AKÜ FEBİD 11 (2011) 021302 (9-18) AKU J. Sci. 11 (2011) 021302 (9-18)
On The Euclidean and the Spectral Norms of Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel Matrices
Hasan ÖĞÜNMEZ
Department of Mathematics, Faculty of Science and Arts, Kocatepe University, Afyon-TURKEY e-posta: hogunmez@aku.edu.tr
Geliş Tarihi: 20 Ekim 2011; Kabul Tarihi: 05 Ocak 2012
Key words Quaternion Cauchy-
Toeplitz Matrices, Quaternion Cauchy-
Hankel Matrices, Euclidean Norm, Spectral Norm. 15A45,
15A60
Abstract
In this study, we have established upper and lower bounds for the Euclidean and the Spectral norms of Quaternion Cauchy-Toeplitz (T) and Quaternion Cauchy-Hankel (H) matrices respectively. Besides,
by assuming complex matrix Tn A1A2j , we have defined the matrix
1 2
2
] 1
[ A A
A
Tn A
and similarly, by assuming complex matrix Hn B1B2j, we have also defined another matrix
1 2
2
] 1
[ B B
B
Hn B . Then we have obtained the bounds of the spectral norms for these matrices.
Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel Matrislerinin Euclidean ve Spektral Normları Üzerine
Anahtar Kelimeler Quaternion Cauchy- Toeplitz Matrices, Quaternion Cauchy-
Hankel Matrices, Euclidean Norm, Spectral Norm. 15A45,
15A60
Özet
Bu çalışmada, sırasıyla Quaternion Cauchy-Toeplitz (T), Quaternion Cauchy-Hankel (H) Matrislerinin Spektral ve Euclidean normlar için alt ve üst sınarlar elde ettik. Ayrıca, Tn A1 A2j , kompleks
matrisi yardımıyla
1 2
2
] 1
[ A A
A
Tn A matrisini ve benzer şekilde Hn B1B2j, kompleks
matrisi yardımıyla
1 2
2
] 1
[ B B
B
Hn B matrisini tanımlayıp bu matrislerin spektral normlar için
sınırlar elde ettik.
© Afyon Kocatepe Üniversitesi
1. Introduction and Preliminaries
In quantum physic, the family of quaternions plays an important role. But in mathematics they generally play a role in algebraic systems, skew fields or noncommutative division algebras, matrices in commutative rings take attention but, matrices with quaternion entries has not been investigated very much yet. But in recent times quaternions are in order of day.
The main obstacles in the study of quaternion matrices, as expected come from the noncommutative multiplication of quaternions.
One will find that working on a quaternion matrix problem is often equivalent to dealing with a pair
of complex matrices [Zhang(1997), Lee(1949)].
Recently, the studies concern with matrix norms, has been given by several authors, see for instance [Moenck(1977),Mathias(1990),Visick(2000),Zielke (1988),Horn and Johnson(1991), Bozkurt(1996), Solak and Bozkurt(2003),Türkmen and Bozkurt (2002)] and references cited therein. In this paper, we have obtained some a lower and an upper bounds for the Euclidean and spectral of Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel Matrices. Now, we need the following definitions and preliminaries.
Definition1. Let C and R denote the fields of the complex and real numbers respectively. Let Q be a
Afyon Kocatepe University Journal of Sciences
four-dimensional vector space over R with an ordered basis, denoted by e ,,i j and k. A real quaternion, simply called quaternion, is a vector
k j i
e 1 2 3
0 x x x
x
x Q
with real coefficients x0,x1,x2 and
x
3 .Besides the addition and the scalar multiplication of the vector space Q over R, the product of any two quaternions e ,,i j and k are defined by the requirement that e act as a identity and by the table
2 1
2
2 j k
i
. ,
, jk kj i ki ik j k
ji
ij
Let Mmn(Q), simply Mn(Q) when
m n
, denote the collection of allm n
matrices with quaternion entries.Definition 2. Let
A A
1 A
2j
Mn(Q), where2 1
, A
A
aren n
complex matrices. We shall call the2 n 2 n
complex matrix,
1 2
2
1
A A A A
uniquely determined by A, the complex adjoint matrix or adjoint of the quaternion matrix A [Lee(1949)].
