on Sums and Products of K-Tripotent Matrices

on Sums and Products of K-Tripotent Matrices

S. Krishnamoorthy and P. S. Meenakshi

Head of the Department of Mathematics, Ramanujan Research Centre, Government Arts College (Autonomous), Kumbakonam-612 001, India.

[*]Research Scholar, Ramanujan Research Centre,

Government Arts College (Autonomous), Kumbakonam-612 001, India.

Abstract: In this paper, the sums and products of k-Tripotent matrices are discussed and some related results are obtained.

Keywords: Tripotent matrices, k-Tripotent matrices, Sums and Products of k-Tripotent matrices.

Ams Subject Classification number: 15A09; 15A15;15A57

1 Introduction

R.D.Hill and S.R.Waters [1] have developed a theory of k-real and k-hermitian matrices as a generalization of secondary real and secondary hermitian matrices. S.Krishnamoothy and P.S.Meenakshi [2] have studied the basic concepts of k-Tripotent matrices as generalization of k- Tripotent matrices. J.A.Erdos [3] has initiated the study on products of idempotent matrices.

Throughout the paper, let , denote the unitary space of order n and be the space

all complex n×n matrices. Let k be a fixed product of disjoint transportation in the set of all permutation on {1,2n}. Hence it is involutory (i.e. K2=I identity permutation). If Kis the associated permutation matrix of k then it clearly satisfies the following properties.

K2=I and K= KT = =K*

2 Sums and Products of k-Tripotent matrices

Theorem 2.1. Let A and B be two k-Tripotent matrices then A B

A B

is k-Tripotent if and only if

Proof. Let A and B be k-Tripotent matrices, Therefore Assume that A B

KA3K A and KB3K B .

Now A B KA3K KB3K

K A3 B3 K

K(A B)3 K

if A B

Hence A B is k-Tripotent matrices.

Conversely, Assume A B

A B K A B3 K

is k-Tripotent matrices.

K A3 B3 3ABA BK

KA3K KB3K K

3ABA BK

A B K

3ABA BK

Hence K 3ABA BK 0 , which implies that A B .

Generalization. Let

n

n

A1 A2…………….An be k-Tripotent matrices then Ai is k-Tripotent if and only

i1

if Ai Aj At 0 for i , j and t in {1,2.n}.

i jt

Proof. Let

n 3

n n n

K Ai K K Ai Aj At K

i1 i1 j1 t 1

n

n n n

K A3

K K A A2 A A K

i i s j t

i1 i1

s1

j,t 1

j t

n n

n n

n n

K A A A K K

A A2

K K A A A K

j i t

t s

t i j

it i j

it i j

j 1 i,t 1

t 1 s1

t 1 i, j 1

n n n

n n n n

KA3 K K A A2 K K A A A K K A A2 K

i i s

i j t j s

i1

i1

n n

s1

i1

n n

j,t 1

j t

j1

n n

s1

K A A A K K

A A2

K K A A A K

j i t

t s

t i j

it i j

it i j

j 1 i,t 1

t 1 s1

t 1 i, j 1

Since s are k-Tripotent matrices, we have

n 3 n

n n

n n n n

K A K A K

A A2

K K A A A K K A A2 K

i

i i s

i j t j s

i1

i1 i1 s1 i1

n n n n

j,t 1

jt

j 1 s1

n n

K A A A K K

A A2

K K A A A K

j i t

t s

t i j

it i j

it i j

j 1 i,t 1

t 1 s1

t 1 i, j 1

(2.1.)

Here i,j,t {1,2.n}

Assume that

n

n

Ai is k- Tripotent matrices. From (2.1.) we have,

i1

n n n n

n n n n

A A K A A2 K K A A A K K A A2 K

i i i s

i j t j s

i1

i1

i1

n n

s1

i1

n n

j,t 1

j t

j 1

n n

s1

K A A A K K

A A2

K K A A A K

j i t

t s

t i j

it i j

it i j

j 1 i,t 1

t 1 s1

t 1 i, j 1

n n n n

n n n n

A A K A A2 K K A A A K K A A2 K

i i i s

i j t j s

i1

i1

i1

s1

n n

i1

j,t 1

jt

n n

j 1

s1

n n

K A A A K K

A A2

K K A A A K

j i t

t s

t i j

it i j

it i j

j 1 i,t 1

t 1 s1

t 1 i, j 1

n n

n n n n

0 K

A A2 K K A A A K K A A2 K

i s

i j t j s

i1

s1

n n

i1

j ,t 1

j t

n n

j 1

s1

n n

K A A A K K

A A2

K K A A A K

j i t

t s

t i j

it i j

it i j

j 1 i,t 1

t 1 s1

t 1 i, j 1

Hence, it follows that,

n n

n n

n n

n n

K A A2 K K A A A K K A A2 K K A A A K

i s

1. j t

2. s

j i t

j t it

j t it

i1

s1

i1

j ,t 1

j 1

s1

j 1 i,t 1

n n

n n

K A A2 K K A A A K 0

t s t i j

But,

t 1

s1

t 1

i, j 1 i j

n n

n n

n n

n n

K A A2 K K A A A K K A A2 K K A A A K

i s

1. j t

2. s

j i t

j t it

j t it

i1

s1

i1

j ,t 1

j 1

s1

j 1 i,t 1

n n

n n

K A A2 K K A A A K

A A A

t s

t i j

i j t

t 1 s1 t 1

i, j 1 i j

i j t

Hence

Ai Aj At 0 .

