-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 .
A. R. Meenakshi, Fuzzy Matrix: Theory and Applications, MJP, Chennai, India, 2008.
H. Schwerdtfeger, Introduction to Linear Algebra and the Theory of Matrices, P. Noordhoff, Groningen, The Netherlands, 1962.