Download The Rational Matrices

The Rational Matrices
Keles H.
Karadeniz Technical University, Faculity of Science,
Department of Mathematics, Campus of Kanuni, Trabzon,
TURKEY (Tel: +90462-377-2568; e-mail: [email protected]).
Abstract: In this paper, we give the research about rational properties of two matrices with the same
dimension. We will see that the properties here are analogous to the case in real numbers. This study is
about their properties and some examples are given.
Keywords: matrix, division, matrices division, rational matrices.
, ∈ ℚℳ
Proof. Let
1. INTRODUCTION
Let
with B ≠ 0 . Then the matrix
A and B ∈ M ( ℝ )
+
Let us start with the definition as follows.
 i  is called the division of by and denoted by


 B  n × n
A
B j
÷ or
. The set of rational matrices are showen by
ℚℳ = , ∈ ℳℝ, || ≠ 0 .
The
term
rational
in
the
set
1
ℚℳ = , ∈ ℳℝ, || ≠ 0 refers to the fact that a
rational matrix represents a ratio of two matrices. The set of
rational matrices is include by ℳℝ. A rational matrix is a
matrix like , where , are matrices. If = 0 then this
division is not defined.
= 0 ∈ ℚℳ and
= ∈ ℚℳ.
Theorem 1. Let
Proof. Let
,
=
!
.
∈ ℚℳ . Then
.
"() #
/
+
'
(
* ) % *() &
% &÷% &=
|$|
∈ ℚℳ .
$
,
$
.
|$| -() +( || *
)
.
.
"() #
/
+
'
(
* ) % *() &
= ||' + ,
|$|
∈ ℚℳ . Then
|$|
$
Theorem 2. Let
ii. = .
, ∈ ℚℳ, || ≠ 0. Then
$
% & ÷ % & = % &% & .
We can easily get
+
=
!
Proof. Let
.
Sequare matrix of reel numbers is denoted with ℳℝ or
ℳ.
1
,
!
$
, ∈ ℚℳ. Then
% & ÷ % & = || $ .
= || .
= || " + #
$
$
Let , , ∈ ℳℝ, || ≠ 0. Then
Lemma 2. Let
2. THE RATIONAL MATRICES
% & ÷ % & = ||' + ,
Corollary 1. Let
i.
= || " + #
=
We will give the Lemma without proof[1].
Lemma 1.
= || + || Example 1.
$
$
, ∈ ℚℳ, || ≠ 0.
$
0
1
2
3
2 67 0
=4 5
3
4 5 = 4 54 5 .
1
1
Let = 4
1
3
2
0
1
2
3
5, = 4
1
1
2
5,
1
.
=4
2
1
1
5
5 and > = 4
1
2
−1
=4
2
⟹
$
.
+
*
5
=4
−1
=4
−1
1
4 54 5 = 4
$
5
4
−1
Lemma 3. Let
i. % & =
ii. % & =
.
E
.
7
5
3
0 −1
5, = 4
1 $
1
$
−4
3
5⇒ =4
3
−1
4
5=
−1
.
+
*
−4
5
3
4
5,
−1
3
−1
iii. If || ≠ 0 from Lemma 1.
4 −1
54
−1 2
4
5,
−1
0
5=
1
iii.
B
∈ ℚℳ, ∈ ℳℝ. Then
 n a C (B) ⋯
sk
k1
∑
k =1

⋮
⋮

 n
 ∑ ask Ck 1 ( B ) ⋯
 k =1
$
$
I
A
i
B
+ 0 = ,0 +
.
=
 n


 ∑ ask ckn  Cs 1 ( B ) 
 k =1


⋮

n
n

∑  ∑ ask ck 1  Csn ( B )
s=1
k =1
For
$
I
J
.
.
∑a
sk
j

A=

1
A
B
B
% &% & =
1
j
B

2
A
B
i
j
 =
A
2
B
.
$
+
%*&
.
n
 A  = 1 Ai n ?
 
 
n B j 
B
B
Example 2. Let = 4
Ck 1 ( B )
[ i ]A =
∈ ℚℳ. Then
∈ ℚℳ and K ∈ ℤ! . Is then

  c11 ⋯ c1n 
k =1

⋮
⋮ 
 ⋮ ⋮

n
  c ⋯ c 
nn
∑ ask Ck1 ( B ) n1
k =1
n
,
$
ii. If = from i)
B
J
Fig. 1. Matrix A.



+
∈ ℚℳ. Then
+% + &=% + &+
 A C

B
B
=
=
()
$
= 4 5 . 4 5.
=
A=
+
Lemma 4. Let
 n  n a c  C ( B) ⋯
 ∑ sk k 1  s1
∑

s=1  k =1

⋮
⋮

n  n

 ∑  ∑ ask ck 1  Csn ( B ) ⋯

 s=1  k =1
A
, ,
I
$ J
iv. %& % $ & is rational matrix.
Proof. If = F( , = G( and = H( then
AC
i.
Let us the following lemma without proof.
iii. If || ≠ 0, also
i.
Theorem 3. Let
ii.
.
= || = || || = 4 5 . 4 5 .
2
1
1
1
5, = 4
1
1
0
5 and K = 2.
1
Fig. 4. Matrix
A
2
B
2
.
then
2
 A ≠ A .
 
 B  B2
2
Note. In general, for rational matrices we will get,
n
 A ≠ A
.
 
 B  Bn
Fig. 2. Matrix B.
Then
2
=M1
1
1
O
3
% & =4
−2
0
1 0
1
1 1 N = 4 2 15
2
1 1
−1 0
1
1 1
E
2
5
3
5 and E = 4
5,
1
−7 −4
ii. %&
6
6Q
=
=
,
∈ ℚℳ and K, P ∈ ℤ! , then
Theorem 4. If
i. %&
n
$
.
-
.
Q
6
Q
= R% & S = % & .
Proof. i. For the proof see paper [Keleş, H. ( 2011)].
ii. It is see clearly.
Theorem 5. Let
,
$
∈ ℚℝ . Then
.
"() #
/
+
'
(
* ) % *() &
% & ÷ % & = ||' + ,
Proof. If
Fig. 3. Matrix
()
A
2
.
,
$
$
∈ ℚℝ .
|$|
$
.
|$| -() % & ÷ % & = ||
+( *
)
.
.
"() #
/
+
(
'
* ) % *() &
= ||' + ,
|$|
.
B
REFERENCES
Keleş, H. ( 2011). Matrix Division. International
Association of Engineers (IAENG. London. U.K. ).
Keleş, H. ( 2008). Lineer Cebire Giriş. Akademi Yayınevi.
Trabzon.Turkey. 2008.
Bolian, L. and Hong-Jian L. ., ( 2000). Matrices in
Combinatorics and Graph Theory. Kluwer Academic
Publishers. The Netherlands.