AKÜ FEBİD 11 (2011) 021301 (1-7) AKU J. Sci. 11 (2011) 021301 (1-7)
On the
pNorms of Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel Matrices
Hasan Öğünmez
Department of Mathematics, Faculty of Science and Arts, Kocatepe University, Afyon-TURKEY Geliş Tarihi:21 Ekim 2012; Kabul Tarihi: 12 Ocak 2012
Key words Quaternion Cauchy-
Toeplitz Matrices, Quaternion Cauchy-
Hankel Matrices, Euclidean Norm,
pnorm.
15A45, 15A60
Abstract
In this study, we have established upper and lower bounds for the p norms of the matrices T and
Hwhere T and Hare Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel Matrices respectively.
Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel Matrislerinin
pNormları Üzerine
Anahtar kelimeler Quaternion Cauchy- Toeplitz Matrices, Quaternion Cauchy-
Hankel Matrices, Euclidean Norm,
pnorm.
15A45, 15A60
Özet
Bu çalışmada, T , H sırasıyla Quaternion Cauchy-Toeplitz ve Quaternion Cauchy-Hankel Matrisleri olmak üzere, T ve H matrislerinin p normlar için alt ve üst 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 matrices norms,
has been given by several authors, see for instance [Moenck(1977),Mathias(1990),Visick(2000),Zielke (1988),Horn and Johnson(1991),Bozkurt(1998), Bozkurt(1996),Bozkurt(1996),Türkmen and Bozkurt (2002)] and references cited therein. In this paper, we have obtained some a lower and an upper bounds for the p of Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel Matrices. Now, we need the following definitions and preliminaries.
Definition 1. Let C and R denote the fields of the complex and real numbers respectively. Let Q be a 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
Afyon Kocatepe University Journal of Sciences
with real coefficients
x
0, x
1, x
2 andx
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
M
n(Q) whenm n
, denote the collection of allm n
matrices with quaternion entries.Definition 2. Let A A1 A2jMn(Q), where
2 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
[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
[Moencko(1977)]. Throughout the paper Z and R will represent the sets of positive integers and positive real numbers, respectively.
2. Matrices of Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel
Definition 3. The matrices in
x
quaternion which definitions have gave from Definition 1, form l t ,,
2 Z and p1,2,,n, r 1,2,,n
and m p r
r x l p r x
t p
x
x
1 , 3 1 1
1 , 2
1 1 1
0 0,
defining as below
r m p
r l p
r t p
T 1 1 1
k j
i
is called Quaternion Cauchy-Toeplitz matrix. By the similar way
r m p
r l p
r t p
H 1 1 1
k j
i
is called Quaternion Cauchy-Hankel matrix.
In this section we are going to find upper and lower bounds for the Euclidean norm and the p norms of Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel matrices in (2.1) and (2.2).
2.1. The p Norms of Quaternion Cauchy-Toeplitz and Quaternion Cauchy-Hankel Matrices
Lets give some upper and lower bounds following theorem, for p norms, of in definition (2.1) and (2.2) Quaternion Cauchy-Toeplitz and Quaternion
Cauchy-Hankel matrices.
Theorem 1. For p norm of definition (2.1) Quaternion Cauchy-Toeplitz matrix 2t ,,l mZ let p be positive even integer
pp p
p p
m m
p
l l
p
t t
p p
m p m
p
l l
p
t t
p p
p
p p
p p
p p
m l t T n
T n p
p
p p
p p
1 2
1
1 1
1 1
)!
1 (
1
1 1
)!
1 (
1
1 1
)!
1 ( 2 1 2 2
1 1
)!
1 (
1
1 1
)!
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
) 3 . 2 (
are valid as upper and lower bounds.
Proof From definition p norm
m m m m
p
m m
m p nm
m m
m p nm
l l
l l
p
t t
l p nl
l l
l p nl
t t
t t
p
t t
t p tn
t t
t p tn
n
s p
p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
m l t n
s n m
l t n T
p
p p
sm m sl
l st t sm
m sl
l st t
1 1
1 1
)!
2 (
1
1 1
)!
1 (
) 1 (
1 1
)!
1 (
) 1 (
1 1
1 1
)!
2 (
1
1 1
)!
1 (
) 1 (
1 1
)!
1 (
) 1 (
1 1
1 1
)!
2 (
1
1 1
)!
1 (
) 1 (
1 1
)!
