International Journal of Computational Engineering Research||Vol, 03||Issue, 6||
K-RANGE SYMMETRIC INCLINE MATRICES P.Shakila Banu Department of Mathematics, Karpagam University,Coimbatore-641 021.
ABSTRACT The concept of generalized k-symmetric incline matrices is introduced as a development of the complex k - EP matrix and k -EP fuzzy matrix. A set of necessary and sufficient conditions are determined for a incline matrix to be k-range symmetric. Further equivalent characterization of k-range symmetric matrices are established for incline matrices and also the existence of various g-inverses of a matrix in £n has been determined. Equivalent conditions for various g-inverses of a k-range symmetric matrix to be k-range symmetric are determined. Generalized inverses belonging to the sets A {1,2} , A{1,2,3} and A{1,2,4} of a k-range symmetric matrix A are characterized. MS classification: 16Y60, 15B33.
KEY WORDS: Incline matrix, k-range symmetric, G-inverse.
I.
INTRODUCTION:
Incline is an algebraic structure and is a special type of a semiring. Inclines are additively idempotent semirings in which products are less than or equal to either factor. The notion of inclines and their applications are described comprehensively in Cao, Kim and Roush [2]. Kim and Roush [3] have studied the existence and construction of various g-inverses for matrices over the Fuzzy algebra analogous to that for complex matrices [1]. In [5], the authors have discussed the existence and construction of various g-inverses and Moore-Penrose inverse associated with a matrix over an incline whose idempotent elements are linearly ordered. In [4], the equivalent conditions for EP-elements over an incline are determined. In [6], the equivalent conditions for EP matrix are discussed.
II.
PRELIMINARIES:
In this section, some basic definition and required results are given. Definition 2.1: A non empty set £ with two binary operations ‘+’ and ‘·‘ is called an incline if it satisfy the following conditions. 1) (£,+) is a semilattice. 2) (£,·) is a semigroup. 3) x(y+z) = xy+xz for all x,y,zє£ 4) x+xy = x and y+xy=y for x,y є£ Definition 2.2: For a matrix A є £nm. Consider the following 4 equations 1) AXA=A 2) XAX=X 3) (AX)T=AX 4) (XA)T =XA. Here , AT is the transpose of A. X is said to be a inverse of A and X єA{1} if X satisfies λ-equation , where λ is a subset of {1,2,3,4}.In particular, if λ = {1,2,3,4} then X is called the Moore-Penrose inverse of A and it denoted as A†. Definition 2.3[6]: A є £n is range symmetric incline matrix if and only if R(A)=R(A T). Lemma 2.4 [5]: Let A є £mn be a regular matrix. AATAA† exists and A†=AT. III. K-RANGE SYMMETRIC INCLINE MATRICES Definition 3.1: A matrix A є £n is said to be k-symmetric if A=KATK. Note 3.2: Throughout, let ‘k’ be a fixed product of disjoint transpositions in Sn = {1,2,,…,n} and K be the associated permutation matrix. We know that KKT = KTK=In, K=KT,K2=I —˃ 3.1 R(A) =R(KA), C(A)=C(AK) —˃ 3.2 www.ijceronline.com
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K-Range Symmetric Incline... Definition 3.3: A matrix A є £n is said to be k-range symmetric if R(A)=R(KATK),where R(A)={x є £n / x = yA for some y є £n}. Lemma 3.4: Let A є £mn be a regular matrix. For A† exists then (KA)† exists. Proof A† exists AATA=A [By Lemma 2.4] KAATA=KA (KA)(KAT)(KA)=KA KAT є (KA){1} [By Defn.2.2 ] (KA)† exists. Theorem 3.5: Let A є £n, the following are equivalent (i) A is k- range symmetric (ii)KA is range symmetric (iii)AK is range symmetric (iv)AT is k-range symmetric (v)C(KA)=C((KA)T) (vi)R(A)=R(ATK) (vii)R(AT)=R(AK) (viii)A=HKATK for H є £n (ix)A=KATKH for H є £n (x)AT=KAKH1 for H1 є £n (xi)AT=H1AK for H1 є £n Proof (i) (ii) (iii) A is k-range symmetric R(A)=R(KATK) R(KA)=R((KA)T) KA is range symmetric KKAKT is range symmetric [By 3.1.] AK is range symmetric Thus (i) (ii) (iii) hold. (i) (iv) A is k-range symmetric KA is range symmetric (KA)T is range symmetric ATK is range symmetric AT is k-range symmetric Thus (i) (iv) hold. (ii) (v) KA is range symmetric R(KA)=R((KA)T) C((KA)T)=C(KA) Thus (ii) (v) hold. (ii) (vi) KA is range symmetric R(KA)=R((KA)T) R(A)=R(ATK) Thus (ii) (vi) hold. (iii) (vii) AK is range symmetric R(AK)=R((AK)T) R(AK)=R(AT) Thus (iii) (vii) hold. (i) (viii) (x) A is k-range symmetricR(A)=R(KATK) A=HKATK for H є £n AT=KAKH1 [where H1=HT] Thus (i) (viii) (x) hold. (iii) (ix) (xi) AK is range symmetric R(AK)=R((AK)T) R(AK)=R(AT) C((AK)T)=C(A) A=KATH for H є £n www.ijceronline.com
