k -Kernel Symmetric Matrices

Volume 2009, Article ID 926217,8pages doi:10.1155/2009/926217

Research Article

k -Kernel Symmetric Matrices

A. R. Meenakshi and D. Jaya Shree

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

Correspondence should be addressed to D. Jaya Shree,shree [email protected] Received 30 March 2009; Accepted 21 August 2009

Recommended by Howard Bell

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

Copyrightq2009 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.

1. Introduction

Throughout we deal with fuzzy matrices that is, matrices over a fuzzy algebraFwith support 0,1under max-min operations. Fora, b ∈ F,ab max{a, b},a·b min{a, b}, letF_mn be the set of allm×nmatrices overF, in shortFnn is denoted asFn. ForA ∈ Fn, letA^T, A,RA,CA,NA, andρAdenote the transpose, Moore-Penrose inverse, Row space, Column space, Null space, and rank ofA, respectively.Ais said to be regular ifAXA A has a solution. We denote a solution X of the equationAXA A by A⁻ and is called a generalized inverse, in short, g-inverse ofA. HoweverA{1}denotes the set of all g-inverses of a regular fuzzy matrix A. For a fuzzy matrix A, ifA exists, then it coincides withA^T 1, Theorem 3.16. A fuzzy matrix A is range symmetric ifRA RA^Tand Kernel symmetric ifNA NA^T {x :xA 0}. It is well known that for complex matrices, the concept of range symmetric and kernel symmetric is identical. For fuzzy matrixA ∈ F_n,Ais range symmetric, that is,RA RA^TimpliesNA NA^Tbut converse needs not be true2, page 217. Throughout, letk-be a fixed product of disjoint transpositions inSn 1,2, . . . , n and, K be the associated permutation matrix. A matrixA a_ij ∈ F_n is k-Symmetric if aij akjki fori, j 1 ton. A theory fork-hermitian matrices over the complex field is developed in 3 and the concept ofk-EP matrices as a generalization of k-hermitian and EPor equivalently kernel symmetric matrices over the complex field is studied in4–6.

Further, many of the basic results onk-hermitian and EP matrices are obtained for thek- EP matrices. In this paper we extend the concept ofk-Kernel symmetric matrices for fuzzy matrices and characterizations of ak-Kernel symmetric matrix is obtained which includes the result found in2as a particular case analogous to that of the results on complex matrices found in5.

2. Preliminaries

Forx x1, x2, . . . , x_n∈ F_1×n, let us define the functionκx x_k1, x_k2, . . . , x_kn^T ∈ F_n×1. SinceK is involutory, it can be verified that the associated permutation matrix satisfy the following properties.

SinceK is a permutation matrix,KK^T K^TK I_n andK is an involution, that is, K²I, we haveK^T K.

P.1KK^T,K²I, andκx KxforA∈ Fn, P.2NA NAK,

P.3ifAexists, thenKA AKandAK KA P.4Aexist if and only ifA^Tis a g-inverse ofA.

Definition 2.1see2, page 119. ForA∈ F_nis kernel symmetric ifNA NA^T, where NA {x/xA0 andx∈ F_1×n}, we will make use of the following results.

Lemma 2.2 see2, page 125. For A, B ∈ Fn and P being a permutation matrix,NA NB⇔NPAP^T NPBP^T

Theorem 2.3see2, page 127. ForA∈ Fn, the following statements are equivalent:

1Ais Kernel symmetric,

2PAP^Tis Kernel symmetric for some permutation matrixP, 3there exists a permutation matrixPsuch thatPAP^T _D0

0 0

with det D >0.

3. k-Kernel Symmetric Matrices

Definition 3.1. A matrixA∈ Fnis said to bek-Kernel symmetric ifNA NKA^TK Remark 3.2. In particular, whenκi ifor eachi1 ton, the associated permutation matrix Kreduces to the identity matrix andDefinition 3.1reduces toNA NA^T, that is,Ais Kernel symmetric. IfAis symmetric, thenAisk-Kernel symmetric for all transpositionskin S_n.

Further, A is k-Symmetric implies it is k-kernel symmetric, for A KA^TK automatically implies NA NKA^TK. However, converse needs not be true. This is, illustrated in the following example.

