ON THE NORMS OF SOME SPECIAL MATRICES WITH GENERALIZED FIBONACCI SEQUENCE

Journal of Applied Mathematics & Informatics.
2015.
Sep,
33(5_6):
593-605

- Received : June 08, 2014
- Accepted : June 22, 2015
- Published : September 30, 2015

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In this study, we define
r
−circulant, circulant, Hankel and Toeplitz matrices involving the integer sequence with recurrence relation
U_{n}
=
pU
_{n-1}
+
U
_{n-2}
, with
U
_{0}
=
a
,
U
_{1}
=
b
. Moreover, we obtain special norms of above mentioned matrices. The results presented in this paper are generalizations of some of the results of
[1
,
10
,
11]
.
AMS Mathematics Subject Classification : 15A45, 15A60, 15A36, 11B39.
. Solak found bounds for the special norms of circulant matrices
[10]
. In
[12]
the authors determined the upper and lowers bounds for Cauchy-Toeplitz and Cauchy Hankel matrices. In
[8]
bounds of circulant,
r
−circulant, semi-circulant and Hankel matrices with tribonacci sequence obtained. In
[6]
, the author pre-sented some results about circulant, negacyclic and semi-circulant matrices with the modified Pell, Jacobsthal and Jacobsthal- Lucas numbers. Shen and Cen found the bounds of spectral norm of Fibonacci and Lucas numbers
[11]
. The generalized Fibonacci sequence is defined as:
with initial conditions
U
_{0}
=
a
,
U
_{1}
=
b
, where
a
and
b
are positive integer.
It is clear that (1) can be written as:
where
F_{n}
is called the
n
th term of
p
-Fibonacci sequence and defined by
Generally from equation (3), we have
Equation (2) can be written as:
A matrix A = A
_{r}
= (a
_{ij}
) ∈
M_{n,n}
(ℂ) is called
r
−circulant on generalized se-quence, if it is of the form
where
r
∈ ℂ. If
r
=1, then matrix A is called circulant.
A matrix A = (a
_{ij}
) ∈
M_{n,n}
(ℂ) is called semi-circulant on generalized Fi-bonacci sequence, if it is of the form
A Hankel matrix on generalized Fibonacci sequence is defined as:
H = (h
_{ij}
) ∈
M_{n,n}
(ℂ), where h
_{ij}
= U
_{i}
+
_{j}
_{−1}
. Similarly, a matrix A = (a
_{ij}
) ∈
M_{n,n}
(ℂ) is Toeplitz matrix on generalized Fibonacci sequence (1), if it is of the form a
_{ij}
=
U_{i−j}
. The
ℓ_{p}
norm of a matrix A = (a
_{ij}
) ∈
M_{n,n}
(ℂ) is defined by
If
. The Euclidean (Frobenius) norm of the matrix A is defined as:
The spectral norm of the matrix A is given as:
where
γ_{i}
are the eigenvalues of the matrix (Ā)
^{t}
A.
The following inequality between Euclidean and spectral norm holds
[13]
Definition 1.1
(
[9]
). Let A = (a
_{ij}
) and B = (b
_{ij}
) be
m
×
n
matrices. Then, the Hadamard product of A and B is given by
Definition 1.2
(
[10]
). The maximum column length norm
c
_{1}
(.) and maximum row length norm
r
_{1}
(.) for
m
×
n
matrix A = (a
_{ij}
) is defined as
Theorem 1.3
(
[7]
).
Let A
= (
a_{ij}
),
B
= (
b_{ij}
)
and C
= (
c_{ij}
)
be p
×
q matrices. If C
=
A
◦
B, then
║
C
║
_{2}
≤
r
_{1}
(
A
)
c
_{1}
(
B
).
The following lemmas describe the properties of
p
-Fibonacci sequence.
Lemma 1.4
(
[5]
).
Let F_{n} be the n-th term of p-Fibonacci sequence then,
Lemma 1.5
(
[5]
).
The sum of square of first n terms of p-Fibonacci sequence is given by
The following lemmas describes the properties of generalized Fibonacci se-quence
U_{n}
.
Lemma 1.6.
The sum of first n terms of generalized Fibonacci sequence U_{n} is given as:
Lemma 1.7
(
[5]
).
The sum of square of first n terms of the sequence U_{n} is given by:
Lemma 1.8.
Sum of product of consecutive terms of generalized Fibonacci se-quence is given as:
where
Proof
. From equation (2), we have.
By lemmas (1.4) and (1.5), we get
Theorem 1.9.
For all n
≥ 1
where R_{n} and S_{n} are defined in lemma (1.4) and (1.5) respectively.
Proof
. From Lemma (1.7) and (1.8), we have
Lemma 1.10.
For all n
> 1
Proof
. From equation (5), we obtain
On the other hand, from equation (3), we have
Thus, we have
r
−circulant, circulant and sem-circulant on generalized Fibonacci sequence
U_{n}
.
Theorem 2.1.
Let A
=
A_{r}
(
U
_{0}
,
U
_{1}
,...,
U
_{n−1}
)
be r−circulant matrix
.
Proof
. The
r
−circulant matrix A on the sequence (1) is given as:
and from the definition of Euclidean norm, we have
Here we have two cases depending on
r
.
Case 1. If |
r
| ≥ 1, then from equation (8), we have
