International Journal of Mechanical Engineering and Technology (IJMET)
(IJM
Volume 10, Issue 1, January 2019,
201 pp. 450–456, Article ID: IJMET_10_01_046
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nd ISSN Online: 0976-6359
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THE DIAGONALIZATION MATRIX OF THE ⊗ ( ≡∗
Dunya Mohamed Hamed
College of Education,
Education University of Mustansiriyha, Baghdad, Iraq
,
)
ABSTRACT
The aim of this paper is to determine the diagonalization of the ⊗ (≡
( ∗ T , ), where
(≡∗ T , ) is the tensor product
product of the matrix of the rational valued character table
⊗
of the group T , by itself nn times, where p, q are prime number, p> and q | p−1.
Keywords: Tensor Product, The rational character table, the group T
,
Cite this Article: Dunya Mohamed Hamed,
Hamed The Diagonalization Matrix of The ⊗ (
∗
≡
ernational Journal of Mechanical Engineering and Technology,
Technology 10(1),
, ), International
2019, pp. 450–456.
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The tensor product of two matrices and the rational character table of the group T , has been
given in respectively [1], [5].
In this work, we found two matrices P, Q and we give some concepts that we shall use to
determine the diagonal matrix of the tenser product of the matrix
matrix rational character table of
group T , of n- times of itself where p, q are prime number, p> and q | p−1.
−1.
1. INTRODUCTION
Preliminaries
Some definition and basic concepts have been given in this section
Definition (2-1), [1]: Let A∈M K , B∈M
a B a B ⋯a B
a B a B ⋯a B
A⊗B=
⋯ ⋮
⋮
⋮
a B a B ⋯a B
Where,
A=
a
a
a
a
⋯a
⋯a
⋯ ⋮
⋯a
b
b
⋮
b
⋯b
⋯b
⋯ ⋮
⋯b
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450
a
⋮
a
⋮
b
b
and B=
⋮
b
K , we define a matrix A⊗ B ∈M
K by :
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The Diagonalization Matrix of The ⊗ ( ≡∗
Thus,
α
α
A⊗B= ⋮
α
Where
α
α
⋮
α
a b
a b
α =
⋮
a b
a b
a b
⋮
a b
a b
a b
α## =
⋮
a b
a b
a b
⋮
a b
a b
a b
α #=
⋮
a b
⋯α
⋯α
⋯ ⋮
⋯α
a b
a b
⋮
a b
⋯a
⋯a
⋯
⋯a
⋯a
⋯a
⋯
⋯a
⋯a
⋯a
⋯
⋯a
,
)
b
b
⋮
b
b
b
⋮
b
b
b
⋮
b
and k=nm
3 0 0
−1 0
Example (2-2):Consider A=$
), B=*−1 1 0 , , then
0 2
0 2 −1
−3 0 0 0
0 0
/ 1 −1 0 0
0 04
. 0 −2 1 0
0 03
A ⊗ B =.
0 03
0
0 0 6
.0
0 0 −2 2 0 3
-0
0 0 0 4 −225
Proposition (2-3), [1]:
5
Let A ,Aˋ be two different matrices inM K and B , Bˋ be two different matrices inM
then
1- A + Aˋ ⊗ B = A⊗ B + Aˋ ⊗ B .
2- A⊗ B . Aˋ ⊗Bˋ = AAˋ ⊗ BBˋ .
3- det A⊗ B = det A
. det B
.
K ,
Definition (2-4), [2]:
Let T be a matrix representation of finite group G over a field F , then the character χ of T is
a mapping χ : G ⟶ F define by χ(g)=Tr( T(g) ) refers to the trace of the matrix T(g) .
Clearly χ(1) = n , which is called the degree of χ . Also, characters of degree 1 are called
linear characters.
