤∗퐵 if and only if there exist 푈∈ℂ푚푥푏,푉∈ ℂ푛푥푏 satisfying 푈∗푈=퐼푏=푉 ∗푉, for which 퐴=푈 퐷1 0 0 0 푉 ∗ and 퐵=푈 퐷1 0 0 퐷2 푉 ∗ (1.8) where 퐷1 and 퐷2 are positive definite diagonal matrices of degree a and 푏−푎, respectively. For 퐴,퐵∈ℂ푛 퐻, the matrix 푈 in (1.8) may be replaced by 푉, but then 퐷1 and 퐷2 represent any nonsingular real diagonal matrices. Lemma 1.11[1] Let 퐴,퐵∈ℂ푚푥푛 and let 푎=푟 퐴 <푟 퐵 =푏. Then 퐴≤−퐵 if only if there exist U ∈ℂ푚푥푏, 푉∈ℂ푛푥푏, satisfying 푈 ∗푈=퐼푏=푉 ∗푉, for which 퐴=푈 퐷1 0 0 0 푉 anda 퐵=푈 퐷1+푅 퐷2푆 푅퐷2 퐷2푆 퐷2 푉∗ (1.9) Where 퐷1 and 퐷2 are positive definite diagonal matrices of degree 푎 and 푏−푎, while 푅∈ℂ푎푥푏−푎 and 푆∈ℂ푏−푎푋푎 are arbitrary. For 퐴,퐵∈ℂ푛 퐻 the matrices 푈 and 푆 in (1.9) may be replaced by 푉 and 푅 ∗ respectively, but then 퐷1 and 퐷2 represent any nonsingular real diagonal matrices. Lemma 1.12 Let 퐴퐵,퐵퐵∈ℂ푛 퐻 be star-ordered as 퐴퐵≤∗퐵퐵. Then 퐴퐵≤퐿퐵퐵 if and only if 훾 퐴퐵 =훾 퐵퐵 , Where 훾 (.) denotes the number of negative eigenvalues of a given bimatrix. Proof Case (i) Let 푟 퐴퐵 = 푟 퐵퐵 that is, 푟 퐴1 = 푟 퐵1 and 푟 퐴2 = 푟 퐵2 . The result is trivial. Case (ii) Let 푟 퐴퐵 < 푟 퐵퐵 that is, 푟 퐴1 <푟 퐵1 and 푟 퐴2 < 푟 퐵2 Lemma (1.1) ensures that if 퐴퐵≤∗퐵퐵, then 퐵퐵−퐴퐵= 퐵1∪퐵2 −(퐴1∪퐴2) = 퐵1−퐴1 ∪ 퐵2−퐴2 = 푈1 퐷11 0 0 퐷12 푉1∗−푈1 퐷11 00 0 푉1∗ ∪ 푈2 퐷21 0 0 퐷22 푉2∗−푈2 퐷21 00 0 푉2∗

4. On Some Partial Orderings for Bimatrices
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=(푈1∪푈2) 퐷11∪퐷12 0 0 퐷12∪퐷22 푉1∗∪푉2∗ − 푈1∪푈2 퐷11∪퐷21 0 0 0 푉1∗∪푉2∗ =푈퐵 퐷퐵1 0 0 퐷퐵2 푉퐵 ∗−푈퐵 퐷퐵1 0 0 0 푉퐵 ∗ 퐵퐵−퐴퐵=푈퐵 0 0 0 퐷퐵2 푉퐵 ∗ Hence it is seen that the order 퐴퐵≤퐿퐵퐵 is equivalent to the non-negative definiteness of 퐷퐵2, that is, 훾 퐷2 =0.Consequently, the result follows by noting that 훾 퐴퐵 =훾 퐷퐵1 and 훾 퐵퐵 =훾 퐷퐵1 +훾 퐷퐵2 .