and from lemma (1.7), we get.
By inequality (7), we obtain
On the other hand, let us define two new matrices C and D as :
Then it is easy to see that A = C ◦ D, so from definition (1.2)
Now using theorem (1.3), we obtain
Combine inequalities (9) and (10), we get following inequality
Case 2. If |
r
| ≤ 1, then we have
By inequality (7), we get
On the other hand, let the matrices C′ and D′ be defined as:
such that A = C′ ◦ D′, then by definition (1.2), we obtain
and
Again by applying theorem (1.3)
and combing inequality (11) and (12), we obtain the required result.
Remark 2.2.
The above theorem is the generalization of the result
[11]
. If put
p
= 1,
U
_{0}
= 0 and
U
_{1}
= 1 then
U_{n}
=
U
_{n−1}
+
U
_{n−2}
, which is same as
F_{n}
=
F
_{n−1}
+
F
_{n−2}
with initial conditions
F
_{0}
= 0 and
F
_{1}
= 1.
Theorem 2.3.
Let A be the circulant matrix on generalized Fibonacci sequence.
Proof
. Since by definition of circulant matrix, the matrix A is of the form
and form the definition of Euclidean norm, one can get,
By inequality (7), we get
Let matrices B and C be defined as:
Then the row norm and column norm of B and C are given as:
Using theorem (1.3), we have
Combine (14) and (15), we get
Remark 2.4.
Above result is the generalization of Solak 's work
[10]
, in which the author found the upper and lower bounds for the Euclidean and spectral norms of circulant matrices.
Theorem 2.5.
Let A be an n
×
n semi-circulant matrix A
= (
a_{ij}
)
with the generalized Fibonacci numbers then,
Proof
. For the semi-circulant matrix A = (a
_{ij}
) with the Generalized Fibonacci sequence numbers we have
From the definition of Euclidean norm, we have
Using lemma (1.9), we get the required result
Theorem 3.1.
If A
= (
a_{ij}
)
is an n
×
n Hankel matrix with a_{ij}
=
U
_{i}
_{+}
_{j}
_{−1}
, then
where T_{n} is defined in lemma (1.9).
Proof
. From the definition of Hankel matrix, the matrix A is of the form
So, we have
Theorem 3.2.
If A
= (
a_{ij}
)
is an n
×
n Hankel matrix with a_{ij}
=
U
_{i}
_{+}
_{j}
_{−1}
then, we have
Proof
. From theorem (3.1) and inequality (7), we have
Let us define two new matrices
It can be easily seen that A = M ◦ N. Thus we get
Using the theorem (1.3), we have
Theorem 3.3.
If A
= (
a_{ij}
)
is an n
×
n Hankel matrix with a_{ij}
=
U
_{i}
_{+}
_{j}
_{−1}
. Then we have
║
A
║
_{1}
= ║
A
║
_{∞}
=
U
_{2n+1}
−
U
_{n+1}
.
Proof
. From the definition of the matrix A , we can write
by lemma (1.6), we have
Similarly, the row norm of the matrix A can be computed as:
Theorem 3.4.
The bounds of spectral norms of the Toeplitz matrix A are given as:
and
where T
_{n−1}
and T
_{−(n−1)}
are defined in lemma (1.9) and (1.10) respectively.
Proof
. The Toeplitz matrix A define by the sequence (1) is given as
From the definition of Euclidean norm , we have
From Lemma (1.9) and (1.10), we have
Using inequality (7), we obtain
On the other hand , let us consider the matrices.
such that, A =
C
◦
D
. Then using definition (1.2)
By theorem (1.3), we obtain the desired result
Remark 3.5.
Norms of Toeplitz matrix with Fibonacci and Lucas numbers have been calculated by Akbulak and Bozkurt
[1]
. Theorem (3.4) is a generalization of their paper.
Zahid Raza received M.Sc. from the University of Punjab and Ph.D from Abdus Salam School of Mathematical Sciences Government College University Lahore, Pakistan. Since 2010 he has been working at National University of Computer & Emerging Sciences, Lahore campus. Currently, he is working as an associate professor of mathematics in the faculty of Sciences and Humanities at Lahore Campus. His research interests include algebraic combinatorics and elementary number theory.
Department of Mathematics, National University of Computer & Emerging Sciences, La-hore campus B-Block Faisal Town, Lahore, Pakistan.
E-mail: zahid.raza@nu.edu.pk
Muhammad Msim Ali received MS mathematics from National University of Computer & Emerging Sciences. He is currently a faculty member at Punjab Group of Colleges, Lahore. His research interests are computational mathematics and combinatorial number theory.
Department of Mathematics, National University of Computer & Emerging Sciences, La-hore B-Block Faisal Town, Lahore, Pakistan.
E-mail: masimali99@gmail.com

