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

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.