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1. International Journal of Computational Engineering Research||Vol, 03||Issue, 6||
www.ijceronline.com ||June|2013|| Page 34
K-RANGE SYMMETRIC INCLINE MATRICES
P.Shakila Banu
Department of Mathematics,
Karpagam University,Coimbatore-641 021.
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(AT
).
Lemma 2.4 [5]: Let A є £mn be a regular matrix. AAT
AA†
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=KAT
K.
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
= KT
K=In, K=KT
,K2
=I —˃ 3.1
R(A) =R(KA), C(A)=C(AK) —˃ 3.2
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.

2. K-Range Symmetric Incline...
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Definition 3.3: A matrix A є £n is said to be k-range symmetric if R(A)=R(KAT
K),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 AAT
A=A [By Lemma 2.4]
KAAT
A=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(AT
K)
(vii)R(AT
)=R(AK)
(viii)A=HKAT
K for H є £n
(ix)A=KAT
KH 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(KAT
K)
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
AT
K 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(AT
K)
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 symmetricR(A)=R(KAT
K)
A=HKAT
K 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=KAT
H for H є £n

3. K-Range Symmetric Incline...
www.ijceronline.com ||June|2013|| Page 36
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 symmetricKA is normal
 (KA)(KA)T
=(KA)T
(KA)
KAAT
K=AT
KKA
KAAT
K=AT
A (by 3.1)
AAT
K=KAT
A
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(AT
A).
Proof
AT
is a g-inverse of A =˃A =AAT
A =>R(A)=RAAT
A) R(AT
A) 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(KAT
K) =>R(A)=R(AT
K)
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(KAT
K) =>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)T
KK) [XA is k-range symmetric]
=R(AT
XT
K) =R(XT
K)=R((KX)T
)
R((KA)T
)=R(AT
K)=R(XT
AT
K)=R((KAX)T
)

4. K-Range Symmetric Incline...
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=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(XT
AT
K) = 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 symmetricX 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(AT
XT
)= R(XT
) = R((KX)T
).
KA is range symmetric R(KA) =R((KA)T
)R((KX)T
)=R(KX)
KX is range symmetricX 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] A. Ben Israel, T. N. E. Greville (2003) Generalized Inverses : Theory and Applications,2nd
Ed., Springer, New York.
[2] Z. Q. Cao, K. H. Kim, F. W. Roush (1984) Incline algebra and Applications, John Wiley and Sons, New York.
[3] K. H. Kim, F. W. Roush (1980) Generalized Fuzzy Matrices, Fuzzy Sets Sys.,4, 293-315.
[4] AR. Meenakshi, S. Anbalagan (2010) EP Elements in an incline, International Journal of Algebra,4,541-550.
[5] AR. Meenakshi, P. Shakila Banu (2010) g-inverses of matrices over a regular incline, Advances in Algebra,3,33-42.
[6] AR. Meenakshi, P. Shakila Banu (2012) EP Matrices over an Incline, International Journal of Mathematical Science,11, 175-182.