IJMMSInternational Journal of Mathematics and Mathematical Sciences1687-04250161-1712Hindawi Publishing Corporation92621710.1155/2009/926217926217Research Articlek-Kernel Symmetric MatricesMeenakshiA. R.Jaya ShreeD.BellHowardDepartment of MathematicsKarpagam College of EngineeringCoimbatore 641032India200924082009200930032009210820092009Copyright © 2009This 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.

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.

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 An, 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 An, 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  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 . 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  as a particular case analogous to that of the results on complex matrices found in .

2. Preliminaries

For x=(x1,x2,,xn)1×n, let us define the function κ(x)=(xk(1),xk(2),,xk(n))Tn×1. Since K is involutory, it can be verified that the associated permutation matrix satisfy the following properties.

Since K is a permutation matrix, KKT=KTK=In and K is an involution, that is, K2=I, we have KT=K.

K=KT, K2=I, and κ(x)=Kx For An ,

N(A)=N(AK),

if A+ exists, then (KA)+=A+K and (AK)+=KA+

A+ exist if and only if AT is a g-inverse of A.

Definition 2.1 (see [<xref ref-type="bibr" rid="B6">2</xref>, page <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M81"><mml:mrow><mml:mn>119</mml:mn></mml:mrow></mml:math></inline-formula>]).

For An is kernel symmetric if N(A)=N(AT), where N(A)={x/xA=0  and  x1×n}, we will make use of the following results.

Lemma 2.2 (see [<xref ref-type="bibr" rid="B6">2</xref>, page <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M85"><mml:mrow><mml:mn>125</mml:mn></mml:mrow></mml:math></inline-formula>]).

For A,Bn and P being a permutation matrix, N(A)=N(B)N(PAPT)=N(PBPT)

Theorem 2.3 (see [<xref ref-type="bibr" rid="B6">2</xref>, page <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M89"><mml:mrow><mml:mn>127</mml:mn></mml:mrow></mml:math></inline-formula>]).

For An, the following statements are equivalent:

A is Kernel symmetric,

PAPT is Kernel symmetric for some permutation matrix P,

there exists a permutation matrix P such that PAPT=[D000] with detD>0.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M97"><mml:mrow><mml:mi>k</mml:mi></mml:mrow></mml:math></inline-formula>-Kernel Symmetric MatricesDefinition 3.1.

A matrix An is said to be k-Kernel symmetric if N(A)=N(KATK)

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=KATK automatically implies N(A)=N(KATK). However, converse needs not be true. This is, illustrated in the following example.

Example 3.3.

Let A=[000.60.5100.50.30],K=KATK=[000.60.3100.50.50].

Therefore, A is not k-symmetric.

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

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

Lemma 3.4.

For An, A+ exists if and only if (KA)+ exists.

Proof.

By [1, Theorem 3.16], For Amn 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 AATA=AATAAT=AT. Hence AT is a 2 inverse of A. Both AAT and ATA are symmetric. Hence AT=A+: A+existsAATA=AKAATA=KA(KA)(KA)T(KA)=KA(KA)T(KA){1}(KA)+,exists(By,P.4).

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 An, the following statements are equivalent:

A is k-Kernel symmetric,

KA is Kernel symmetric,

AK is Kernel symmetric,

N(AT)=N(KA),

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

Lemma 3.6.

Let An, then any two of the following conditions imply the other one,

A is Kernel symmetric,

A is k-Kernel symmetric,

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

Proof.

However, (1) and (2) (3): Aisk-KernelsymmetricN(A)=N(KATK)N(A)=N(KAT)(By,P.2)Hence,(1)and(2)N(AT)=N(A)=N((AK)T). Thus (3) holds.

Also (1) and (3) (2):

AisKernelsymmetricN(A)=N(AT)Hence,(1)and(3)N(A)=N((AK)T)N(AK)=N((AK)T)(By,P.2)AKisKernelsymmetricAisk-Kernelsymmetric(byTheorem(3.5)). Thus (2) holds.

However, (2) and (3) (1):

Aisk-KernelsymmetricN(A)=N(KATK)N(A)=N((AK)T)(by,P.2)Hence(2)and(3)N(A)=N(AT). 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:

B is k-Kernel symmetric,

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

K=[K100K2] where K1 and K2 are permutation matrices of order r and n-r, respectively,

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 y0 and partitioned in conformity with that of B=[D000]. Clearly, zN(B)=N((BT))=N((BK)T). Let us partition K as K=[K1K3K3TK2], Then KBT=[K1K3K3TK2][DT000]=[K1DT0K3TDT0]. Now

z=[0  y]N(B)=N(KBT)[0  y][K1DT  0K3TDT0]=0yK3TDT=0   Since N(DT)=0, it follows that yK3T=0.