1 (
) 1 (
2 2 2
1 1 2
2 2
, 2 1
, 2 ,
2 1
, 2
, 1 1
, 1
, 1 1
, 1
, 2 1
, 2 ,
2 1
, 2
, 1 1
, 1
, 1 1
, 1
, 2 1
, 2 ,
2 1
, 2
, 1 1
, 1
, 1 1
, 1 ) (
) ( ) (
) 4 . 2 (
2
1 2
)2 1 (
2 )2
1 (
2 )2 1 (
2 )2
1 (
2 )2
1 (
2 )2 1 (
2
is written. If we divide both of side of equality with
n
and if take a limit forn
.
m m m m
p n
m m
m p n
nm
m m
m p n
nm
l l
l l
p n
t t
l p n
nl
l l
l p n
nl
t t
t t
p n
t t
t p n
tn
t t
t p n
tn n
p n p n
n p p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
n p p
m l t T
p
1 1
1 1
)!
2 (
1
1 1
)!
1 (
) 1 (
1 1
)!
1 (
) 1 (
1 1
1 1
)!
2 (
1
1 1
)!
1 (
) 1 (
1 1
)!
1 (
) 1 (
1 1
1 1
)!
2 (
1
1 1
)!
1 (
) 1 (
1 1
)!
1 (
) 1 ( 2 2 1 2
, 2 1
, 2 ,
2 1
, 2
, 1 1
, 1
, 1 1
, 1
, 2 1
, 2 ,
2 1
, 2
, 1 1
, 1
, 1 1
, 1
, 2 1
, 2 ,
2 1
, 2
, 1 1
, 1
, 1 1
, 1 )
( lim
lim
2
is written and we obtain,
m m
p
l l
p
t t
p p
n p
p p
p p
p p
m l t T
p
1 1
)!
1 (
1
1 1
)!
1 (
1
1 1
)!
1 ( 2 1 2 1 2
1 , 1 1
, 1
1 , 1 1
, 1
1 , 1 1
, 1 )
( )
5 . 2 (
2
in this case, if we take a root of p. degree both of side of (2.6) inequality. We obtain,
pp p
m m
p
l l
p
t t
p p
p p
p p
p p
m l t T
n
1 2
1
1 1
)!
1 (
1
1 1
)!
1 (
1
1 1
)!
1 ( 2 1 2 2
1 , 1 1
, 1
1 , 1 1
, 1
1 , 1 1
, 1 )
( )
6 . 2 (
it's already upper bound for
n
pT
p1
.
Lets obtain lower bound now. Let p be positive even integer, if we throw out first term of (2.5), right hand has been gotten smaller, so
p p p
sm m sl
l st
t
sm m sl
l st
t n
s p
T p n
n
s p
1
2 1
2 2 2
2 2
2
2 2 2
2 2
1 2
1
) 1 ( ) 1 ( ) 1 (
) 1 ( ) 1 ( ) 1 ) ( 1 1 ( 2
If we take a limit for
n
.
pp
m m
p
l l
p
t t
p p
p p
p p
p p
p p
T n
1 1
1 1
)!
1 (
1
1 1
)!
1 (
1
1 1
)!
1 (
1 1 2
1 , 1 1
, 1
1 , 1 1
, 1
1 , 1 1
, 1 )
7 . 2 (
is obtained. It's already lower bound for
n
pT
p1
. Hence, proof has been completed from (2.6) and (2.7).
Theorem2. For p norm of definition(2.2) Quaternion Cauchy-Hankel matrix 2t ,,l mZ let p be positive even integer
p
m p m
l p l
t p t
p
m p m
m m l
p l l
l t
p t t
t H p
p p
p
1
1 1 , 1 2
1 , 1 2
1 , )! 2
2 (
) 1 (
1 1 , 1 1
1 1 , 1 1
1 1 , 1 1
)!
1 (
) 1 (
1 1
is valid for upper bound.
Proof. From definition of p norm
2 2
2 2 2
2 2
2 1
1
2 2 2
2 2
2
1
) ) 1 (
1 ( ) ) 1 (
1 ( ) ) 1 (
1 ) ( (
) ) 1 ( 1 ( ) ) 1 ( 1 ( ) ) 1 ( 1 ) (
8 . 2 (
p p
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 p p
if we compute sum where both of side of equality. We obtain
m
m m
m p
l l l
l p
t t t
t p
m m m
m m
m
l l l
l l
l
t t t
t t
t p
m s
m l
s l t
s t n
s
p n
p
p n
p
p n
p
p n
p
p n
p
p n
p s
p
1 1
)!
