ON THE NORMS OF SOME SPECIAL MATRICES WITH GENERALIZED FIBONACCI SEQUENCE

Abstract

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 U₀ = a, U₁ = 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].

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 . 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 U0 = a, U1 = b, where a and b are positive integer.

It is clear that (1) can be written as:

where Fn is called the nth term of p-Fibonacci sequence and defined by

Generally from equation (3), we have

Equation (2) can be written as:

A matrix A = Ar = (aij) ∈ Mn,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 = (aij) ∈ Mn,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 = (hij) ∈ Mn,n(ℂ), where hij = Ui+j−1. Similarly, a matrix A = (aij) ∈ Mn,n(ℂ) is Toeplitz matrix on generalized Fibonacci sequence (1), if it is of the form aij = Ui−j . The ℓp norm of a matrix A = (aij) ∈ Mn,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 γi are the eigenvalues of the matrix (Ā)tA.

The following inequality between Euclidean and spectral norm holds [13]

Definition 1.1 ([9]). Let A = (aij) and B = (bij) be m × n matrices. Then, the Hadamard product of A and B is given by

Definition 1.2 ([10]). The maximum column length norm c1(.) and maximum row length norm r1(.) for m × n matrix A = (aij) is defined as

Theorem 1.3 ([7]). Let A = (aij), B = (bij) and C = (cij) be p × q matrices. If C = A ◦ B, then ║C║2 ≤ r1(A)c1(B).

The following lemmas describe the properties of p-Fibonacci sequence.

Lemma 1.4 ([5]). Let Fn 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 Un.

Lemma 1.6. The sum of first n terms of generalized Fibonacci sequence Un is given as:

Lemma 1.7 ([5]). The sum of square of first n terms of the sequence Un 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 Rn and Sn 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

 

2. r−circulant, circulant and semi-circulant

In this section, we shall give main results related to r−circulant, circulant and sem-circulant on generalized Fibonacci sequence Un.

Theorem 2.1. Let A = Ar(U0,U1,...,Un−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, U0 = 0 and U1 = 1 then Un = Un−1 + Un−2, which is same as Fn = Fn−1 + Fn−2 with initial conditions F0 = 0 and F1 = 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 = (aij) with the generalized Fibonacci numbers then,

Proof. For the semi-circulant matrix A = (aij) 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

 

3. Hankel and Toeplitz matrix norm

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

Theorem 3.1. If A = (aij) is an n × n Hankel matrix with aij = Ui+j−1, then

where Tn 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 = (aij) is an n × n Hankel matrix with aij = Ui+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 = (aij) is an n × n Hankel matrix with aij = Ui+j−1. Then we have ║A║1 = ║A║∞ = U2n+1 − Un+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 Tn−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.