Example 3.3. Let

A

⎡

⎢⎢

⎣

0 0 0.6 0.5 1 0 0.5 0.3 0

⎤

⎥⎥

⎦, K

⎡

⎢⎢

⎣ 0 0 1 0 1 0 1 0 0

⎤

⎥⎥

⎦

KA^TK

⎡

⎢⎢

⎣

0 0 0.6 0.3 1 0 0.5 0.5 0

⎤

⎥⎥

⎦.

3.1

Therefore,Ais notk-symmetric.

For thisA,NA {0}, sinceAhas no zero rows and no zero columns.

NKA^TK {0}. HenceAisk-Kernel symmetric, butAis notk-symmetric.

Lemma 3.4. ForA∈ F_n,Aexists if and only ifKAexists.

Proof. By1, Theorem 3.16, ForA ∈ F_mn ifA exists then A A^T which impliesA^T is a g-inverse ofA. Conversely ifA^T is a g-inverse ofA, thenAA^TAA⇒A^TAA^TA^T. Hence A^Tis a 2 inverse ofA. BothAA^TandA^TAare symmetric. HenceA^TA:

Aexists⇐⇒AA^TAA

⇐⇒KAA^TAKA

⇐⇒KAKA^TKA KA

⇐⇒KA^T ∈KA{1}

⇐⇒KA,exists

By,P.4 .

3.2

For sake of completeness we will state the characterization of k-kernel symmetric fuzzy matrices in the following. The proof directly follows fromDefinition 3.1and byP.2.

Theorem 3.5. ForA∈ Fn, the following statements are equivalent:

1Aisk-Kernel symmetric, 2KAis Kernel symmetric, 3AKis Kernel symmetric, 4NA^T NKA, 5NA NAK^T,

Lemma 3.6. LetA∈ Fn, then any two of the following conditions imply the other one, 1Ais Kernel symmetric,

2Aisk-Kernel symmetric, 3NA^T NAK^T.

Proof. However,1and2⇒3:

Aisk-Kernel symmetric⇒NA N

KA^TK ⇒NA N

KA^T By,P.2 Hence, 1and 2 ⇒N

A^T

NA N

AK^T .

3.3

Thus3holds.

Also1and3⇒2:

Ais Kernel symmetric⇒NA N A^T Hence, 1and3 ⇒NA N

AK^T ⇒NAK N

AK^T By,P.2 ⇒AK is Kernel symmetric

⇒Aisk-Kernel symmetric

by Theorem3.5 .

3.4

Thus2holds.

However,2and3⇒1:

Aisk-Kernel symmetric⇒NA N

KA^TK ⇒NA N

AK^T by,P.2 Hence2and3 ⇒NA N

A^T .

3.5

Thus,1holds.

Hence, Theorem.

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

Lemma 3.7. LetB_D0

0 0

, whereDisr×r fuzzy matrix with no zero rows and no zero columns, then the following equivalent conditions hold:

1Bisk-Kernel symmetric, 2NB^T NBK^T, 3K

K1 0 0 K2

whereK1andK2are permutation matrices of order r andn-r, respectively, 4k k1k2wherek1is the product of disjoint transpositions onSn {1,2, . . . , n}leaving

r1, r2, . . . , nfixed andk2is the product of disjoint transposition leaving1,2, . . . , r fixed.

Proof. Since D has no zero rows and no zero columns ND ND^T {0}. Therefore NB NB^T/{0}andBis Kernel symmetric.

Now we will prove the equivalence of1,2, and3. Bis k-Kernel symmetric ⇔ NB^T NBK^Tfollows from By Lemma3.6.

Choosez 0 ywith each component ofy /0 and partitioned in conformity with that ofB _D0

0 0

. Clearly,z ∈NB NB^T NBK^T. Let us partitionKasK _K

1K3

K^T₃ K2

, Then

KB^T

K1 K3

K₃^T K2

D^T 0 0 0

K1D^T 0 K^T₃D^T 0

. 3.6

Now

z 0y

∈NB N KB^T

⇒ 0 y

K1D^T 0 K^T₃D^T 0

0

⇒yK^T₃D^T 0

3.7

SinceND^T 0, it follows thatyK₃^T 0.

Since each component ofy /0 under max-min compositionyK^T₃ 0, this impliesK₃^T 0⇒K30.

Therefore

K

K1 0 0 K2

. 3.8

Thus,3holds, Conversely, if3holds, then

KB^T

K1D^T 0

0 0

, N KB^T

NB. 3.9

Thus1⇔2⇔3holds.

However,3⇔4: the equivalence of3and4is clear from the definition ofk.