II. Star Partial Ordering
Theorem (3) of Baksalary and Pukel sheim [3] asserts that, for any A,B ∈ℂ푛 ≥ , A ≤∗B⟺ A2≤∗B2 ⇒퐴 퐵=퐵 퐴 (2.1) This result is revisited here with the emphasis laid on the question which from among four implications comprised in (2.1) continues to be valid for bimatrices not necessarily being bihermitian non negative definite. Theorem2.1 Let 퐴퐵∈ℂ푛 퐸푃 and 퐵퐵∈ℂ푛푋푛.Then 퐴퐵≤∗퐵퐵⇒퐴퐵 2≤∗퐵퐵 2 and 퐴퐵퐵퐵=퐵퐵퐴퐵 (2.2) Proof Since 퐴퐵=퐴1∪퐴2∈ℂ푛 퐸푃⟺퐴퐵퐴퐵 †=퐴퐵 †퐴퐵 That is, 퐴1퐴1†=퐴1†퐴1 and 퐴2퐴2†=퐴2†퐴2 Now, 퐴퐵 2퐴퐵 †=퐴12 퐴1†∪퐴22퐴2† =퐴1 퐴1퐴1† ∪퐴2 퐴2퐴2† =퐴1∪퐴2 퐴퐵 2퐴퐵 †=퐴퐵 Also, 퐴퐵 †퐴퐵 2=퐴1†퐴12 ∪퐴1†퐴12 = 퐴1†퐴1 퐴1∪ 퐴2+ 퐴2 =퐴1∪퐴2 퐴퐵 †퐴퐵 2=퐴퐵 ⇒퐴퐵 2퐴퐵 †=퐴퐵 †퐴퐵 2 And 퐴퐵 2 †= 퐴12 ∪퐴22 † = 퐴1퐴1 †∪ 퐴2퐴2 † =퐴1†퐴1†∪퐴2†퐴2† = 퐴1†∪퐴2† 2 퐴퐵 2 †= 퐴퐵 † 2 Consequently, in view of (1.3), 퐴퐵퐵퐵=퐴1퐵1∪퐴2퐵2 = 퐴12 퐴1†퐵1∪퐴22퐴2†퐵2 =퐴12 퐴1†퐴1∪퐴22퐴2†퐴2

5. On Some Partial Orderings for Bimatrices
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퐴퐵퐵퐵=퐴퐵 2 And 퐵퐵퐴퐵=퐵1퐴1∪퐵2퐴2 =퐵1퐴1†퐴12 ∪퐵2퐴2†퐴22 =퐵1퐴1†퐴12 ∪퐴2퐴2†퐴22 퐵퐵퐴퐵=퐴퐵 2 ⇒퐴퐵퐵퐵=퐵퐵퐴퐵=퐴퐵 2 Moreover, 퐴퐵 2 †퐵퐵 2= 퐴12 ∪퐴22 † 퐵12∪퐵22 = 퐴12 †퐵12∪ 퐴22 †퐵22 = 퐴12 † 퐴12 ∪ 퐴22 †퐴22 = 퐴12 ∪퐴22 † 퐴12 ∪퐴22 퐴퐵 2 †퐵퐵 2= 퐴퐵 2 † 퐴퐵 2 and 퐵퐵 2 퐴퐵 2 †=퐵12 퐴12 †∪퐵22 퐴22 † =퐵1 퐴1 퐴1† 2∪퐵2 퐴2 퐴2† 2 =퐴12 퐴12 † ∪퐴22 퐴22 † 퐵22 퐴퐵 2 †=퐴퐵 2 퐴퐵 2 † ⟹퐴퐵 2≤∗퐵퐵 2 Note2.2 Implication 2.2 is not reversible. Example2.3 Consider the bimatrices, 퐴퐵= 1 10 0 ∪ 2 20 0 and 퐵퐵= 1 11 −1 ∪ 2 22 −2 In which the order 퐴퐵≤∗퐵퐵 does not entail either of the conditions 퐴퐵 2≤∗퐵퐵 2, 퐴퐵 퐵퐵=퐵퐵 퐴퐵. On the otherhand, if 퐴퐵= 1 00 0 ∪ 2 00 0 and 퐵퐵= 0 11 0 ∪ 0 22 0 then 퐴퐵 2 ≤∗퐵퐵 2, but 퐴퐵 ∗퐴퐵≠퐴퐵 ∗퐵퐵 and 퐴퐵 퐵퐵≠ 퐵퐵 퐴퐵. This showing that even for bihermitian matrices the star