1. Introduction
“Special matrices” is a widely studied subject in the research area of matrix analysis. Especially, special matrices whose entries are well known number sequences have become a very interesting research subject in recent years and many authors have obtained some good results in this area. Some researches denoted the norms of the special matrices involving famous number sequences. The authors found lower bounds, upper bounds and exact values for the spectral norms of these matrices.
Our aim in this study is to obtain some norm bounds more closure than those in literature to the exact value of matrix A’s spectral norm. Lots of articles which concern estimates for spectral norms of special matrices have been written so far.
Solak has studied the norms of circulant matrices with fibonacci and lucas numbers in [1] , Türkmen and Gökbaş have made a similar study by using the r-circulant matrix with pell and pell-lucas numbers in [2] , Shen and Cen have made a similar study by using the same special matrix with k-fibonacci and k-lucas numbers in [3] , Akbulak and Bozkurt found lower and upper bounds for the spectral norms of toeplitz matrices with classical fibonacci and lucas numbers entries in [4] , Shen gave upper and lower bounds for the spectral norms of toeplitz matrices with k-fibonacci and k-lucas numbers entries in [5] , Akbulak and Bozkurt have made a similar study by using the hankel matrix with fibonacci and lucas numbers in [6] , Gökbaş and Türkmen gave upper and lower bounds for the spectral norms of r-toeplitz matrices involving fibonacci and lucas numbers in [7] , Bozkurt and Tam obtained determinants and inverse of circulant matrices with jacobsthal and jacobsthal-lucas numbers in [8] .
The Fibonacci and Lucas sequences
and
are defined by the recurrence relations (Table 1)
The following sum formulas the Fibonacci and Lucas numbers are well known [9] :
The Euclidean norm of the matrix A is defined as
The singular values of the matrix A is
where
is an eigenvalues of matrix
and
is conjugate transpose of the matrix A. The square roots of the maximum eigenvalues of
are called the spectral norm of A and are induced by
.
Table 1. The Fibonacci and Lucas sequence have been given.
The following imequality holds,
Define the maximum row length norm
and the maximum column length norm
of any matrix A by
and
respectively. Let A, B and C be
matrices. If
then
for the matrices
and
the Hadamard Product of these matrices is defined as
[10] .
Let us give some lemmas which we will use in our result.
Lemma 1.1: Be a Hankel matrix whose entries determined by
where
stands for nth Fibonacci numbers [6] .
Lemma 1.2: Be a Hankel matrix whose entries determined by
where
stands for nth Fibonacci numbers [11] .
Lemma 1.3: Be a Hankel matrix whose entries determined by
where
stands for nth Lucas numbers [6] .
Lemma 1.4: Be a Hankel matrix whose entries determined by
where
stands for nth Lucas numbers [11] .
2. Result
In this section, we define an
r-Hankel matrix
whose entries are
or
where
and
denote the usual Fibonacci and Lucas numbers, respectively. We obtain some inequalities related to
.
Definition 2.5: A matrix
is called a r-Hankel matrix if it is of the form
Obviously, the r-Hankel matrix H is determined by parameter r and its first row elements
, thus we denote
. For r =1, the matrix H is called a Hankel matrix.
Theorem 2.6: Let
be a Hankel matrix satisfying
, where
.
1)
,
2)
,
where
is the spectral norm and
denotes the nth Fibonacci number.
Proof. The matrix A is of the form
Then,
Hence, when
we obtain
On the other hand, let the matrices B and C be
matrices satisfying
.
and
Since
and
we have
when
we also obtain
On the other hand, let the matrices B and C be
matrices satisfying
.
and
Since
and
we have
Thus, the proof is concluded.
Theorem 2.7: Let
be a Hankel matrix satisfying
, where
.
1)
,
2)
,
where
is the spectral norm and
denotes the nth Lucas number.
Proof. The matrix A is of the form
Then,
Hence, when
we obtain
On the other hand, let the matrices B and C be
matrices satisfying
.
and
Since
and
we have
when
we also obtain
On the other hand, let the matrices B and C be
matrices satisfying
.
and
Since
and
we have
Thus, the proof is concluded.
3. Numerical Examples
Example 3.8: Let
be a r-Hankel matrix where
. It can easily be seen that the values obtained in theorem 2.6. are more closure than those obtained in lemma 1.1. to exact values in from Table 2.
Example 3.9: Let
be a r-Hankel matrix where
. It can easily be seen that the values obtained in theorem 2.7. are more closure than those obtained in lemma 1.3. to exact values in from Table 3.
Table 2. Numerical results of
.
Table 3. Numerical results of
.
4. Conclusion
In this paper, we firstly define the r-Hankel matrix with entries Fibonacci and Lucas numbers. Then we introduce the Euclidean norm equality and the spectral norm inequalities of these matrices. Furthermore, we compared our finding with the exact value of matrix A’s spectral norm. In the future we shall further develop some kinds of r-Hankel matrix such as right r-Hankel, left r-Hankel and geometric r-Hankel. Also we will observe upper and lower bounds for the spectral norm of these matrices involving famous number sequences.