2 (
1
1 1
)!
2 (
1
1 1
)!
2 (
1
1 1
1
1 1
1
1 1
1 )!
1 (
1
) ) 1 ( 1 ( ) ) 1 ( 1 ( ) ) 1 ( 1 1 (
1 , 2 1
, 2
1 , 2 1
, 2
1 , 2 1
, 2
1 , 1 1
, 1
1 , 1 1
, 1
1 , 1 1
, 1
2 2 2 2
2 2
2
and
p m m mn p m m mn
p
l nl p l
l nl p l
p
t tn p t
t tn p t
p
m mn p m
m mn p m
m mn m
l nl p l
l nl p l
l nl l
t tn p t
t tn p t
t tn t p
p
m s n
m l
s n
l t
s n s t
n n s
1 1 , 1 2
1 , )! 2
2 ( 1
1 1 , 1 2
1 , )! 2
2 ( 1
1 1 , 1 2
1 , )! 2
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
2 ) 2 ) 1 ( 1 (
2 ) 2
) 1 ( 1 (
2 ) 2
) 1 ( 1 ( ) 2 ( 1 1
if we take limit of (2.8) for
n
. We use properties of polygamma function.0 )
(
2
2 2
2 2
2 1 2
1 (1 ( 1) ) (1 ( 1) ) (1 ( 1) )
lim )
9 . 2
(
p
m s n
m l
s n
l t
s n
s t
n
n
n s
and
tt ll mm
p
m m m
m
l l l
l t
t t
t p
m s
m l
s l t
s t n
n s
p p
p
p
p p
s
p p
p
1 1
1 )!
2 (
) 1 (
1 1
1 1
1 1
)!
1 (
) 1 (
) ) 1 ( 1 ( ) ) 1 ( 1 ( ) ) 1 ( 1 1 (
1 , 2 1
, 2 1
, 2
1 , 1
1 , 1 1
, 1 lim
) 10 . 2 (
1 1
2 2 2 2
2 2
2
Hence, we obtain
m
m l
l t
t p
m m m
m l
l l
l t
t t
t p p p
p p
p
p p
p H
p p
1 1
1 )!
2 (
) 1 (
1 1
1 1
1 1
)!
1 (
) 1 (
1 , 2 1
, 2 1
, 2
1 , 1 1
, 1 1
, 1
1 1
from equalities of (2.9) and (2.10). If we root of p. both of side of inequality, we obtain
pp p
m m l
l t
t p
m m m
m l
l l
l t
t t
t p p
p p
p
p p
p H
1 1 1
1 1
1 )!
2 (
) 1 (
1 1
1 1
1 1
)!
1 (
) 1 (
1 , 2 1
, 2 1
, 2
1 , 1 1
, 1 1
, 1
Hence, the proof is completed.
References
Zhang, F., (1997), Quaternions and matrices of quaternions, Linear Algebra and its Applications, 251:21-57.
Lee, H.C., (1949), Eigenvalues of canonical forms of matrices with quaternion coefficent, Proc R.I.A., 52, Sect.
Moenck, R., (1977), On Computing Closed forms for Summations., Proc. MACYSYMA user conf., 225-236.
Mathias, R., (1990), The Spectral Norm of a Nonnegative Matrix, Linear Algebra and its Applications, 131: 269-284.
Visick, G., (2000), A Quantitative Version of Observation That The Hadamart Product is A Principal Submatrix of The Kronecker Product., Linear Algebra and its Applications, 304: 45-68.
Zielke, G., (1988), Some Remarks on the Matrix Norms, Condition Numbers and Error Estimates for Linear Equations., Linear Algebra and its Applications, 110: 29-41.
Horn, R.A. and Johnson, C.R., (1991), Topics in Matrix Analysis., Cambridge University Pres.
Bozkurt, D., (1998), On the
l
p Norms of Cauchy- Toeplitz Matrices., Linear and Multilinear Algebra, 44: 341-346.Bozkurt, D., (1996), On the Bounds of Cauchy- Toeplitz Matrices., Siberian Journal of Differential Equations, 3(2): 135-142.
Bozkurt, D., (1996), On the Bounds for the lp norm of almost Cauchy-Toeplitz Matrix,.Turkish Journal of Math., 20(4): 545-552.
Türkmen, R. and Bozkurt, D., (2002), On the Bounds for the Norms of Cauchy-Toeplitz and Cauchy- Hankel Matrices., Applied Mathematics and Computation, 132: 633-642.