order between 퐴퐵 2 푎푛푑 퐵퐵 2 and does not entail the star order between 퐴퐵 and 퐵퐵 and the commutativity of these bimatrices which are the other two implications contained in (2.1). It is pointed out that the two conditions on the right-hand side of (2.2) are insufficient for 퐴퐵≤∗퐵퐵. When there is no restriction on 퐴퐵, a similar conclusion is obtained in the case of combining the two orders 퐴퐵≤∗퐵퐵 and 퐴퐵 2≤∗퐵퐵 2.The bimatrices, 퐴퐵= 0 10 0 ∪ 0 20 0 and 퐵퐵= 0 11 0 ∪ 0 22 0 and their squares are star ordered, but 퐴퐵 퐵퐵≠퐵퐵 퐴퐵. However, the combination of the order 퐴퐵≤∗퐵퐵 with the commutativity condition appears sufficient for 퐴퐵 2 ≤∗퐵퐵 2 for all quadratic bimatrices. Theorem2.4 Let 퐴퐵,퐵퐵∈ℂ푛푥푛. Then 퐴퐵≤∗퐵퐵 and 퐴퐵 퐵퐵=퐵퐵 퐴퐵⟹퐴퐵 2 ≤∗퐵퐵 2

6. On Some Partial Orderings for Bimatrices
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Proof On account of (1.2), it follows that if 퐴퐵≤∗퐵퐵 and 퐴퐵 퐵퐵=퐵퐵 퐴퐵 then, 퐴퐵 2 ∗퐵퐵 2= 퐴12 ∪퐴22 ∗ 퐵12∪퐵22 = 퐴12 ∗∪ 퐴22 ∗ 퐵12∪퐵22 = 퐴12 ∗퐵12∪ 퐴22 ∗퐵22 =퐴1∗ 퐴1∗ 퐴1 퐵1∪퐴2∗ 퐴2∗퐴2 퐵2 = 퐴1∗ 2 퐴1 퐵1∪ 퐴2∗ 2퐴2 퐵2 =퐴1 ∗ 퐴1∗ 퐵1 퐴1∪퐴2∗ 퐴2∗퐵2 퐴2 = 퐴12 ∗퐴12 ∪ 퐴22 ∗퐴22 퐴퐵 2 ∗퐵퐵 2= 퐴퐵 2 ∗퐴퐵 2 and 퐵퐵 2 퐴퐵 2 ∗= 퐵12∪퐵22 퐴12 ∪퐴22 ∗ =퐵12 퐴12 ∗∪퐵22 퐴22 ∗ =퐵1 퐵1퐴1 ∗ 퐴1∗ ∪퐵2 퐵2퐴2 ∗ 퐴2∗ =퐵1 퐴1퐴1 ∗ 퐴1 ∗∪퐵2 퐴2퐴2 ∗ 퐴2 ∗ =퐴1 퐵1퐴1 ∗ 퐴1 ∗∪퐴2 퐵2퐴2 ∗ 퐴2 ∗ =퐴1 퐴1퐴1 ∗ 퐴1 ∗∪퐴2 퐴2퐴2 ∗ 퐴2 ∗ =퐴12 퐴1퐴1 ∗∪퐴22 퐴2퐴2 ∗ =퐴12 퐴12 ∗∪퐴22 퐴22 ∗ 퐵퐵 2 퐴퐵 2 ∗=퐴퐵 2 퐴퐵 2 ∗
III. Minus Partial Ordering
Baksalary and Pukelsheim [3, Theorem 2] showed that for ,퐵∈ℂ푛 ≥ , the following three implications hold. 