k -Kernel Symmetric Matrices

Abstract

In this paper we present equivalent characterizations of k -Kernel symmetric Matrices. Necessary and sufficient conditions are determined for a matrix to be k -Kernel Symmetric. We give some basic results of kernel symmetric matrices. It is shown that k-symmetric implies k -Kernel symmetric but the converse need not be true. We derive some basic properties of k -Kernel symmetric fuzzy matrices. We obtain k-similar and scalar product of a fuzzy matrix.

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Recommended by Howard Bell

Department of Mathematics, Karpagam College of Engineering, Coimbatore 641032, India

Received 30 March 2009; Accepted 21 August 2009

1. Introduction

Throughout we deal with fuzzy matrices that is, matrices over a fuzzy algebra ... with support [0,1 ] under max-min operations. For a,b∈... , a+b=max {a,b} , a·b=min {a,b} , let _...mn be the set of all m×n matrices over ... , in short _...nn is denoted as _...n . For A∈_...n , let ^AT , ^A+ , R(A) , C(A) , N(A) , and ρ(A) denote the transpose, Moore-Penrose inverse, Row space, Column space, Null space, and rank of A , respectively. A is said to be regular if AXA=A has a solution. We denote a solution X of the equation AXA=A by ^A- and is called a generalized inverse, in short, g -inverse of A . However A{1} denotes the set of all g -inverses of a regular fuzzy matrix A. For a fuzzy matrix A, if ^A+ exists, then it coincides with ^AT [1, Theorem 3.16 ]. A fuzzy matrix A is range symmetric if R(A)=R(^AT ) and Kernel symmetric if N(A)=N(^AT )={x:xA=0} . It is well known that for complex matrices, the concept of range symmetric and kernel symmetric is identical. For fuzzy matrix A∈_...n , A is range symmetric, that is, R(A)=R(^AT ) implies N(A)=N(^AT ) but converse needs not be true [2, page 217 ]. Throughout, let k -be a fixed product of disjoint transpositions in _Sn =1,2,...,n and, K be the associated permutation matrix. A matrix A=(_aij )∈_...n is k -Symmetric if _aij =_ak(j)k(i) for i,j=1 to n . A theory for k -hermitian matrices over the complex field is developed in [3] and the concept of k -EP matrices as a generalization of k -hermitian and EP (or) equivalently kernel symmetric matrices over the complex field is studied in [4-6]. Further, many of the basic results on k -hermitian and EP matrices are obtained for the k -EP matrices. In this paper we extend the concept of k -Kernel symmetric matrices for fuzzy matrices and characterizations of a k -Kernel symmetric matrix is obtained which includes the result found in [2] as a particular case analogous to that of the results on complex matrices found in [5].

2. Preliminaries

For x=(_x1 ,_x2 ,...,_xn ) ∈_...1×n , let us define the function κ(x)=(_xk(1) ,_xk(2) ,...,_xk(n)^)T ∈_...n×1 . Since K is involutory, it can be verified that the associated permutation matrix satisfy the following properties.

Since K is a permutation matrix, K^KT =^KT K=_In and K is an involution, that is, ^K2 =I , we have ^KT =K .

(P.1): K=^KT , ^K2 =I , and κ(x)=Kx For A∈_...n ,

(P.2): N(A)=N(AK) ,

(P.3): if ^A+ exists, then (KA⁾⁺ =^A+ K and (AK⁾⁺ =K^A+

(P.4): ^A+ exist if and only if ^AT is a g -inverse of A .

Definition 2.1 (see [2, page 119 ]).

For A∈_...n is kernel symmetric if N(A)=N(^AT ) , where N(A)={x/xA=0 and x∈_...1×n } , we will make use of the following results.

Lemma 2.2 (see [2, page 125 ]).

For A,B∈_...n and P being a permutation matrix, N(A)=N(B)...N(PA^PT )=N(PB^PT )

Theorem 2.3 (see [2, page 127 ]).

