Research Article | Open Access
-Kernel Symmetric Matrices
In this paper we present equivalent characterizations of -Kernel symmetric Matrices. Necessary and sufficient conditions are determined for a matrix to be -Kernel Symmetric. We give some basic results of kernel symmetric matrices. It is shown that k-symmetric implies -Kernel symmetric but the converse need not be true. We derive some basic properties of -Kernel symmetric fuzzy matrices. We obtain k-similar and scalar product of a fuzzy matrix.
Throughout we deal with fuzzy matrices that is, matrices over a fuzzy algebra with support  under max-min operations. For , , , let be the set of all matrices over , in short is denoted as . For , let , , , , , and denote the transpose, Moore-Penrose inverse, Row space, Column space, Null space, and rank of , respectively. is said to be regular if has a solution. We denote a solution of the equation by and is called a generalized inverse, in short, -inverse of . However denotes the set of all -inverses of a regular fuzzy matrix A. For a fuzzy matrix A, if exists, then it coincides with [1, Theorem ]. A fuzzy matrix A is range symmetric if and Kernel symmetric if . It is well known that for complex matrices, the concept of range symmetric and kernel symmetric is identical. For fuzzy matrix , is range symmetric, that is, implies but converse needs not be true [2, page ]. Throughout, let -be a fixed product of disjoint transpositions in and, be the associated permutation matrix. A matrix is -Symmetric if for to . A theory for -hermitian matrices over the complex field is developed in  and the concept of -EP matrices as a generalization of -hermitian and EP (or) equivalently kernel symmetric matrices over the complex field is studied in [4–6]. Further, many of the basic results on -hermitian and EP matrices are obtained for the -EP matrices. In this paper we extend the concept of -Kernel symmetric matrices for fuzzy matrices and characterizations of a -Kernel symmetric matrix is obtained which includes the result found in  as a particular case analogous to that of the results on complex matrices found in .
For , let us define the function . Since is involutory, it can be verified that the associated permutation matrix satisfy the following properties.
Since is a permutation matrix, and is an involution, that is, , we have .(P.1), , and For ,(P.2),(P.3)if exists, then and (P.4) exist if and only if is a -inverse of .
Definition 2.1 (see [2, page ]). For is kernel symmetric if , where , we will make use of the following results.
Lemma 2.2 (see [2, page ]). For and being a permutation matrix,
Theorem 2.3 (see [2, page ]). For , the following statements are equivalent:(1) is Kernel symmetric,(2) is Kernel symmetric for some permutation matrix ,(3)there exists a permutation matrix such that with .
3. -Kernel Symmetric Matrices
Definition 3.1. A matrix is said to be -Kernel symmetric if
Remark 3.2. In particular, when for each to , the associated permutation matrix reduces to the identity matrix and Definition 3.1 reduces to , that is, is Kernel symmetric. If is symmetric, then is -Kernel symmetric for all transpositions in .
Further, is -Symmetric implies it is -kernel symmetric, for automatically implies . However, converse needs not be true. This is, illustrated in the following example.
Example 3.3. Let
Therefore, is not -symmetric.
For this , , since has no zero rows and no zero columns.
. Hence is -Kernel symmetric, but is not -symmetric.
Lemma 3.4. For , exists if and only if exists.
Proof. By [1, Theorem 3.16], For if exists then which implies is a -inverse of . Conversely if is a -inverse of , then . Hence is a 2 inverse of . Both and are symmetric. Hence :
For sake of completeness we will state the characterization of -kernel symmetric fuzzy matrices in the following. The proof directly follows from Definition 3.1 and by (P.2).
Theorem 3.5. For , the following statements are equivalent:(1) is -Kernel symmetric,(2) is Kernel symmetric,(3) is Kernel symmetric,(4),(5),
Lemma 3.6. Let , then any two of the following conditions imply the other one, (1) is Kernel symmetric,(2) is -Kernel symmetric,(3).
Proof. However, () and () ⇒ ():
Thus () holds.
Also () and () ⇒ ():
Thus () holds.
However, () and () ⇒ ():
Thus, () holds.
Hence the theorem.
Toward characterizing a matrix being -Kernel symmetric, we first prove the following lemma.
Lemma 3.7. Let , where is fuzzy matrix with no zero rows and no zero columns, then the following equivalent conditions hold: (1) is -Kernel symmetric,(2),(3) where and are permutation matrices of order r and , respectively,(4) where is the product of disjoint transpositions on leaving fixed and is the product of disjoint transposition leaving fixed.
Proof. Since has no zero rows and no zero columns . Therefore and is Kernel symmetric.
Now we will prove the equivalence of (),(), and (). is -Kernel symmetric follows from By Lemma (3.6).
Choose with each component of and partitioned in conformity with that of . Clearly, . Let us partition as , Then Now
Since , it follows that .
Since each component of under max-min composition , this implies .
Therefore Thus, () holds, Conversely, if () holds, then
However, : the equivalence of () and () is clear from the definition of .
Definition 3.8. For , is -similar to if there exists a permutation matrix such that .
Theorem 3.9. For and (where as defined in Lemma 3.7). Then the following are equivalent:(1) is -Kernel symmetric of rank ,(2) is -similar to a diagonal block matrix with ,(3) and with and .
Let scalar product of and is defined by . For any subset
Remark 3.10. In particular, when reduces to the identity matrix, then Theorem 3.9 reduces to Theorem 2.3. For a complex matrix , it is well known that , where is the orthogonal complement of . However, this fails for a fuzzy matrix hence this decomposition fails for Kernel fuzzy matrix. Here we shall prove the partial inclusion relation in the following.
Theorem 3.11. For , if , then and .
Proof. Let , since , for atleast one . Suppose (say) then under the max-min composition implies, the row of , therefore, the column of . If , then there exists such that . Since column of , it follows that, component of , that is, which is a contradiction. Hence and .
For any , for some . For any , and Therefore, z , .
Remark 3.12. We observe that the converse of Theorem 3.11 needs not be true. That is , if , then and need not be true. These are illustrated in the following Examples.
Example 3.13. Let
since has no zero columns, .
Example 3.14. Let
For this ,
Therefore, for , but .
Therefore, is not contained in .
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Copyright © 2009 A. R. Meenakshi and D. Jaya Shree. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.