i jt

Conversely, Assume that

Ai Aj At 0

i jt

n n

n n n n

But

A A A K

A A2 K K A A A K K A A2 K

i j t

i s

i j t j s

i j t

i1

s1

i1

j,t 1

j t

j 1

s1

n n

n n

n n

K A A A K K

A A2

K K A A A K

j i t

t s

t i j

it i j

it i j

j 1 i,t 1

t 1 s1

t 1 i, j 1

n n

n n

n n

n n

Hence K

A A2 K K A A A K K A A2 K K A A A K

i s

i j t

j i t

i1

s1

i1

j,t 1

j 1

s1

j 1 i,t 1

n n

n n

j t it

K A A2 K K A A A K 0

t s t i j

t 1

s1

t 1

i, j 1 i j

.(2.2.)

Substitute (2.2.) in (2.1.), then we have ,

n 3 n

K Ai

K Ai

i1

n

i1

Hence Ai is k-Tripotent matrices.

i1

1 0 0 1 0 0

Example 2.2. Let

A i 1 i and B i 1 i clearly A and B are k-Tripotent

0 0 1

0 0

1

matrices. Let K be the associated permutation matrix such as,

0 0 1

K 0 1 0 Here A

1 0 0

B . Hence A+B is also a k-Tripotent matrices.

Theorem 2.3. Let A and B be k-Tripotent matrices. If AB BA then A B is also k-Tripotent matrix.

Proof. Let A and B be two k-Tripotent matrices. Since KA3K A and KB3K B . Assume that

AB BA.

AB KA3 KKB3 K

KAAAKKBBBK

KAAABBBK

KAABABBK if AB BA

KABABABK

K AB3 K , Hence the matrix A B is k-Tripotent matrix.

Generalization. If

n

A1 A2…………….An be k-Tripotent matrix belonging to a commuting family of

matrices then

Ai

i1

is a k-Tripotent matrices.

Proof. Let

A1 A2…………….An be k-Tripotent matrices.

n 3

K Ai

K K A1 A2 …………….An A1 A2…………….An A1 A2…………….An

i1

1 2

1 2

n

n

K A3 A3……………A3 K

KA3KKA3 K……………KA3 K

1 2 n

A3 A3……………A3

1 2 n

n

n

Ai

i1

n

n

Hence the matrices Ai

i1

is k-Tripotent .

1 0 0 1 0 0

Example 2.4. Let

A i 1 i and B i 1 i

0 0 1

0 0

1

0 0 1

K 0 1 0 where K is the associated permutation matrix.

1 0 0

Clearly A and B are k-Tripotent matrices.

1 0 0 1 0 0

AB 0 1 0 and BA 0 1 0

i.e AB BA. Hence A B is also a

0 0

1

0 0

1

k-Tripotent matrices.

Remark 2.5. If A and B are two k-Tripotent matrices then, A + B is k-Tripotent if and only if [ A , B ] =3 A B [ A + B ]. A B is k-Tripotent if [ A B ]=0. Where [ A , B ] be the commutator of the matrices A and B.

Lemma 2.6. Let A be a k-Tripotent matrix. Then A is Tripotent if and only if AK KA , where K is the associated permutation matrix of k.

Proof. Let A be a k-Tripotent matrix.

Assume that AK KA

Pre-multiply by K , we have

KAK A,

A3 A

Hence A is Tripotent.

Conversely, Assume that A is Tripotent matrices.

A3 A

KAK A

Pre-multiply by K , AK KA .

if A is k-Tripotent

1/ 2 1/ 2

Example 2.7. Let

A 1/ 2 1/ 2

is a tripotent matrix and it also commutes with the

0 1

1 0

1 0

associated permutation matrix K , i.e AK KA .

Note 2.8. Lemma 2.6. fails if we relax the condition of commutability of matrices A and K . A is not Tripotent then AK KA in such cases.

References

1. R.D Hill, S.R.Waters, On k-real and k-hermitian matrices, Linear.Alg. Appl.169(1992) 17-29.

2. S.Krishnamoorthy and P.S. Meenakshi, On k-Tripotent matrices, International J. of Math. Sci&Engg. Appls. 7 (1)(2013) 101-105.

3. J.A. Erdos, On products of idempotent matrices, Glasgow Math.J. 8(1967) 118-122.