Now we give some preliminaries related to our study. Let A be any
n n
matrix. The p norms of the matrix A are defined as).
1 (
/ 1
1 1
p a
A
p p ij n
j m
i p
If p, then
. max lim
, ij
j p i
n A a
A
The well-known Euclidean norm of matrix A is
2 / 1 2
1
1
ij n
j m
i
E a
A
and also the spectral norm of matrix A is
) ( max
2 1
A A
A
i Hn
i
where A is
m n
andA
H is the conjugate transpose of the matrix A. The following inequality holds:E
E A A
A
n 2
) 1 1 . 1 (
[Zielke (1988)]. A function is called a psi (or digamma) function if
} ) ( {ln )
( )
2 . 1
( x
dx x d
where
. )
(
10
dt t e
x
t xx
The
n
th derivatives of a function is called a polygamma function. ) ( ln )
( )
, ( ) 3 . 1
(
x
dx d dx x d dx x d
n n n
If
n 0
then( 0 , x ) ( x )
dxln ( x )
d
. Onthe other hand, if
a 0
,b
is any number andn
is positive integer, then0 ) , ( lim )
4 . 1
(
a n b
n
[Moenck(1977)]. Throughout the paper Z and R will represent the sets of positive integers and positive real numbers, respectively.
Let A and B be
n n
matrices. The Hadamard product of A and B is defined by].
[ )
5 . 1
( AB aijbij
If
is any norm onn n
matrices, thenB
A B A )
6 . 1 (
[Visick(2000)].
Define the maximum column length norm cj() and the maximum row length norm
r
i( )
of any matrix A bym i
a A
c ij
m
j i
j( ) max 1,2, ,
) 7 . 1 (
2 / 1 2
1
and
n j
a A
r ij
n
i j
i( ) max 1,2, ,
) 8 . 1 (
2 / 1 2
1
respectively [Horn and Johnson(1991)]. Let A, B and
C
bem n
matrices. IfA B C
then) ( ) ( )
9 . 1
( A
2 r
iB c
jC
[Mathias(1990)].2. Matrices of Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel
Definition 3. The matrices in
x
quaternion from Definition 1, for 2t ,,l mZ andn
p1,2,, , r 1,2,,n and
r p r
p r
p l m
t
x x
x
x
1
1
1 1 3 1 2 1 1
0
0 , , ,
r m p
r l p
r t p
T 1 1 1
) 1 . 2
( i j k
is called Quaternion Cauchy-Toeplitz matrix.
r m p
r l p
r t p
H 1 1 1
) 2 . 2
( i j k
is called Quaternion Cauchy-Hankel matrix.
In this section we are going to find an upper and lower bounds for the Euclidean norm and the spectral norms of Quaternion Cauchy-Toeplitz and
Quaternion Cauchy-Hankel matrices in (2.1) and (2.2).
2.1. Euclidean and Spectral Norms of Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel Matrices
Theorem 1. For Euclidean norm of T Quaternion Cauchy-Toeplitz matrix which is defined in (2.1),
) csc csc
(csc )
3 . 2
(
2 t 2 l 2 mE
n
T
is valid upper bound, where 2t ,,l mZ. Proof. From Euclidean norm the following
2 2 2
2 2
2
2 2 2
2 2
1 2
1 2 2 2 2
) 1 ( ) 1 ( ) 1 (
) 1 ( ) 1 ( ) 1 ) ( ( ) (
sm m sl
l st
t
sm m sl
l st
s t n m
l t n T
n
s E
is obtained. If we divide both of sides by
n
and if we take upper bound of right hand side to infinity, we obtain
t l m
n
n s E
sm m sl
l st
t
sm m sl
l st
t n
m s l t n T
2 2 2 22 2
2 2
2 2
2 2
2 2
2 1 2
1 2
2 2 2
csc csc
csc
) 1 ( ) 1 ( ) 1 (
) 1 ( ) 1 ( ) 1 ) ( 1 ( 1 lim
or 1 T 2E 2
csc2 t csc2 l csc2 m
,n
if we take
m l
E t
n T
csc2 csc2 csc21
then we obtain
t l m
E
n
T csc
2 csc
2 csc
2 which is an upper bound for Euclidean norm of Quaternion Cauchy-Toeplitz matrix.
