International Journal of Computational Mathematics

Volume 2014 (2014), Article ID 795175, 6 pages

http://dx.doi.org/10.1155/2014/795175

Research Article

On the Spectrum and Spectral Norms of -Circulant Matrices with Generalized -Horadam Numbers Entries

Lele Liu

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

Received 17 May 2014; Accepted 25 August 2014; Published 31 August 2014

Academic Editor: Chengpeng Bi

Copyright © 2014 Lele Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This work is concerned with the spectrum and spectral norms of -circulant matrices with generalized -Horadam numbers entries. By using Abel transformation and some identities we obtain an explicit formula for the eigenvalues of them. In addition, a sufficient condition for an -circulant matrix to be normal is presented. Based on the results we obtain the precise value for spectral norms of normal -circulant matrix with generalized -Horadam numbers, which generalize and improve the known results.

1. Introduction

There is no doubt that the -circulant matrices have been one of the most interesting research areas in computation mathematics. It is well known that these matrices have a wide range of applications in signal processing, digital image disposal, coding theory, linear forecast, and design of self-regress.

There are many works concerning estimates for spectral norms of -circulant matrices with special entries. For example, Solak [1] established lower and upper bounds for the spectral norms of circulant matrices with Fibonacci and Lucas numbers entries. subsequently, Ipek [2] investigated some improved estimations for spectral norms of these matrices. Bani-Domi and Kittaneh [3] established two general norm equalities for circulant and skew circulant operator matrices. Shen and Cen [4] gave the bounds of the spectral norms of -circulant matrices whose entries are Fibonacci and Lucas numbers. In [5] they defined -circulant matrices involving -Lucas and -Fibonacci numbers and also investigated the upper and lower bounds for the spectral norms of these matrices.

Recently, Yazlik and Taskara [6] define a generalization of the special second-order sequences such as Fibonacci, Lucas, -Fibonacci, -Lucas, generalized -Fibonacci and -Lucas, Horadam, Pell, Jacobsthal, and Jacobsthal-Lucas sequences. For any integer number , the generalized -Horadam sequence is defined by the following recursive relation:

where and are scaler-value polynomials, . The following are some particular cases.

(i)If , and , , the -Fibonacci sequence is obtained:

(ii)If , and , , the -Lucas sequence is obtained:

(iii)If , and , , the Fibonacci sequence is obtained:

(iv)If , and , , the Lucas sequence is obtained:

(v)If , and , , the Jacobsthal sequence is obtained:

In [7], the authors present new upper and lower bounds for the spectral norm of an -circulant matrix , and they study the spectral norm of circulant matrix with generalized -Horadam numbers in [8]. In this paper, we first give an explicit formula for the eigenvalues of -circulant matrix with generalized -Horadam numbers entries using different methods in [7]. Afterwards, we present a sufficient condition for an -circulant matrix to be normal. Based on the results, the precise value for spectral norms of normal -circulant matrix whose entries are generalized -Horadam numbers is obtained, which generalize and improve the main results in [1, 2, 4, 5].

2. Preliminaries

In this section, we present some known lemmas and results that will be used in the following study.

Definition 1. For any given , the -circulant matrix , denoted by , is of the form

It is obvious that the matrix turns into a classical circulant matrix for .

Lemma 2 (see [9]). Let be an -circulant matrix; then the eigenvalues of are given by

where is the th root of unity.

Let us take any matrix of order ; it is well known that the spectral norm of matrix is

where is the conjugate transpose of and is the eigenvalue of .

For a normal matrix (i.e., ), we have the following lemma.

Lemma 3 (see [10]). Let be a normal matrix with eigenvalues . Then the spectral norm of is

The following lemma can be found in [11].

Lemma 4 (see [11], Abel transformation). Suppose that and are two sequences, and ; then

3. Spectrum of -Circulant Matrix with Generalized -Horadam Numbers

We start this section by giving the following lemma.

Lemma 5. Suppose that is a generalized -Horadam sequence defined in (1). The following conclusions hold.

(1)If , then

(2)If , then

Proof. (1) According to (1), we have

Changing the summation index in (14), we have

By direct calculation, together with recursive relation (1), one can obtain that

Therefore we immediately obtain (12) from .

(2)Suppose that ; we first illustrate that . Let ; then . Combining (1) and , one can obtain that

which shows that is a constant sequence, and therefore

Evaluating summation from to , we have

Changing the summation index in (19) gives

Therefore

In view of assumptions and , we know that . Thus we obtain (13) from (21).

From Lemma 5 we have the following theorem.

Theorem 6. Let be an -circulant matrix with eigenvalues ; then for the following hold.

(1)If , then

(2)If , then

Proof. According to Lemma 2, we have

Using Abel transformation (Lemma 4), we have

(1)In the light of (12) and (25), one can obtain that

It is clear that

Substituting (27) into (26), we obtain that

Therefore we have

We immediately obtain formula (22) from (29).

(2)Taking into account (13) and (25), we have

It follows that

Therefore we obtain (23). This concludes the proof.

4. Spectral Norms of Normal -Circulant Matrices

In this section, we consider the spectral norms of normal -circulant matrix whose entries are generalized -Horadam numbers. Our results generalize and improve the results in [1, 2, 4, 5]. The following lemma can be found in [9], and we give a concise proof.

Lemma 7. Let be an -circulant matrix. If , then is normal matrix.

Proof. It is well known that

If , then

That is, . According to (32), we obtain that

Therefore , which shows that is normal.

According to Theorem 6 and Lemma 7, we have the following theorem.

Theorem 8. Suppose that is an -circulant matrix. If and , then the spectral norm of is

    The following theorem simplifies and generalizes the results of Theorem 2.2 in [12].

Theorem 9. Let be a circulant matrix; then

Proof. Suppose that ; it follows from Lemma 7 that is normal. Notice that

It follows from Lemma 3 that . According to Theorem 6, if and , we obtain that

Similarly, if , it follows that

This completes the proof.

Taking into account formulae (4)–(6), we have the following corollary.

Corollary 10. Let be a circulant matrix; then

Corollary 11. Let be a circulant matrix; then

Corollary 12. Let be a circulant matrix; then

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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