Example (2-5):
In symmetric group S@ =< x,y : x =y @ =1,xy= y x> , define the representation T : S@ ⟶ GL(2 , ₵ )
EF
0 1
w 0
such that : T(x)= $
) and T(y)= $
) , where w = D G@ , then the character χ of T is
1 0
0 w
χ (T(x) )= 0+0=0 , χ (T(y) )= w +w = −1
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451
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Balasem A. Al-Quraishi, Nor Zelawati Binti Asmuin, Mohammed Najeh Nemah and Salih Meri A
Definition (2-6), [2]:
The character afforded by irreducible representation is called irreducible character;
otherwise it is called compound character.
Example (2-7): Linear characters are irreducible character.
Definition (2-7), [3]:
A class function on a group G is a function f : G ⟶₵ which is constant on conjugate classes ,
that is f(x O y x ) = f(y) , ∀ x ,y ∈ G , if all values of f are in Z , then it is called Z-valued
class function .
Proposition (2-8), [3]: characters are class function.
Proof:
Let T be matrix representation and χ character of T, then
χ (x O y x ) = Tr( T (x O y x ) ) = Tr ( T (x O ) T(y ) T(x ) )
= Tr (T (x O ) T(x) T(y))
=Tr (T(y)) =χ(y)
Proposition (2-9), [5]:
Let p and q be two prime numbers such that p>q and q|p-1, then the rational character table of
the group T , is
(≡∗
,
)=
K
K
K@
1
1
1
g
q−1
q−1
−1
g
g@ p−1 −1 0
Definition (2-10), [3]::
A rational valued character θof G is a character whose values are in Z,That is θ(x) ∈
Z , ∀x ∈ G .
Theorem (2-11), [6]:
Let M be anm × n matrix with entries in a principal domain R, then thereexist matricesP,Q, D
such that:
1- P and Q are invertible
2- QMP O =D
3- D is diagonal matrix
4- If we denoted D`` by d` , then there exists a natural number r , 0 ≤ r ≤ min(m ,n)
such that j>r implies dd = 0 and j≤ r implies dd ≠ 0 and 1 ≤ j ≤ r implies dd
divides ddf .
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The Diagonalization Matrix of The ⊗ ( ≡∗
,
)
Definition (2-12), [6]:
Let M be a matrix with entries in a principal domain R , be equivalent to a matrixD= diag{ d
, d , ......., dh ,0 ,0 , ......., 0} such that dd ∕ ddf for 1 ≤ j ≤ r , we call D the invariant factor
matrix of M and ij , ik , ......., il the invariant factor of M .
Theorem (2-12), [6]:Let Mbe a matrix with entries in a principal domain R, thenthe invariant
factor areunique.
Theorem (2-13), [4]:
Let A, B are two matrices nonsingular matrices of degree n, m respectively over principal
domain R, and let
P A Q = D(A) =Diag { d (A), d (A), ⋯, d (A)},
P B Q = D(B) =Diag { d (B), d (B), ⋯, d (B)}, be the invariant factor matrices of A and B,
Then, (P ⊗P ).(A⊗B). ( Q ⊗Q ) =D (A)⊗D (B)
And, from this the invariant factor matrices of A⊗ B can be written down
Let H and L be P and P −groups respectively, where P and P are distinct primes, we
know that:
≡(H×L)= ≡(H) ⊗ ≡(L) ,since gcd(P , P )=1 , we have
≡∗(H×L) = ≡∗ (H) ⊗≡∗ (L).
The Diagonal Matrix of The ⊗( ≡∗
,
):
In this section, we found two matrices P and Qto determine the diagonal matrix of the⊗
(≡∗ T , ), where ⊗ (≡∗ T , ) denote to the tenser product of the matrix rational character
table of groupT , by itself n- times. Where p, q are prime number, p> and q | p−1.
We apply theorem (2-13) to determine the diagonal of ⊗(≡∗ T
1
p
1
1
0
0
Let p=*−1 −1 0 ,and Q=*−1 0 −1,
−p −p 0
−1 0 −1
be two matrices which is the invariant factor matrix for ≡∗ T
≡ T
∗
where
1
1
1
q
−
1
q
−
1
−1
=*
,
p − 1 −1
0
,
Hence, by theorem (2-13) we get
P. (≡∗ T
,
).