퐴≤−퐵 and 퐴2≤−퐵2 ⟹ 퐴퐵=퐵퐴 (3.1) 퐴≤−퐵 and 퐴퐵=퐵퐴 ⟹ 퐴2≤−퐵2 (3.2) 퐴2≤−퐵2 and 퐴퐵=퐵퐴 ⟹ 퐴≤− 퐵 (3.3) Now,we extend this implications for bimatrices also by the following theorems. Theorem 3.1 For any 퐴퐵,퐵퐵∈ℂ푛푥푛 if 퐴퐵≤− 퐵퐵 and 퐴퐵퐵퐵=퐵퐵퐴퐵 then 퐴퐵 2≤−퐵퐵 2 Proof First notice that, 퐴퐵≤− 퐵퐵 and 퐴퐵퐵퐵=퐵퐵퐴퐵 (3.4) ⟹퐴퐵퐵퐵=퐴1퐵1∪퐴2퐵2=퐴12 ∪퐴22=퐵1퐴1∪퐵2퐴2=퐵퐵퐴퐵 (3.5) On account of 1.7, it follows that, 퐴퐵퐵퐵=퐴1퐵1∪퐴2퐵2 =퐴1 퐵1† 퐴1 퐵1∪퐴2 퐵2† 퐴2퐵2 =퐴1 퐵1†퐵1 퐴1∪퐴2 퐵2†퐵2 퐴2 =퐴1퐴1∪퐴2퐴2

7. On Some Partial Orderings for Bimatrices
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=퐴12 ∪퐴22 퐴퐵퐵퐵=퐴퐵 2 퐵퐵 퐴퐵=퐵1 퐴1∪퐵2퐴2 =퐵1 퐴1 퐵1† 퐴1∪퐵2 퐴2 퐵2† 퐴2 = 퐴1 퐵1 퐵1† 퐴1 ∪ 퐴2 퐵2 퐵2† 퐴2 = 퐴1 퐴1∪ 퐴2 퐴2 =퐴12 ∪퐴22 퐵퐵 퐴퐵=퐴퐵 2 ⟹퐴퐵퐵퐵=퐴퐵 2=퐵퐵 퐴퐵 The conditions on the right hand sides of (1.7) and (3.5) lead to the equalities, 퐵퐵 2 퐵퐵 2 †퐴퐵 2= 퐵12∪퐵22 퐵12∪퐵22 † 퐴12 ∪퐴22 =퐵12 퐵12 †퐴12 ∪퐵22 퐵22 †퐴22 =퐵12 퐵12 †퐵12퐵1† 퐴1∪퐵22 퐵22 †퐵22퐵2† 퐴2 = 퐵12퐵1†퐴1∪퐵22 퐵2†퐴2 =퐵1퐴1∪퐵2퐴2 =퐵퐵퐴퐵 ⇒퐵퐵 2 퐵퐵 2 †퐴퐵 2=퐴퐵 2 퐴퐵 2 퐵퐵 2 †퐵퐵 2=퐴12 퐵12 †퐵12∪퐴22 퐵22 †퐵22 =퐴1퐵1 퐵12 †퐵12∪퐴2퐵2 퐵22 †퐵22 =퐴1퐵1†퐵12 퐵12 †퐵12∪퐴2퐵2†퐵22 퐵22 †퐵22 =퐴1퐵1†퐵12∪퐴2퐵2†퐵22 =퐴1퐵1∪퐴2퐵2 =퐴퐵퐵퐵 ⇒퐴퐵 2 퐵퐵 2 †퐵퐵 2=퐴퐵 2 (iii) 퐴퐵 2 퐵퐵 2 †퐴퐵 2=퐴12 퐵12 †퐴12 ∪퐴22 퐵22 †퐵22 =퐴1퐵1 퐵12 †퐵1퐴1∪퐴2퐵2 퐵22 †퐵2퐴2 =퐴1퐵1†퐵12퐵1†퐴1∪퐴2퐵2†퐵22퐵2†퐴2 =퐴1 퐵1†퐵1 퐵1퐵1† 퐴1∪퐴2 퐵2†퐵2 퐵2퐵2† 퐴2 =퐴12 ∪퐴22 퐴퐵 2 퐵퐵 2 †퐴퐵 2=퐴퐵 2 According to (1.7), which shows that 퐴퐵 2≤−퐵퐵 2 Lemma 3.2 Let 퐴퐵휖ℂ푛 퐸푃.If the Moore – Penrose inverse 퐵퐵 † of some 퐵퐵휖ℂ푛 퐻 is a generalized inverse of a bimatrix 퐴퐵, that is, 퐴퐵퐵푩 †퐴퐵=퐴퐵, Then 퐴퐵휖ℂ푛 퐻. Proof

8. On Some Partial Orderings for Bimatrices
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.9| Sept. 2014 | 73|