For A∈_...n , the following statements are equivalent:

(1) A is Kernel symmetric,

(2) PA^PT is Kernel symmetric for some permutation matrix P ,

(3) there exists a permutation matrix P such that PA^PT =[D000] with det D>0 .

3. k -Kernel Symmetric Matrices

Definition 3.1.

A matrix A∈_...n is said to be k -Kernel symmetric if N(A)=N(K^AT K)

Remark 3.2.

In particular, when κ(i)=i for each i=1 to n , the associated permutation matrix K reduces to the identity matrix and Definition 3.1 reduces to N(A)=N(^AT ) , that is, A is Kernel symmetric. If A is symmetric, then A is k -Kernel symmetric for all transpositions k in _Sn .

Further, A is k -Symmetric implies it is k -kernel symmetric, for A=K^AT K automatically implies N(A)=N(K^AT K) . However, converse needs not be true. This is, illustrated in the following example.

Example 3.3.

Let [figure omitted; refer to PDF]

Therefore, A is not k -symmetric.

For this A , N(A)={0} , since A has no zero rows and no zero columns.

N(K^AT K)={0} . Hence A is k -Kernel symmetric, but A is not k -symmetric.

Lemma 3.4.

For A∈_...n , ^A+ exists if and only if (KA⁾⁺ exists.

Proof.

By [1, Theorem 3.16], For A∈_...mn if ^A+ exists then ^A+ =^AT which implies ^AT is a g -inverse of A . Conversely if ^AT is a g -inverse of A , then A^AT A=A[implies]^AT A^AT =^AT . Hence ^AT is a 2 inverse of A . Both A^AT and ^AT A are symmetric. Hence ^AT =^A+ : [figure omitted; refer to PDF]

For sake of completeness we will state the characterization of k -kernel symmetric fuzzy matrices in the following. The proof directly follows from Definition 3.1 and by (P.2).

Theorem 3.5.

For A∈_...n , the following statements are equivalent:

(1) A is k -Kernel symmetric,

(2) KA is Kernel symmetric,

(3) AK is Kernel symmetric,

(4) N(^AT )=N(KA) ,

(5) N(A)=N((AK^)T ) ,

Lemma 3.6.

Let A∈_...n , then any two of the following conditions imply the other one,

(1) A is Kernel symmetric,

(2) A is k -Kernel symmetric,

(3) N(^AT )=N((AK^)T ) .

Proof.

However, (1 ) and (2 ) [implies] (3 ): [figure omitted; refer to PDF] Thus (3 ) holds.

Also (1 ) and (3 ) [implies] (2 ):

[figure omitted; refer to PDF] Thus (2 ) holds.

However, (2 ) and (3 ) [implies] (1 ):

[figure omitted; refer to PDF] Thus, (1 ) holds.

Hence the theorem.

Toward characterizing a matrix being k -Kernel symmetric, we first prove the following lemma.

Lemma 3.7.

Let B=[D000] , where D is r×r fuzzy matrix with no zero rows and no zero columns, then the following equivalent conditions hold:

(1) B is k -Kernel symmetric,

(2) N(^BT )=N((BK^)T ) ,

(3) K=[_K1 00_K2 ] where _K1 and _K2 are permutation matrices of order r and n-r , respectively,

(4) k=_k1k2 where _k1 is the product of disjoint transpositions on _Sn = {1,2,...,n} leaving (r+1,r+2,...,n) fixed and _k2 is the product of disjoint transposition leaving (1,2,...,r) fixed.

Proof.

Since D has no zero rows and no zero columns N(D)=N(^DT )={0} . Therefore N(B)=N(^BT )≠{0} and B is Kernel symmetric.

Now we will prove the equivalence of (1 ),(2 ), and (3 ). B is k -Kernel symmetric ...N(^BT )=N((BK^)T ) follows from By Lemma (3.6).

Choose z=[0 y] with each component of y≠0 and partitioned in conformity with that of B=[D000] . Clearly, z∈N(B)=N((^BT ))=N((BK^)T ) . Let us partition K as K=[_K1K3^K3T_K2 ] , Then [figure omitted; refer to PDF] Now

[figure omitted; refer to PDF] Since N(^DT )=0 , it follows that y^K3T =0 .

