Norms and Spread of the Fibonacci and Lucas RSFMLR Circulant Matrices

Abstract

Circulant type matrices have played an important role in networks engineering. In this paper, firstly, some bounds for the norms and spread of Fibonacci row skew first-minus-last right (RSFMLR) circulant matrices and Lucas row skew first-minus-last right (RSFMLR) circulant matrices are given. Furthermore, the spectral norm of Hadamard product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is obtained. Finally, the Frobenius norm of Kronecker product of a Fibonacci RSFMLR circulant matrix and a Lucas RSFMLR circulant matrix is presented.

1. Introduction

Circulant type matrices have been put on the firm basis with the work in [1–4] and so on. Circulant type matrices have significant applications in networks systems. In [5], some preliminary results on the dynamical behaviours of some specific nonmonotone Boolean automata networks which are called xor circulant networks were showed. In [6], the authors proposed a special class of the feedback delay network using circulant matrices. In [7], the impact of interior symmetries on the multiplicity of the eigenvalues of the Jacobian matrix at a fully synchronous equilibrium for the coupled cell systems associated with homogeneous networks was analyzed by Aguiar and Ruan, which was based on the circulant adjacency matrices of the networks induced by these interior symmetries. Exploiting the circulant structure of the channel matrices, the realistic near fast fading scenarios with circulant frequency selective channels were analysed by Eghbali et al. in [8]. The existence of doubly periodic travelling waves in cellular networks involving the discontinuous Heaviside step function by circulant matrix was studied by Wang and Cheng in [9].

The Fibonacci and Lucas sequences and are defined by the recurrence relations [10, 11]: If we start from , then Fibonacci and Lucas sequences are given by In [10], their Binet forms are given by

The following sum formulations for the Fibonacci and Lucas numbers are well known [11]: Lately, some authors studied the problems of the norms of some special matrices [11–21]. The author [11] found upper and lower bounds for the spectral norms of Toeplitz matrices such that and . In [13], the authors obtain upper and lower bounds for the spectral norms of matrices and , where and are -Fibonacci and -Lucas sequences, respectively, and they also give the bounds for the spectral norms of Kronecker and Hadamard products of these special matrices, respectively [14]. Solak and Bozkurt [16] have found out upper and lower bounds for the spectral norms of Cauchy-Toeplitz and Cauchy-Hankel matrices. Solak [18–20] has defined and as circulant matrices, where and ; then he has given some bounds for the and matrices concerned with the spectral and Euclidean norms.

In this paper, we define two kinds of special matrices as follows.

A Fibonacci row skew first-minus-last right (RSFMLR) circulant matrix is defined as a square matrix of the form

A Lucas row skew first-minus-last right (RSFMLR) circulant matrix is defined as a square matrix of the form Obviously, the RSFMLR circulant matrix is determined by its first row, and RSFMLR circulant matrix is a circulant matrix [22].

We define as the basic RSFMLR circulant matrix; that is, It is easily verified that has no repeated roots in its splitting field and is both the minimal polynomial and the characteristic polynomial of the matrix . In addition, is nonderogatory and satisfies and .

As we all know, letting be a RSFMLR circulant matrix with the first row , it is clear that Thus, is a RSFMLR circulant matrix if and only if for some polynomial . The polynomial will be called the representer of the RSFMLR circulant matrix . By (11), it is clear that is a RSFMLR circulant matrix if and only if commutes with ; that is, .

In addition to the algebraic properties that can be easily derived from the representation (11), we mention that RSFMLR circulant matrices have very nice structure. The product of two RSFMLR circulant matrices is a RSFMLR circulant matrix and is a RSFMLR circulant matrix too.

Let be an matrix. The Euclidean (or Frobenius) norm, the spectral norm, the maximum column sum matrix norm, and the maximum row sum matrix norm of the matrix are, respectively [11], where denotes the conjugate transpose of . The following inequality holds: Let and be matrices. The Hadamard product of and is defined by . If is any norm on matrices, then [18, 23] Kronecker product of and is given to be [18] Then [18] Let be an matrix with eigenvalues , . The spread of is defined as [24, 25] An upper bound for the spread due to Mirsky [24] states that where denotes the Frobenius norm of and is the trace of .

2. Norms and Spread of Fibonacci RSFMLR Circulant Matrices

Theorem 1. Let be a Fibonacci RSFMLR circulant matrix, where denote Fibonacci numbers given by (1); then two kinds of norms of are given by

Proof. The matrix is of the form (8), by (14), (15); then we have Since the Fibonacci sequences are defined by the recurrence relations (1), then we obtain To sum up, we can get Then which completes the proof.

Theorem 2. Let be a Fibonacci RSFMLR circulant matrix, where denote Fibonacci numbers given by (1); then where

Proof. Since and given by (1), the matrix is of the form We know that from equivalent norms. By (5), we can get Then where We have On the other hand, suppose that Then We can get Furthermore, We obtain The other result is obtained as follows: which completes the proof.

Theorem 3. Let be a Fibonacci RSFMLR circulant matrix, where denote Fibonacci numbers given by (1); then the bound for the spread of is where

Proof. The trace of is . By Theorem 2 and inequation (21), we have where We can get where which completes the proof.

3. Norms and Spread of Lucas RSFMLR Circulant Matrices

Theorem 4. Let be a Lucas RSFMLR circulant matrix, where denote Lucas numbers given by (2); then two kinds of norms of are given by

Proof. The matrix is of the form (9), by (14), (15); then we get Since the Lucas sequences are defined by the recurrence relations (2), then we obtain To sum up, we can get Then which completes the proof.

Theorem 5. Let be a Lucas RSFMLR circulant matrix, where denote Lucas numbers given by (2); then where

Proof. Since and , the matrix is of the form We know that from equivalent norms. By (6), we can get Then We have where On the other hand, supposing that then We obtain We have We get The other result is obtained as follows: which completes the proof.

Theorem 6. Let be a Lucas RSFMLR circulant matrix, where denote Lucas numbers given by (2); then where

Proof. The trace of is . By Theorem 5 and by inequation (21), we have where We obtain where which completes the proof.

Corollary 7. Let be a Fibonacci RSFMLR circulant matrix and let be a Lucas RSFMLR circulant matrix, where and denote Fibonacci numbers and Lucas numbers, respectively; then the spectral norm of Hadamard product of and satisfies the following inequality:

Proof. The proof is trivial by Theorems 2 and 5; we obtain By inequation (17), we have which completes the proof.

Corollary 8. Let be a Fibonacci RSFMLR circulant matrix and let be a Lucas RSFMLR circulant matrix, where and denote Fibonacci numbers and Lucas numbers, respectively; then the Frobenius norm of Kronecker product of and is where

Proof. Since the proof is trivial by Theorems 2 and 5, we obtain By (19), then where which completes the proof.

4. Conclusion

In this study, we define matrices of the following forms: let be a Fibonacci RSFMLR circulant matrix and let be a Lucas RSFMLR circulant matrix. Firstly, we get lower and upper bounds for the spectral norms of these matrices. Upper bounds for the spread of the matrix and the matrix are given. Afterwards, we obtain some corollaries related to norms of Hadamard and Kronecker products of these matrices. Based on the existing problems in [26–28], we will explore solving these problems by circulant matrices technology.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is supported by the Development Project of Science & Technology of Shandong Province (Grant no. 2012GGX10115) and the AMEP of Linyi University, China.