Corollary 1. For spectral norm of T Quaternion Cauchy-Toeplitz matrices defined in (2.1),
2 2
2
2
csc csc
csc )
4 . 2
(
t
l
m T
is valid lower bound, where 2t ,,l mZ.
Proof. Following relation, for spectral norm of T matrix will be 1 .
T 2
n T E Then we obtain
. csc
csc
csc
22 2
2t
l
m T
Now for in definition (2.1) of spectral norm of T Quaternion Cauchy-Toeplitz matrix, we have obtained upper bound to give as a theorem before lets give some definition end concepts.
Definition 4. Ji
1nn (i1,2,,n) let it be a squareJ
matrix with all entries 1.Now,
nr r p
t p
A
1 1 1 ,
i and
nr r p p r
p m
l
A
1 , 1
2 1 1
i write Tn A1A2j. From Definition 2,
1 2
2
)
15 . 2
( A A
A T
nA
is being occured as complex matrix in
2 n 2 n
. From definition (2.5) lets give the following theorem.Theorem 2. Let
t l m 2
in (2.5). Then the upper bound for the spectral norm of Quaternion Cauchy- Toeplitz matrix is
2 . 1 2
, 1 1 2 2
ln 2
2 1 2
1 2 2
ln 1 2 2 2 )
6 . 2 (
2 1
2 2
n n
n Tn
Proof. Consider
2 n 2 n
complex matrix in (2.5), withn
r r p A p
1 2 ,
1 1
i,
n n n
n
n n
n n
n n
A
i i
i i
i i
i i
i i
i i
i i
i i
2 2 2
2 2 2
3 2 3
2 2 1 2
2
5 2
2 3 2
2
5 2
2 7
2 2 3
2
3 2
2 5
2 2
1
and with
n
r r p p r
A p
1 2 ,
1 2
1 1
2
i,
n
n nn
n n
n n
n n
A
i i
i i
i i
i i
i i
i i
i i
i i
1 2 1
1 1
1 2 1
2 1
1
1 1
1 2 1
1 1
1 2 1
2
3 2 3
2 2 1
2 2
5 2
2 3
2 2
5 2
2 7
2 2 3
2
3 2
2 5
2 2
2
in this case, we write
2 n 2 n
matrix as.
i i
i
i i
i
i i
i i
i i
i i
i
i i
i
T
n n
n n
n n
n n
n n
n n
n n
n n
n
2 1
2 1
1
2 1
1 2 1
2 2
1 1
2 1
2
1 2 1
1 2
1 1
2 1
2
1 1
2 1
2 2
2
3 2
2 1
2 2 3
2 2 1
2 2
5 2
2 3
2 5
2 2 3
2
3 2
2 3
2 2
3 2
2 1
2 2 3
2 2 1
2 2
5 2
2 3
2 5
2 2 3
2
3 2
2 3
2 2
i i
i
i i
i
i i
i
i i
i
i i
i
i i
i
Then, the Hadamard Product (1.5) of
Tn BnDn
for
i i
n J J
A
B A1 2
and
1 2
2 1
A A
A Dn A
.1 2
2 1 2
1
A A
A A J
J
A T A
i i
n
From (1.7) and (1.8), we obtain
2
1 2
1 2 1 2
1 1
1
1 2
1 3
1 1
2 1 3
2 1 1
2 ln 1 2 2 2
1 2 2 2
1 2 2 2 2 1
2 2 )
(
n B
r
s n
s
s n s
in this case we write
2
211 2
1
1( ) 2 2 2 1 ln2 2
) 7 . 2
( r Bn n
similarly
21
12
2
) 1 2 (
1 1 3 1 2
1 1
) 1 2 (
1 9
3 1 1 2
1 3
2 1 1
, 1 2 2
ln 2
2 2
1 2 1
2 )
(
2
2
n n
D c
s n
s s n
s
s s n
Then
2
121 2
2 1
1
( ) 2 ln 2 2 1 ,
) 8 . 2
( c D
n n n
we can see the following relation and we utilize from (1.9) that from (2.7) and (2.8).
