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, letAT,

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 withAT 1, Theorem 3.16. A fuzzy matrix A is range symmetric ifRA RATand Kernel symmetric ifNA NAT {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 RATimpliesNA NATbut 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

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, . . . , xn∈ F1×n, let us define the functionκx xk1, xk2, . . . , xknT ∈ Fn×1.

SinceK is involutory, it can be verified that the associated permutation matrix satisfy the following properties.

SinceK is a permutation matrix,KKT KTK I_n andK is an involution, that is,

K2_{I, we haveK}T _K.

P.1KKT,K2_{I, andκx KxforA∈ F}

n,

P.2NA NAK,

P.3ifAexists, thenKA AKandAK KA

P.4Aexist if and only ifATis a g-inverse ofA.

Definition 2.1see2, page 119. ForA∈ Fnis kernel symmetric ifNA NAT, where

NA {x/xA0 andx∈ F1×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⇔NPAPT _NPBPT

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

1Ais Kernel symmetric,

2PAPTis Kernel symmetric for some permutation matrixP,

3there exists a permutation matrixPsuch thatPAPT D0 0 0

with det D >0.

3.

k

-Kernel Symmetric Matrices

Definition 3.1. A matrixA∈ Fnis said to bek-Kernel symmetric ifNA NKATK

Remark 3.2. In particular, whenκi ifor eachi1 ton, the associated permutation matrix

Kreduces to the identity matrix andDefinition 3.1reduces toNA NAT, that is,Ais Kernel symmetric. IfAis symmetric, thenAisk-Kernel symmetric for all transpositionskin

Sn.

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

⎤ ⎥ ⎥ ⎦

KAT_K ⎡ ⎢ ⎢ ⎣

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.

NKAT_{K {0}. HenceAisk-Kernel symmetric, butAis notk-symmetric.}

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

Proof. By1, Theorem 3.16, ForA ∈ F_mn ifA exists then A AT which impliesAT is a g-inverse ofA. Conversely ifAT is a g-inverse ofA, thenAATAA⇒ATAATAT. Hence

AT_{is a 2 inverse ofA. BothAA}T_andAT_{Aare symmetric. HenceA}T_A:

A_{exists⇐⇒AA}T_AA

⇐⇒KAAT_AKA

⇐⇒KAKAT_{KA KA}

⇐⇒KAT ∈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,

4NAT NKA,

5NA NAKT,

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

1Ais Kernel symmetric,

2Aisk-Kernel symmetric,

Proof. However,1and2⇒3:

Aisk-Kernel symmetric⇒NA NKATK

⇒NA NKAT _By,P.2

Hence, 1and 2 ⇒NATNA NAKT.

3.3

Thus3holds.

Also1and3⇒2:

Ais Kernel symmetric⇒NA NAT

Hence, 1and3 ⇒NA NAKT

⇒NAK NAKT _By,P.2

⇒AK is Kernel symmetric

⇒Aisk-Kernel symmetric by Theorem3.5.

3.4

Thus2holds.

However,2and3⇒1:

Aisk-Kernel symmetric⇒NA NKATK

⇒NA NAKT by,P.2

Hence2and3 ⇒NA NAT.

3.5

Thus,1holds.

Hence, Theorem.

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

Lemma 3.7. LetBD0 0 0

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

1Bisk-Kernel symmetric,

2NBT NBKT,

3KK1 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

Proof. Since D has no zero rows and no zero columns ND NDT {0}. Therefore

NB NBT_{/{0}andBis Kernel symmetric.}

Now we will prove the equivalence of1,2, and3. Bis k-Kernel symmetric ⇔

NBT _NBKT_{follows from By Lemma3.6.}

Choosez 0 ywith each component ofy /0 and partitioned in conformity with that ofB D_{0 0}0. Clearly,z ∈NB NBT NBKT. Let us partitionKasK _KKT1K3

3 K2

, Then

KBT

K1 K3

KT

3 K2

DT ₀

0 0

K1DT 0

KT

3DT 0

. 3.6

Now

z0y∈NB NKBT

⇒0 y

K1DT 0

KT

3DT 0

0

⇒yKT

3DT 0

3.7

SinceNDT 0, it follows thatyK₃T 0.

Since each component ofy /0 under max-min compositionyKT₃ 0, this impliesK₃T 0⇒K30.

Therefore

K

K1 0

0 K2

. 3.8

Thus,3holds, Conversely, if3holds, then

KBT

K1DT 0

0 0

, NKBT_{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 KPTKBP.

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 matrixD0 0 0

with detD >0,

Proof. 1⇔2.

By usingTheorem 2.3andLemma 3.7the proof runs as follows.

Aisk-Kernel symmetric⇐⇒KAis Kernel symmetric :

⇐⇒PKAPT

E 0 0 0

with detE >0

for some permutation matrixP By Theorem2.3

⇐⇒AKPT

E 0 0 0

P

⇐⇒AKPT_KK

E 0 0 0

P By P.1

⇐⇒AKPT_K

K1 0

0 K2

E 0 0 0

P

⇐⇒AKPT_K

K1E 0

0 0

P

⇐⇒AKPT_K D 0 0 0 P. 3.10

ThusAisk-similar to a diagonal block matrixD0 0 0

, whereDK1Eand detD >0 .

However,2⇔3:

AKPT_K

K1E 0

0 0 P K PT

1 P3T

PT

2 P4T

K1 0

0 K2

D 0

0 0

P1 P2

P3 P4

K PT 1 PT 2 K1D

P1 P2

KGLGT_{, whereG} PT 1 PT 2

, GT P1 P2

, LK1D∈ Fr

GT_GP

1 P2

PT

1

PT

2

P1P1TP2P2TIr, L∈ Fr.

3.11

Letx, y ∈ F1×nA˙ scalar product ofxandyis defined byxyT x, y. For any subset

S∈ F1×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}

Cn_{NA⊕RAthis decomposition fails for Kernel fuzzy matrix. Here we shall prove the} partial inclusion relation in the following.

Theorem 3.11. ForA∈ Fn, ifNA/{0}, thenRAT⊆NA⊥andRAT/F1×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 ofAT 0. Ifx ∈RAT, then there exists y ∈ F1×n such thatyAT x. Sinceith column of

AT _{0, it follows that,ith component ofx0, that is,xi 0 which is a contradiction. Hence}

x /∈RAT_andRAT_/F

1×n.

For anyz∈RAT,zyATfor somey∈ F1×n. For anyx∈NA,xA0 and

x, zxzT

xyATT

xAyT

0.

3.12

Therefore, z∈NA⊥,RAT⊆NA⊥.

Remark 3.12. We observe that the converse of Theorem 3.11 needs not be true. That is , if

RAT_/F

1×n, thenNA/{0}andNA⊥⊆RATneed 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, RAT {x, y, z: 0≤x≤0.6,0≤y≤1,0≤z≤0.5}. Therefore,RAT/F1×3.

Example 3.14. Let

A

⎡ ⎢ ⎢ ⎣

1 1 0

0 1 0

0 0 0

⎤ ⎥ ⎥

For thisA,

NA {0,0, z:z∈ F},

NA⊥_{x, y,0:x, y∈ F,} 3.15

Here,RAT {x, y,0: 0≤y≤x≤1}_/F1×3.

Therefore, forx > y∈ F,x, y,0∈NA⊥butx, y,0/∈RAT. Therefore,NA⊥is not contained inRAT.

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