It is mentioned that, 퐴퐵휖ℂ푛 퐸푃 if and only if , 퐴퐵퐴푩 †=퐴푩 †퐴퐵 Then, on the other hand 퐴퐵퐴푩 †퐵푩 †퐴푩 ∗ =퐴1퐴1†퐵1퐴ퟏ ∗ ∪퐴2퐴2†퐵2퐴ퟐ ∗ = 퐴1퐵1퐴1퐴1† ∗ ∪ 퐴2퐵2퐴2퐴2† ∗ = 퐴1퐴1† ∗ ∪ 퐴2퐴2† ∗ =퐴1퐴1†∪퐴2퐴2† 퐴퐵퐴푩 †퐵푩 †퐴푩 ∗=퐴푩 †퐴퐵 (3.6) Further, 퐴퐵퐴푩 †퐵푩 †퐴푩 ∗ =퐴1퐴1†퐵1†퐴ퟏ ∗∪퐴2퐴2†퐵2†퐴ퟐ ∗ =퐴1†퐴ퟏ ∗∪퐴2†퐴ퟐ ∗ 퐴퐵퐴푩 †퐵푩 †퐴푩 ∗=퐴푩 †퐴푩 ∗ Comparing (3.6) with (3.7) we get 퐴푩 † 퐴퐵=퐴푩 †퐴푩 ∗ Pre-multiplying by 퐴퐵, 퐴퐵퐴푩 †퐴퐵=퐴퐵퐴푩 †퐴푩 ∗ ⇒ 퐴1퐴1†퐴1 ∪ 퐴2퐴2†퐴2 =퐴1퐴1†퐴ퟏ ∗∪퐴2퐴2†퐴ퟐ ∗ 퐴1∪퐴2=퐴ퟏ ∗∪퐴ퟐ ∗ 퐴퐵=퐴푩 ∗ ⇒퐴퐵휖ℂ푛 퐻 Theorem 3.3 Let 퐴퐵∈ℂ푛 퐸푃and 퐵퐵∈ℂ푛 퐻. Then 퐴≤− 퐵 and 퐴2≤퐿퐵2⟺퐴퐵≤∗퐵퐵 (3.8) Proof Without loss of generality, 퐴퐵 may be assumed to be Hermitian. From (1.6) it is clear that and that 푟 퐴퐵 ≤푟 퐵퐵 and that the equality holds only in the trivial case when 퐴퐵=퐵퐵. Therefore, assume that 푎=푟 퐴퐵 <푟 퐵퐵 =푏 that is 푟 퐴1 =푟 퐴2 =푎 and 푟 퐵1 =푟 퐵2 =푏 If 퐴퐵≤∗퐵퐵, then the fact that 퐴퐵,퐵퐵∈ℂ푛 퐻 enables representing these bimatrices in the forms described in the second part of Lemma(1.10). Hence the ⟸ part of (3.8) follows. For the proof of the converse implication observe that, on account of the first two equalities in (1.7) 퐴퐵 2≤퐿퐵퐵 2⟺퐵퐵 † 퐴퐵 2 퐵퐵 †≤퐿퐵퐵 † 퐵퐵 2 퐵퐵 † ⟺퐵퐵 †퐴퐵 † 퐵퐵 †퐴퐵 ∗ ≤퐿퐵퐵 †퐵퐵 (3.9) By conditions (1.1), the Moore-Penrose inverse of a hermitian bimatrix 퐵퐵 of the form specified in the second part of Lemma (1.11) admits the representation 퐵퐵 †=푉퐵 퐷퐵1−1 −퐷퐵1−1 푅퐵 −푅퐵∗ 퐷퐵1−1 퐷퐵2−1+ 푅퐵∗ 퐷퐵1−1 푅퐵 푉퐵 ∗ and hence

9. On Some Partial Orderings for Bimatrices
| IJMER | ISSN: 2249–6645 | www.ijmer.com | Vol. 4 | Iss.9| Sept. 2014 | 74|
퐵퐵 †퐴퐵 퐵퐵 †퐴퐵 ∗ =푉퐵 퐼퐵푎 0−푅퐵∗ 0 퐼퐵푎 − 푅퐵 0 0 푉퐵 ∗ =푉퐵 퐼퐵푎 − 푅퐵 −푅퐵∗ 푅퐵∗ 푅퐵 푉퐵 ∗ (3.10) Since the bimatrix 퐵퐵 † 퐵퐵 =퐵퐵퐵퐵 † represents the orthogonal projector onto ℛ 퐵퐵 =ℛ 푉퐵 , it may be expressed as 푉퐵푉퐵 ∗. Consequently, in view of (3.10) 퐵퐵 † 퐵퐵−퐵퐵 † 퐴퐵 퐵퐵 † 퐴퐵 ∗ =푉퐵 0 푅퐵 푅퐵∗ 퐼퐵(푏−푎) −푅퐵∗ 푅퐵 푉퐵 ∗ (3.11) On the account of (3.9), equality (3.11) shows that supplementing