Since each component of y≠0 under max-min composition y^K3T =0 , this implies ^K3T =0[implies]_K3 =0 .

Therefore [figure omitted; refer to PDF] Thus, (3 ) holds, Conversely, if (3 ) holds, then

[figure omitted; refer to PDF] Thus (1)...(2)...(3) holds.

However, (3)...(4) : the equivalence of (3 ) and (4 ) is clear from the definition of k .

Definition 3.8.

For A,B∈_...n , A is k -similar to B if there exists a permutation matrix P such that A=(K^PT K)BP .

Theorem 3.9.

For A∈_...n and k=_k1k2 (where _k1k2 as defined in Lemma 3.7). Then the following are equivalent:

(1) A is k -Kernel symmetric of rank r ,

(2) A is k -similar to a diagonal block matrix [D000] with det D>0 ,

(3) A=KGL^GT and L∈_...r with det L> 0 and ^GT G=_Ir .

Proof.

(1)...(2) .

By using Theorem 2.3 and Lemma 3.7 the proof runs as follows. [figure omitted; refer to PDF] Thus A is k -similar to a diagonal block matrix [D000] , where D=_K1 E and det D>0 .

However, (2 )... (3 ): [figure omitted; refer to PDF] Hence the Proof.

Let x,y∈_...1×n A... scalar product of x and y is defined by x^yT =...x,y... . For any subset S∈_...1×n , ^{S[perpendicular]} ={y:...x,y...=0, for all x∈S}.

Remark 3.10.

In particular, when κ(i)=i,K reduces to the identity matrix, then Theorem 3.9 reduces to Theorem 2.3. For a complex matrix A , it is well known that N(A^{)[perpendicular]} =R(^A* ) , where N(A^{)[perpendicular]} is the orthogonal complement of N(A) . However, this fails for a fuzzy matrix hence ^Cn =N(A)[ecedil]5;R(A) this decomposition fails for Kernel fuzzy matrix. Here we shall prove the partial inclusion relation in the following.

Theorem 3.11.

For A∈_...n , if N(A)≠{0} , then R(^AT )⊆N(A^{)[perpendicular]} and R(^AT )≠_...1×n .

Proof.

Let x≠0∈N(A) , since x≠0 , _xio ≠0 for atleast one _io . Suppose _xi ≠0 (say) then under the max-min composition xA=0 implies, the ith row of A=0 , therefore, the ith column of ^AT =0 . If x∈R(^AT ) , then there exists y∈_...1×n such that y^AT =x . Since ith column of ^AT =0 , it follows that, ith component of x=0 , that is, _xi =0 which is a contradiction. Hence x∉R(^AT ) and R(^AT )≠_...1×n .

For any z∈R(^AT ) , z=y^AT for some y∈_...1×n . For any x∈N(A) , xA=0 and [figure omitted; refer to PDF] Therefore, z ∈N(A^{)[perpendicular]} , R(^AT )⊆N(A^{)[perpendicular]} .

Remark 3.12.

We observe that the converse of Theorem 3.11 needs not be true. That is , if R(^AT )≠_...1×n , then N(A)≠{0} and N(A^{)[perpendicular]} ⊆R(^AT ) need not be true. These are illustrated in the following Examples.

Example 3.13.

Let [figure omitted; refer to PDF] since A has no zero columns, N(A)={0} .

For this A,R(^AT )={(x,y,z):0≤x≤0.6,0≤y≤1,0≤z≤0.5}.

Therefore, R(^AT )≠_...1×3 .

Example 3.14.

Let [figure omitted; refer to PDF] For this A , [figure omitted; refer to PDF] Here, R(^AT )={(x,y,0):0≤y≤x≤1}≠_...1×3 .

Therefore, for x>y∈... , (x,y,0)∈N(A^{)[perpendicular]} but (x,y,0)∉R(^AT ) .

Therefore, N(A^{)[perpendicular]} is not contained in R(^AT ) .

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