Since each component of y0 under max-min composition yK3T=0, this implies K3T=0K3=0.

Therefore K=[K100K2]. Thus, (3) holds, Conversely, if (3) holds, then

KBT=[K1DT000],N(KBT)=N(B). 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,Bn, A is k-similar to B if there exists a permutation matrix P such that A=(KPTK)BP.

Theorem 3.9.

For An and k=k1k2 (where k1k2 as defined in Lemma 3.7). Then the following are equivalent:

A is k-Kernel symmetric of rank r,

A is k-similar to a diagonal block matrix [D000] with detD>0,

A=KGLGT and Lr with detL>  0 and GTG=Ir.

Proof.

(1)(2).

By using Theorem 2.3 and Lemma 3.7 the proof runs as follows. Aisk-KernelsymmetricKAisKernelsymmetric:PKAPT=[E000]withdetE>0forsomepermutationmatrixP(ByTheorem(2.3))A=KPT[E000]PA=(KPTK)K[E000]P(ByP.1)A=KPTK[K100K2][E000]PA=KPTK[K1E000]PA=KPTK[D000]P. Thus A is k-similar to a diagonal block matrix [D000], where D=K1E and det D>0 .

However, (2)(3): A=KPTK[K1E000]P=K[P1TP3TP2TP4T][K100K2][D000][P1P2P3P4]=K[P1TP2T]K1D[P1P2]=KGLGT,whereG=[P1TP2T],GT=[P1P2],L=K1DrGTG=[P1P2][P1TP2T]=P1P1T+P2P2T=Ir,  Lr. Hence the Proof.

Let x,y1×nȦ scalar product of x and y is defined by xyT=x,y. For any subset S1×n,  S={y:x,y=0,  forallxS}.

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)=R(A*), where N(A) is the orthogonal complement of N(A). However, this fails for a fuzzy matrix hence Cn=N(A)R(A) this decomposition fails for Kernel fuzzy matrix. Here we shall prove the partial inclusion relation in the following.

Theorem 3.11.

For An, if N(A){0}, then R(AT)N(A) and R(AT)1×n.

Proof.

Let x0N(A), since x0, xio0 for atleast one io. Suppose xi0 (say) then under the max-min composition xA=0 implies, the ith row of A=0, therefore, the ith column of AT=0. If xR(AT), then there exists y1×n such that yAT=x. Since ith column of AT=0, it follows that, ith component of x=0, that is, xi=0 which is a contradiction. Hence xR(AT) and R(AT)1×n.

For any zR(AT), z=yAT for some y1×n. For any xN(A), xA=0 and x,z=xzT=x(yAT)T=xAyT=0. Therefore, z N(A), R(AT)N(A).

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)R(AT) need not be true. These are illustrated in the following Examples.

Example 3.13.

Let A=[000.60.5100.50.30] since A has no zero columns, N(A)={0}.

For this A,R(AT)={(x,y,z):0x0.6,0y1,0z0.5}.

Therefore, R(AT)1×3.

Example 3.14.

Let A=. For this A, N(A)={(0,0,z):z},N(A)={(x,y,0):x,y}, Here, R(AT)={(x,y,0):0yx1}1×3.

Therefore, for x>y, (x,y,0)N(A) but (x,y,0)R(AT).

Therefore, N(A) is not contained in R(AT).

KimK. H.RoushF. W.Generalized fuzzy matricesFuzzy Sets and Systems198043293315MR58924410.1016/0165-0114(80)90016-0ZBL0451.20055MeenakshiA. R.Fuzzy Matrix: Theory and Applications2008Chennai, IndiaMJPHillR. D.WatersS. R.On κ-real and κ-Hermitian matricesLinear Algebra and Its Applications19921691729MR115835810.1016/0024-3795(92)90167-9BaskettT. S.KatzI. J.Theorems on products of EPr matricesLinear Algebra and Its Applications1969287103MR0251049ZBL0179.05104MeenakshiA. R.KrishnamoorthyS.On κ-EP matricesLinear Algebra and Its Applications1998269219232MR148352910.1016/S0024-3795(97)00066-9SchwerdtfegerH.Introduction to Linear Algebra and the Theory of Matrices1962Groningen, The NetherlandsP. Noordhoff