the minus order 퐴퐵≤−퐵퐵 by Lowner order 퐴퐵 2≤퐿퐵퐵 2 forces to be 0. Then the bimatrix 퐵퐵 characterized in lemma (1.11) takes the form described in lemma (1.1), thus leading to the conclusion that 퐴퐵≤∗퐵퐵 REFERENCES [1] Jerzy.K. Baksalary, Jan Hauke, Xiaoji Liu, Sanyang Liu, ‘Relationships between partial orders of matrices and their powers’ Linear Algebra and its Applications 379 (2004) 277-287. [2] J.K. Baksalary, S.K. Mitra, ‘Left-star and right-star partial orderings’, Linear Algebra Applications, 149 (1991) 73- 89. [3] Jerzy.K. Baksalary, Oskar Haria Baksalary, Xiaoji Liu, ‘Further properties of the star, left-star, minus partial orderings’ Linear Algebra and its Applications 375 (2003) 83-94. [4] J.K. Baksalary, F. Pukelshiom, G.P.H. Styan, ‘some properties of matrix partial orderings’, Linear Algebra Applications, 119 (1989) 57-85. [5] Jerzy.K. Baksalary, Sujit Kumar Mitra, ‘Left-star and right-star and partial orderings’. Elsevier Science Publishing Co.,Inc.,1991 (73-89). [6] M.P. Drazin, ‘Natural structures on semi groups with involution Bull. Amer. Hath. Soc. 84 (1978) 139-141. [7] R.E.Hartwig, ‘How to partially order regular elements’, Math. Japan.25 (1980) 1-13. [8] R.E. Hartwig, G.P.H. Styan, ‘on some characterizations of the star partial ordering for matrices and rank subtractivity’, Linear Algebra Applications 82 (1986) 145-161. [9] K.S.S. Nambooripal, ‘The natural partial order on a regular semi group’ Proc. Edinburgh Math. Soc. 23 (1980) (249- 260). [10] Ramesh.G,Anbarasi.N, ‘ On inverses and Generalized inverses of Bimatrices’, International Journal of Research in Engineering and Technology, Vol.2, Issue 8, Aug 2014,PP(77-90)