Secondary κ-kernel Symmetric Fuzzy Matrices

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1 Intern. J. Fuzzy Mathematical rchive Vol. 5, No. 2, 24, ISSN: (P), (online) Published on 2 December 24 International Journal of Secondary κ-ernel Symmetric Fuzzy Matrices D.Jaya Shree Department of Mathematics, mrita Vishwa Vidyapeetham, mrita University Bangalore 5635, India. jayashreekce@gmail.com bstract. In this paper, characterizations of secondary κ- kernel symmetric fuzzy matrices are obtained. Relation between s- κ- kernel symmetric, s- kernel symmetric, κ- kernel symmetric and kernel symmetric matrices are discussed. Necessary and sufficient conditions are determined for a matrix to be s- κ- kernel symmetric. eywords: Fuzzy matrices, kernel symmetric, s-κ- kernel symmetric MS Mathematics Subject Classification (2): 5B5, 5B57. Introduction ll matrices considered in this paper are fuzzy matrices, that is, matrices over a fuzzy algebra F with support [, ] under max-min operations. fuzzy matrix is range symmetric if R ( ) R( ) and kernel symmetric if N ( ) ). It is well known that for complex matrix, the concept of range and kernel symmetric are same. However this fails for fuzzy matrices. his motivated us to study on s- κ- kernel symmetric matrices. Lee [] has initiated the study of secondary symmetric matrices, that is matrices whose entries are symmetric about the secondary diagonal. Cantoni and Paul [2] have studied persymmetric matrices, that is matrices which are symmetric about both the diagonals and their applications to communication theory. Hill and Waters [3] have developed a theory of κ-real and κ-hermitian matrices as a generalization of s-real and s- hermitian matrices. development of κ- kernel symmetric fuzzy matrices is made by Meenakshi and Jayashree [5] analogous to that of k-real and k-hermitian of a complex matrix [3]. hroughout let κ-be a fixed product of disjoint transpositions in and be the associated permutation matrix. matrix (a ij ) F n is κ- symmetric if a ij a k ( j) k ( i) for i, j to n. Meenakshi and krishnamoorthy[6] have introduced the concept of s-k hermitian matrices as a generalization of secondary hermitian and hermitian matrices. In this paper, we extend the concept of s- κ- kernel symmetric fuzzy matrices as a particular case of the results on complex matrices found in [7]. 2. Preliminaries hroughout let be the permutation matrix with units in its secondary diagonal and let S n,2,..., n and be the κ be a fixed product of disjoint transpositions in { } 89

2 Jaya Shree associated permutation matrix. For ( ) x x x,..., F n let us define the function, 2 F n. Since is involutory, it can be verified that the associated permutation matrix satisfy the following properties. 2 (P.2.) I,, I and R ( x ) x By the definition of V, (P.2.2) V V, VV V V I n and V 2 I (P.2.3) N ( ) V ), N ( ) ) (P.2.4) ( V ) V,( V) If exists, then (P.2.5) ( V ) V,( V) n V + V Definition 2.. [4] F n is kernel symmetric matrix if and only if N ( ) ). Lemma 2.. [[4] P. 9] For F n and a permutation matrix P, N ( ) B) if and only if N ( PP ) PBP ). Lemma 2.2. [5] matrix F n is κ- kernel symmetric is kernel symmetric is kernel symmetric. x n 3. Secondary κ-kernel symmetric fuzzy matrices Definition 3.. matrix F n is s-symmetric if and only if V V. Definition 3.2. matrix F n is s-kernel symmetric if N( ) V V ). Definition 3.3. matrix F n is s- κ-kernel symmetric if N( ) V V). Lemma 3.. matrix F n is s-kernel symmetric is kernel symmetric is kernel symmetric. Proof. is s-kernel symmetric N( ) V V ) [By Definition 3.2] N ( V ) ( V ) ) [By P.2.2] V is kernel symmetric N( VVV ) VV V ) [By Lemme2.] N ( V) ( V) ) [By P.2.2] V is kernel symmetric. Remark 3.. In particular when κ(i) i for i, 2,, n then the associated permutation matrix reduces to the identity matrix and Definition (3.3) reduces to N( ) V V ) which implies that is s-kernel symmetric matrices. 9