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K-Range Symmetric Incline... AT=H1AK [by taking transpose] Thus (iii) (ix) (xi) hold. Lemma 3.6: For A є £mn, A† exists then the following are equivalent: (i) A is k-range symmetric (ii) (KA)(KA)†=(KA)†(KA) (iii) A† is k-range symmetric (iv) KA is normal. Proof Since A† exists by Lemma (2.4 ) A† =AT and by using Lemma (3.3), (KA)†=A†K exists. Then the proof directly follows from Theorem (3.5). Lemma 3.7: For A є £n, A† exists then the following are equivalent: (i) A is k-range symmetric (ii) AA†K=KA†A (iii) KAA†=A†AK Proof Since A† exists by Lemma(2.4) A is k-range symmetricKA is normal (KA)(KA)T=(KA)T(KA) KAATK=ATKKA KAATK=ATA (by 3.1) AATK=KATA AA†K=KA†A [By Lemma 2.4 ] AA†K=KA†A Thus (i) (ii) holds. Since by (3.1) ,K2=I, the equivalence follows by pre and post multiplying AA†K=KA†A by K. Thus (ii) (iii) holds. Lemma 3.8: AT is a g-inverse of A implies R(A) =R(ATA). Proof AT is a g-inverse of A =˃A =AATA =>R(A)=RAATA) R(ATA) R(A) Therefore R(A)=R(AAT) Theorem 3.9: Let A є £n. Then any two of the following conditions imply the other one: (i) A is range symmetric (ii) A is k-range symmetric (iii) R(A)=R(KA) T
Proof (i)and (ii) => (iii) A is k-range symmetric => R(A)=R(KATK) =>R(A)=R(ATK) Hence (i) and (ii) => R(AT)=R(KA)T Thus (iii) holds. (i)and (iii) => (ii) A is range symmetric => R(A)=R(AT) From (i) and (ii) => R(A)=R(KA)T=>R(KA)=R(KA)T => KA is range symmetric => A is k-range symmetric Thus (ii) holds. (i)and (iii) => (i) A is k-range symmetric => R(A) =R(KATK) =>R(A)=R((KA)T) =>R(A)=R(AT) Thus (i) hold. Theorem 3.10: Let A є £n, X є A{1,2} and AX and XA are k-range symmetric. Then A is k-range symmetric X is k-range symmetric. Proof R(KA)=R(KAXA) R(XA) =R(XKKA) R(KA) Hence, R(KA) =R(XA) =R(K(XA)TKK) [XA is k-range symmetric] =R(ATXTK) =R(XTK)=R((KX)T) R((KA)T)=R(ATK)=R(XTATK)=R((KAX)T) www.ijceronline.com
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K-Range Symmetric Incline... =R(KAX) [AX is k-range symmetric] =R(KX) KA is range symmetric R(KA)=R(KA)T R((KX)T)=R(KX) KX is range symmetric X is k-range symmetric Theorem 3.11: Let A є £n, X є A{1,2,3},R(KA)=R((KX)T). Then A is k-range symmetric X is k-range symmetric. Proof Since X є A{1,2,3} we have AXA=A,XAX=X,(AX)T=AX R((KA)T) = R(XTATK) = R(K(AX)T)= R((AX)T)= R(AX) [(AX)T=AX] = R(X) = R(KX) KA is range symmetric R(KA)=R((KA)T)R((KX)T)=R(KX) KX is range symmetricX is k-range symmetric. Theorem 3.12: Let A є £n, X є A{1,2,4},R((KA)T)=R(KX).Then A is k-range symmetric X is k-range symmetric. Proof Since X є A{1,2,4},we have AXA=A,XAX=X, (XA)T=XA R(KA) = R(A) = R(XA)= R((XA)T)= R(ATXT)= R(XT) = R((KX)T). KA is range symmetric R(KA) =R((KA)T)R((KX)T)=R(KX) KX is range symmetricX is k-range symmetric.
ACKNOWLEDGEMENT This work was supported by Grants-in-Aid of Women’s Of Scientists-A, Department of Science and Technology, New Delhi.
REFERENCES [1] [2] [3] [4] [5] [6]
A. Ben Israel, T. N. E. Greville (2003) Generalized Inverses : Theory and Applications,2nd Ed., Springer, New York. Z. Q. Cao, K. H. Kim, F. W. Roush (1984) Incline algebra and Applications, John Wiley and Sons, New York. K. H. Kim, F. W. Roush (1980) Generalized Fuzzy Matrices, Fuzzy Sets Sys.,4, 293-315. AR. Meenakshi, S. Anbalagan (2010) EP Elements in an incline, International Journal of Algebra,4,541-550. AR. Meenakshi, P. Shakila Banu (2010) g-inverses of matrices over a regular incline, Advances in Algebra,3,33-42. AR. Meenakshi, P. Shakila Banu (2012) EP Matrices over an Incline, International Journal of Mathematical Science,11, 175-182.
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