,
,
Hence, by Theorem (2-13) ⟹ D( (≡∗ T
,
−p
0
).Q = * 0 −qp
0
0
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0
0,
−p
) )= diag {−p , −qp , −p }
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Balasem A. Al-Quraishi, Nor Zelawati Binti Asmuin, Mohammed Najeh Nemah and Salih Meri A
Now, we consider explicitly the case n=2, then
1 0 0
0 0 0
/−1 −1 0 0 0 0
.−1 0 −1 0 0 0
.
.−1 0 0 −1 0 0
1 1 0
P⊗P=. 1 1 0
1 0 1
. 1 0 1
0 0 0
.−1 0 0
. 1 1 0
0 0 0
- 1 0 1
0 0 0
0
0
0
0
0
0
−1
1
1
0 0
0 04
0 03
3
0 03
0 03
,
0 03
0 03
1 03
0 12o×o
1 P
1
1
P
1
P
P P
/ −1 0 −1
−P 0 −P
−1 0 −1 4
.
3
−P
0 −P −P
0 −P −P 0 3
. −P
0
0 −1 −P −13
0
. −1 −P −1
0
1
0
13
Q⊗Q=. 1
0
1
0
0
P
0
P
P
03
0
0
0
. P
0
03
. −P −P −P −P −P −P 0
0
0
P
P
03
. P
0
P
0
-P
0
0
0
0
0 2o×o
P
P
P
And, (≡∗ T
/
.
.
.
.
.(
.
.
.(
.
-
,
)⊗(≡∗ T
,
)=⊗ (≡∗ T , )=
1
1
1
1
1
−1
−
1
− 1 −1
−1
p−1
p−1
−1
0
0
−1
−1
−1
−1
−1
( − 1)
( − 1)
1−
( − 1) 1 −
− 1)(p − 1) 0 ( − 1)(p − 1) 1 −
0
p−1
p−1
−1
−1
−1
1
− 1)(p − 1) 1 − p
1−
1−
0
0
1−p
1
(p − 1)
So, we obtain
(P⊗P). (⊗ (≡∗ T
,
p 0
/
0 qp
.
.0 0
.0 0
)) . (Q⊗Q)=. 0 0
.0 0
.0 0
.0 0
-0 0
0 0 0
0 0 0
p 0 0
0 qp 0
0 0 q p
0 0 0
0 0 0
0 0 0
0 0 0
0
0
0
0
0
qp
0
0
0
Hence, by Theorem (2-13) ⟹ D ( ⊗ (≡∗ T
,p }
,
1
1 14
− 1 − 1 −1 3
p − 1 −1 0 3
−1 −1 −1 3
1 − 1 − −1 3
1−p 1 1 3
0 0 3
0
3
0 0 3
0
0 0 3
0
2o×o
0 0
0 0
0 0
0 0
0 0
0 0
p 0
0 qp
0 0
0
04
03
3
03
03
03
03
03
p 2o×o
))= diag {p , qp ,p , qp ,q p , qp ,p , qp
We, consider explicitly the case n=3, then we obtain
@
(P⊗P⊗P). ( ⊗
(≡∗ T
,
@
) ) . (Q⊗Q⊗Q)=D( ⊗
(≡∗ T
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,
454
))=
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The Diagonalization Matrix of The ⊗ ( ≡∗
−
/
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
-
q
−
q
−
q
−
q
−
k
q
−
q
−
q
−
q
−
q
−
q
−
k
q
−
q
−
k
q
−
q
q
−
k
q
−
q
−
k
,
q
−
)
q
−
q
−
q
−
q
−
q
−
k
q
−
q
Hence, by Theorem (2-13) we get:
@
D( ⊗
(≡∗ T
) )= diag {−p3 ,−qp@ ,− p@ ,−qp@ ,−q p@,−qp@ ,− p@ ,−qp@ ,−p@ ,−q p@ , −q p@,
−qp@ , −q p@, −q@ p@ − q p@ ,−qp@ , −q p@ , −qp@ , −p@ ,−qp@ ,− p@ , qp@ ,−q p@,−qp@ ,− p@ ,−qp@
,−p@ } .