3 Secondary κ-kernel Symmetric Fuzzy Matrices 9 Remark 3.2. For κ, the corresponding permutation matrix reduces to and Definition (3.3) reduces to ) ( ) ( N N which implies that is kernel symmetric. Remark 3.3. We note that s- κ-symmetric matrix is s-κ-kernel symmetric for if is s-κsymmetric then V V Hence ) ( ) ( V V N N which implies that is s- κ -kernel symmetric. However the converse need not be true. his is illustrated in the following example. Example 3.. For κ (,2), is symmetric V V Here therefore is symmetric, κ -symmetric, s- κ -kernel symmetric but not s- κ -symmetric. Example 3.2. For κ (,2), V is symmetric, s-κ-symmetric and hence therefore s- κ-kernel symmetric. Example 3.3. For κ (,2)(3) and V here I V, and V V. Now is s- κ-kernel symmetric but not s- κ -symmetric V V

4 Jaya Shree Hence is not s- κ-symmetric. But N( ) V V) {}. heorem 3.. For F n the following are equivalent () is s- κ -kernel Symmetric (2) is kernel symmetric (3) is kernel symmetric (4) is kernel symmetric (5) is kernel symmetric (6) is κ-kernel symmetric (7) is κ -kernel symmetric (8) is s-kernel symmetric (9) is s-kernel symmetric () () () (4) (5) (9) is s- κ-kernel symmetric [By Definition 3.2 ] [By P.2.3] is kernel symmetric is kernel symmetric [By Lemma 2.] is kernel symmetric is s-kernel symmetric hus () (4) (5) (9) hold. (2) (6) is kernel symmetric is κ -kernel symmetric hus (2) (6) hold. (2) () is kernel symmetric hus (2) () hold. (4) () is kernel symmetric hus (4) () hold. () (4) (7) is s- κ -kernel symmetric 92 [By P.2.3] [By P.2.3]

5 Secondary κ-kernel Symmetric Fuzzy Matrices is kernel symmetric is κ -kernel symmetric. hus () (4) (7) hold. (3) (8) is kernel symmetric is s- κ -kernel symmetric. Hence the heorem. In Particular for, the above heorem reduces to the equivalent condition for a matrix to be secondary kernel symmetric. Corollary 3.. For F n the following are equivalent () is s-kernel symmetric (2) is kernel symmetric (3) is kernel symmetric (4) (5) Lemma 3.2. Let F n, if exists exists exists. exists () + exists [follows from Lemma 3.4 in [8]] exists. Lemma 3.2. Let F n, if exists exists exists. exists () + exists [follows from Lemma 3.4 in [8]] exists. Remark 3.4. For F n, exists exists. heorem 3.2. Let F n. hen any two of the following conditions imply the other one. () is κ -kernel symmetric (2) is s- κ -kernel symmetric (3) () and (2) (3) is s- κ -kernel symmetric [By heorem 3.] 93

6 Jaya Shree [By Lemma 2.] is κ -kernel symmetric [By Lemma 2.] Hence () and (2) hus (3) hold. () and (3) (2) is κ -kernel symmetric Hence () and (3) [By Lemma 2.] is s- κ -kernel symmetric [By heorem 3.] hus (2) hold. (2) and (3) () is s- κ-kernel symmetric [By Lemma 2.] Hence (2) and (3) is κ -kernel Symmetric hus () hold. Hence the theorem. REFERENCES..Lee, Secondary Symmetric, Secondary Skew Symmetric, Secondary Orthogonal Matrices, Period Math, Hungary, 7 (976) C.ntonio and B.Paul, Properties of the eigen vectors of persymmetric matrices with applications to communication theory, IEEE rans. Comm., 24 (976) R.D. Hill and S.R.Waters, On k-real and k-hermitian matrices, Linear lgebra and its pplications, 69 (992) R.Meenakshi, Fuzzy Matrix: heory and pplications, MJP Publishers, Chennai, R.Meenakshi and D.Jaya Shree, On k-kernel symmetric matrices, International Journal of Mathematics and Mathematical Sciences, 29, rticle ID 92627, 8 Pages. 6. R.Meenakshi and S.rishanmoorthy, On Secondary k-hermitian matrices, Journal of Modern Science, (29) R. Meenakshi, S.rishnamoorthy and G.Ramesh, On s-k-ep matrices, Journal of Intelligent System Research, 2(28) R.Meenakshi and D.Jaya Shree, On -range symmetric matrices, Proceedings of the National conference on lgebra and Graph heory, MS University, (29),