2 1
2 1 2
, 1 1 2 2
ln 2
2 1 2
1 2 2
ln 1 2 2 2
2 2
n n
n Tn
is being obtained as upper bound.
Theorem 3. Let 2t ,,l mZ hold in (2.2). Then for the Euclidean norm of Quaternion Cauchy-Hankel matrix H, we have
. )
9 . 2
(
H
EProof. From the definition of Euclidean
2 2
2 2
2 1 2
1
2 2
2 2
2 2
1 2
) ) 1 (
1 ( ) ) 1 (
1 ( ) ) 1 (
1 ( ) (
) ) 1 ( 1 ( ) ) 1 ( 1 ( ) ) 1 ( 1 (
m s n
m l
s n
l t
s n s t
n
m s
m l
s l t
s s t
H
n
s n
s E
is clear. If we calculate right hand of this equality, then
m nm m m
nm m m
nm m l
nl l
l nl l l
nl l t
tn t t
tn t t
tn t
m nm n m
m nm n m
m nm m
l nl n l
l nl n l
l nl l t
tn n t
t tn n t
t tn t m
m m
m m
m
l l l
l l
l t
t t
t t
t
m n m
m n m
m m l
n l
l n l
l l t
n t t
n t t
H E t
1 1 1 1
, ) 1 2 1
( 1 1
1 1 , ) 1 2 1 ( 1 1
1 1 , ) 1 2 1 (
1 1 , 1 1
1 , ) 1 2 1
(
1 1 , 1 1
1 , ) 1 2 1 ( 1 1
, 1
1 1 , ) 1 2 1 ( 1 1
1 1 , ) 1 1 (
1 1 1 1
, ) 1 1 ( 1 1
1 1 , ) 1 1 (
1 1 1 1
, ) 1 1 ( 1 1
1 1 , ) 1 1 ( 1 1
1 1 , ) 1 1 (
2
is being found above. Therefore, we arrive
m
n m l
n l t
n t
n n
n
1 1 1 lim
1 1 lim
1 lim
and
1 0 1 , 1 1 lim
1 , 1 1 lim
1 , 1
lim
m
n m l
n l t
n t
n n
n
This implies,
E
H
which is desired. Now, lets shown
r
t
p r
pB
1 , 1 1
i and1 .
1 , 1
2 1
n
r m p
l
p r p r
B
iThen, obtain
2
.
1
B
jB H
n
From Definition 2,
1 2
2
) 1
10 . 2
( B B
B Hn B
is being occurred as
2 n 2 n
complex matrix. Now, we ready to give the upper bound below.Theorem 4. Let
t l m 2
for spectral norm of H
n , Quaternion Cauchy-Hankel matrix in (2.10) than,
2
215 2
5 9
2 104
2 5 2
1 2 3 4 2
, 1 2 2
ln 2
2 ln ) 2 2 2 ( )
11 . 2 (
n n
n Hn
is valid.