,
We, consider explicitly the case n=4, then we obtain
s
(P⊗P⊗P⊗P). ( ⊗
(≡∗ T
(−p
/
.
.
.
.
.
.
.
.
.
.
.
.
-
,
,
s
) ). (Q⊗Q⊗Q⊗Q)=D( ⊗
(≡∗ T
− p
,
−p )
⊗
q
(⊗
,
))=
( −p
,
− p
,
4
3
3
3
3
3
3
3
3
3
3
3
3
−p )2t
×t
s
s
Therefore,
(P⊗P⊗P⊗P). ( ⊗
(≡∗ T , ) ). (Q⊗Q⊗Q⊗Q)=D( ⊗
(≡∗ T , ) )=
{ ps , qps , ps ,qps ,q ps , qps ,ps ,qps ,ps ,q ps , q ps , qps , q ps ,q@ ps q ps , qps ,q ps ,
ps , qps , ps ,qps ,q ps , qps ,ps ,qps ,ps ,q ps , q ps , qps , q ps ,q@ ps , q ps , qps ,q ps , qps ,
q@ ps ,q ps ,q@ ps , qs ps ,q@ ps , q ps , q@ ps ,q ps , ps ,q ps , ps ,,q ps ,q@ ps , q ps , qps ,q ps ,
ps , qps , ps ,qps ,q ps , qps ,ps ,qps ,ps ,q ps , q ps , qps , q ps ,q@ ps q ps , qps ,q ps ,
ps , qps , ps ,qps ,q ps , qps ,ps ,qps ,ps }.
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455
diag
qps ,
q ps ,
qps ,
qps ,
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−
q
−
q
4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
− q 2kr×kr
Balasem A. Al-Quraishi, Nor Zelawati Binti Asmuin, Mohammed Najeh Nemah and Salih Meri A
The general case for p, q are prime number, p>u and q | p−1 given by the following
proposition.
Proposition: If p> and q | p−1, then
(⊗P). ( ⊗ (≡∗ T
,
(−p, −qp, −p)yz= D( ⊗ (≡∗ T
)). (⊗Q)=diagv(−p , − p , −p) ⊗ wxOj
⊗
,
))
Proof: By an inductive argument, the statement is certainly true for k=1
#
#
Assuming it holds for an arbitrary k, then (⊗
P). ( ⊗
(≡∗ T
By theorem (2-13), we obtain
( #f
(≡∗ T
⊗
,
#
))= ( ⊗
(≡∗ T
(≡∗ T
Hence, D( #f
⊗
,
,
))⊗( (≡∗ T
#
))=D( ⊗
(≡∗ T
,
,
,
#
#
) ) . (⊗
Q) = D( ⊗
(≡∗ T
,
))
))
))⊗D((≡∗ T
,
)). □
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
Hill,V.E.; "Groups Representation and Characters"; Haener Press,A Division of Mac Mil
and Publishing Co.Inc, New York, 1976.
Isaacs, I.M; "Character Theory of Finite Groups"; Academic Press., New York, 1976.
James. G, Liebeck .M," Representation and Characters of Groups", Cambridge .Univ
.Press .London, New York, 1993.
Kirdar. M. S; "The Factor of the Z-Valued Class Function Module the Group of the
Generalized Characters", Ph.D. Thesis .University of Birmingham, 1982.
Shabani H , Ashrafi A.R and Ghorbani M , ; " Rational Character Table of Some Finite
Groups " ; Journal of Algebra System , Vol.3 , No.2 , 2016 .
Singler.L.E, "Algebra", Springer – Verlag , 1976 .
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