1. Introduction and Preliminaries

A lot of research papers on the norms of some special matrices have been writ-ten during the last decade
[1
,
2
,
6
,
10
,
11]
. Akbulak and Bozkurt
[1]
found lower and upper bounds for the spectral norms of Toeplitz matrices
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2.r−circulant, circulant and semi-circulant

In this section, we shall give main results related to
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3. Hankel and Toeplitz matrix norm

In this section, we have calculated the bounds of Hankel and Toeplitz matrix associated with generalized Fibonacci sequence.
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BIO

Akbulak M.
,
Bozkurt D.
(2008)
On the norms of Toeplitz matrices involving Fibonacci and Lucas numbers
Hacettepe J. Math. Stat.
37
89 -
95

Halici S.
(2013)
On some inequality and hankel matrices involving Pell, Pell Lucas numbers
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Kalman D.
,
Mena R.
(2002)
The Fibonacci Numbers
The Mathematical Magazine
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24

Koshy T.
(2001)
Fibonacci and Lucas numbers with applications
Wiley-Interscience Publica-tions

Kilic E.
(2008)
Sums of the squares of terms of sequence {un}
Proc. Indian Acad. Sci.(Math. Sci.)
118
27 -
41
** DOI : 10.1007/s12044-008-0003-y**

Kocer E.G.
(2007)
Circulant, negacyclic and semicirculant matrices with the modified Pell, Ja-cobsthal and Jacobsthal-Lucas numbers
Hacettepe J. Math. Stat.
36
133 -
142

Mathias R.
(1990)
The spectral norm of nonnegative matrix
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** DOI : 10.1016/0024-3795(90)90403-Y**

Raza Z
,
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On the norms of circulant, r- circulant, semi-sirculant and Hankel matrices with tribonacci sequence
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Citing 'ON THE NORMS OF SOME SPECIAL MATRICES WITH GENERALIZED FIBONACCI SEQUENCE
'

@article{ E1MCA9_2015_v33n5_6_593}
,title={ON THE NORMS OF SOME SPECIAL MATRICES WITH GENERALIZED FIBONACCI SEQUENCE}
,volume={5_6}
, url={http://dx.doi.org/10.14317/jami.2015.593}, DOI={10.14317/jami.2015.593}
, number= {5_6}
, journal={Journal of Applied Mathematics & Informatics}
, publisher={Korean Society of Computational and Applied Mathematics}
, author={RAZA, ZAHID
and
ALI, MUHAMMAD ASIM}
, year={2015}
, month={Sep}