Proof. To use (2.10), define
n
r r p B p
1 2 ,
1 1
i,
n n n n
n n
n n
n n
n n
n n
i i
i i
i i i
i
i i
i i
i i
i i
B
1 4
2 1 4
2 5
2 2 3 2
2
1 4
2 3 4
2 3
2 2 1 2
2
5 2
2 3 2
2 9
2 7
2
3 2
2 1 2
2 7
2 5
2
1
and
n
r r p p r B p
1 2 ,
1 2
1 1 2
i,
n n n
n n
n
n n
n n
n n
n n
i i
i i
i i
i i
i i
i i
i i
i i
B
) 1 ( )
1 ( )
1 ( )
1 (
) 1 ( )
1 ( )
1 ( )
1 (
) 1 ( )
1 ( )
1 ( )
1 (
) 1 ( )
1 ( )
1 ( )
1 (
1 4
2 1
4 2 5
2 2 3
2 2
1 4
2 3
4 2 3
2 2 1
2 2
5 2
2 3
2 2 9
2 7
2
3 2
2 1
2 2 7
2 5
2
2
in this case, we write
2 n 2 n
matrix as.
) 1 ( )
1 ( )
1 ( )
1 ( )
1 ( )
1 (
) 1 ( )
1 ( )
1 ( )
1 (
) 1 ( )
1 ( )
1 ( )
1 (
) 1 ( )
1 ( )
1 (
) 1 ( )
1 ( )
1 (
) 1 ( )
1 ( )
1 (
] [
1 4
2 5
2 2 3
2 2 1
4 2 5
2 2 3
2 2
5 2
2 9
2 7
2 5
2 2 9
2 7
2
3 2
2 7
2 5
2 3
2 2 7
2 5
2
1 4
2 5
2 2 3
2 2 1
4 2 5
2 2 3
2 2
5 2
2 9
2 7
2 5
2 2 9
2 7
2
3 2
2 7
2 5
2 3
2 2 7
2 5
2
i i
i i
i i
i i
i i i
i
i i
i i i
i
i i
i i
i i
i i
i i
i i
i i
i i
i i
H
n n
n n
n n
n n
n n
n n
n n
n n
n n
n n
n
finding Hadamard Product
i i
n J J
B
M B1 2
and
1 2
2 1
B B
B Kn B
Since, these two matrices are
2 n 2 n
, we write
Hn MnKn
or
1 2
2 1
2 1
B B
B B
J J
B H B
i i
n
From (1.7) and (1.8), we obtain
2
5 2
1 2 3 4 3 2
1 1
3 2
1 7
1 5 1 3
2 1 7
1 5 2 1
1
2 ln 2 2 2
2 2 2
2 2 2
) (
n M
r
s n
s
s n s
in this case
212 5 2
1 2 3 2 4 ln 2 2 2 ) ( )
12 . 2
( 1
n
M
r n
we found
2
5 2
5 9
2 104
3 2
1 1 3 3 2
1 1
3 2
1 49
1 25 3 1 3 2
1 7
1 5 2 1
1
, 1 2 2
ln 2 2 2
2 2
) (
2
2
n n
K c
s n
s s n
s
s s n
Finally,
2 1
2 5 2
, 5 1 2 2
ln 9 2
) 104 ( )
13 . 2
( 1 2
n n
K
c n
is written. We utilize between these norms following relation and from (1.9). By the way from (2.12) and (2.13)
2 1
2 5 2
, 5 1 2 2
ln 9 2
104
2 5 2
1 2 3 2 4 ln ) 2 2 2 (
2 2
n n
n Hn
is obtained as upper bound which is desired.
3. Numerical Results
In this section, have compared our findings with the known bounds of the norms of matrices in the illustrative examples below. We have found between in theorems we have given real norms of matrices in second section.
Example 1. For t l m2,
E comparative values in Theorem 1.) csc csc
(csc
2 t 2 l 2 mn
3 76.9529898 7
76.6302269 200
7 66.6432440 2
66.2833095 150
5 54.4139800 8
53.9950855 100
4 45.5260027 0
45.0482513 70
2 38.4764949 2
37.9366256 50
0 29.8037648 2
29.1560702 30
6 24.3346720 6
23.5885869 20
3 17.2072116 9
16.2632729 10
9 10.8827961 5
9.60997214 4
2 9.42477796 9
8.02994695 3
3 7.69529898 6
6.11010093 2
2 5.44139809 6
3.46410161 1
E T n