Definition 3.8. ForA, B ∈ Fn,Aisk-similar toBif there exists a permutation matrixP such thatA KP^TKBP.

Theorem 3.9. ForA∈ Fnandk k1k2(wherek1k2as defined inLemma 3.7). Then the following are equivalent:

1Aisk-Kernel symmetric of rank r,

2Aisk-similar to a diagonal block matrix_D0

0 0

with detD >0, 3AKGLG^T andL∈ F_r with detL > 0 andG^TGI_r.

Proof. 1⇔2.

By usingTheorem 2.3andLemma 3.7the proof runs as follows.

Aisk-Kernel symmetric⇐⇒KAis Kernel symmetric :

⇐⇒PKAP^T E 0

0 0

with detE >0

for some permutation matrixP

By Theorem2.3

⇐⇒AKP^T E 0

0 0

P

⇐⇒A

KP^TK K

E 0 0 0

P By P.1

⇐⇒AKP^TK K1 0

0 K2

E 0 0 0

P

⇐⇒AKP^TK

K1E 0

0 0

P

⇐⇒AKP^TK D 0

0 0

P.

3.10

ThusAisk-similar to a diagonal block matrix_D0

0 0

, whereDK1Eand detD >0 . However,2⇔3:

AKP^TK

K1E 0

0 0

P

K

P₁^T P₃^T P₂^T P₄^T

K1 0 0 K2

D 0 0 0

P1 P2

P3 P4

K P₁^T

P₂^T

K1D P1 P2

KGLG^T, whereG P₁^T

P₂^T

, G^T P1 P2

, LK1D∈ Fr

G^TG P1 P2

P₁^T P₂^T

P1P₁^TP2P₂^TIr, L∈ Fr.

3.11

Hence the Proof.

Letx, y ∈ F1×nA˙ scalar product ofxandyis defined byxy^T x, y. For any subset S∈ F_1×n, S^⊥ {y:x, y0, for allx∈S}.

Remark 3.10. In particular, whenκi i, Kreduces to the identity matrix, thenTheorem 3.9 reduces toTheorem 2.3. For a complex matrixA, it is well known thatNA^⊥RA^∗, where NA^⊥is the orthogonal complement ofNA. However, this fails for a fuzzy matrix hence CⁿNA⊕RAthis decomposition fails for Kernel fuzzy matrix. Here we shall prove the partial inclusion relation in the following.

Theorem 3.11. ForA∈ F_n, ifNA/{0}, thenRA^T⊆NA^⊥andRA^T/F_1×n.

Proof. Letx /0∈NA, sincex /0,xio/0 for atleast oneio. Supposexi/0saythen under the max-min compositionxA 0 implies, theith row of A 0, therefore, theith column ofA^T 0. Ifx ∈RA^T, then there exists y ∈ F_1×n such thatyA^T x. Sinceith column of A^T 0, it follows that,ith component ofx0, that is,xi 0 which is a contradiction. Hence x /∈RA^TandRA^T/F1×n.

For anyz∈RA^T,zyA^Tfor somey∈ F_1×n. For anyx∈NA,xA0 and x, zxz^T

x yA^T_T xAy^T 0.

3.12

Therefore, z∈NA^⊥,RA^T⊆NA^⊥.

Remark 3.12. We observe that the converse of Theorem 3.11 needs not be true. That is , if RA^T/F_1×n, thenNA/{0}andNA^⊥⊆RA^Tneed not be true. These are illustrated in the following Examples.

Example 3.13. Let

A

⎡

⎢⎢

⎣

0 0 0.6 0.5 1 0 0.5 0.3 0

⎤

⎥⎥

⎦ 3.13

sinceAhas no zero columns,NA {0}.

For thisA, RA^T {x, y, z: 0≤x≤0.6,0≤y≤1,0≤z≤0.5}.

Therefore,RA^T/F1×3. Example 3.14. Let

A

⎡

⎢⎢

⎣ 1 1 0 0 1 0 0 0 0

⎤

⎥⎥

⎦. 3.14

For thisA,

NA {0,0, z:z∈ F}, NA^⊥

x, y,0

:x, y∈ F

, 3.15

Here,RA^T {x, y,0: 0≤y≤x≤1}/F1×3.

Therefore, forx > y∈ F,x, y,0∈NA^⊥butx, y,0/∈RA^T. Therefore,NA^⊥is not contained inRA^T.

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