# The upper bound estimation on the spectral norm of r-circulant matrices with the Fibonacci and Lucas numbers

## Abstract

Let us define $$A=\operatorname{Circ}_{r}(a_{0},a_{1},\ldots,a_{n-1})$$ to be a $$n\times n$$ r-circulant matrix. The entries in the first row of $$A=\operatorname{Circ}_{r}(a_{0},a_{1},\ldots,a_{n-1})$$ are $$a_{i}=F_{i}$$, or $$a_{i}=L_{i}$$, or $$a_{i}=F_{i}L_{i}$$, or $$a_{i}=F_{i}^{2}$$, or $$a_{i}=L_{i}^{2}$$ ($$i=0,1,\ldots,n-1$$), where $$F_{i}$$ and $$L_{i}$$ are the ith Fibonacci and Lucas numbers, respectively. This paper gives an upper bound estimation of the spectral norm for r-circulant matrices with Fibonacci and Lucas numbers. The result is more accurate than the corresponding results of S Solak and S Shen, and of J Cen, and the numerical examples have provided further proof.

## Introduction

For $$n>0$$, the Fibonacci sequence $$\{F_{n}\}$$ is defined by $$F_{n+1}=F_{n}+F_{n-1}$$, where $$F_{0}=0$$ and $$F_{1}=1$$. If we start by zero, then the sequence is given by

\begin{aligned} \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & \cdots \\ F_{n} & 0 & 1 & 1 & 2 & 3 & 5 & 8 & 13 & 21 & \cdots \end{array} \end{aligned}
(1)

If we deduce from $$F_{n+1}$$ that $$L_{n+1}=L_{n}+L_{n-1}$$, and let $$L_{0}=2$$, $$L_{1}=1$$, then we obtain the Lucas sequence $$\{L_{n}\}$$,

\begin{aligned} \begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & \cdots\\ L_{n} & 2 & 1 & 3 & 4 & 7 & 11 & 18 & 29 & 47 & \cdots \end{array} \end{aligned}
(2)

Furthermore, the sequences $$\{F_{n}\}$$ and $$\{L_{n}\}$$ satisfy the following recursion:

\begin{aligned} F_{n}+L_{n}=2F_{n+1}. \end{aligned}
(3)

### Definition 1.1

A matrix A is an r-circulant matrix if it is of the form

$$A= \begin{pmatrix} a_{0} & a_{1} & \cdots& a_{n-2} & a_{n-1}\\ ra_{n-1} & a_{0} & \cdots& a_{n-3} & a_{n-2}\\ \cdots& \cdots& \cdots& \cdots& \cdots\\ ra_{2} & ra_{3} & \cdots& a_{0} & a_{1}\\ ra_{1} & ra_{2} & \cdots& ra_{n-1} & a_{0} \end{pmatrix}.$$

Obviously, the elements of this r-circulant matrix are determined by its first row elements $$a_{0},a_{1},\ldots,a_{n-1}$$ and the parameter r, thus we denote $$A=\operatorname{Circ}_{r}(a_{0}, a_{1}, \ldots, a_{n-1})$$. Especially when $$r=1$$, we obtain $$A=\operatorname{Circ}(a_{0}, a_{1}, \ldots, a_{n-1})$$.

### Definition 1.2

A matrix A is called a symmetric r-circulant matrix if it is of the form

$$A= \begin{pmatrix} a_{0} & a_{1} & \cdots& a_{n-2} & a_{n-1}\\ a_{1} & a_{2} & \cdots& a_{n-1} & ra_{0}\\ \cdots& \cdots& \cdots& \cdots& \cdots\\ a_{n-2} & a_{n-1} & \cdots& ra_{n-4} & ra_{n-3}\\ a_{n-1} & ra_{0} & \cdots& ra_{n-3} & ra_{n-2} \end{pmatrix}.$$

Obviously, the elements of this r-circulant matrix are determined by its first row elements $$a_{0},a_{1},\ldots,a_{n-1}$$ and the parameter r; thus we denote $$A=\operatorname{SCirc}_{r}(a_{0}, a_{1}, \ldots, a_{n-1})$$. Especially when $$r=1$$, we obtain $$A=\operatorname{SCirc}(a_{0}, a_{1}, \ldots, a_{n-1})$$.

For any $$A=(a_{ij})_{m\times n}$$, the well-known spectral norm of the matrix A is

$$\| A\|_{2}= \sqrt{{\max_{1 \leq i \leq n} \lambda_{i} \bigl( A^{H} A \bigr)} },$$

in which $$\lambda_{i} ( A^{H} A )$$ is the eigenvalue of $$A^{H}A$$ and $$A^{H}$$ is the conjugate transpose of matrix A.

Define the maximum column length norm $$c_{1}(\cdot)$$ and the maximum row length norm $$r_{1}(\cdot)$$ of any matrix A by

$$c_{1}(A)=\max_{j}\sqrt{\sum _{i}|a_{ij}|^{2}}$$

and

$$r_{1}(A)=\max_{i}\sqrt{\sum _{j}|a_{ij}|^{2}},$$

respectively.

Let A, B, and C be $$m \times n$$ matrices. If $$A=B\circ C$$, then in accordance with  we have

\begin{aligned} \|A\|_{2}\leq r_{1}(B)c_{1}(C) \end{aligned}
(4)

and

\begin{aligned} \|A\|_{2}\leq\|B\|_{2}\|C\|_{2}. \end{aligned}
(5)

Here, we define $$B=(b_{ij})_{m\times n}$$, $$C=(c_{ij})_{m\times n}$$, and we let $$B\circ C$$ be the Hadamard product of B and C.

In recent years, many authors (see ) were concerned with r-circulant matrices associated with a number sequence. References  calculate and estimate the Frobenius norm and the spectral norm of a circulant matrix where the elements of the r-circulant matrix are Fibonacci numbers and Lucas numbers; the authors found more accurate results of the upper bound estimated, and the numerical examples also have provided further proof.

### Theorem 1.3

(see )

Let $$A=\operatorname{Circ}(F_{0}, F_{1}, \ldots, F_{n-1})$$ be a circulant matrix, then we have

$$\|A\|_{2}\leq F_{n}F_{n-1},$$

where $$\|\cdot\|_{2}$$ is the spectral norm and $$F_{n}$$ denotes the nth Fibonacci number.

### Theorem 1.4

(see )

Let $$A=\operatorname{Circ}(L_{0}, L_{1}, \ldots, L_{n-1})$$ be a circulant matrix, then we have

$${{\|A\|_{2}\leq \begin{cases} \sqrt{[F_{n}F_{n-1}+4F_{n-1}^{2}+4F_{n-1}F_{n-2}+4]\times [F_{n}F_{n-1}+4F_{n-1}^{2}+4F_{n-1}F_{n-2}+4]}, & n\textit{ odd},\\ \sqrt{[F_{n}F_{n-1}+4F_{n-1}^{2}+4F_{n-1}F_{n-2}]\times [F_{n}F_{n-1}+4F_{n-1}^{2}+4F_{n-1}F_{n-2}-3]}, & n\textit{ even}, \end{cases}}}$$

where $$\|\cdot\|_{2}$$ is the spectral norm, and $$L_{n}$$ and $$F_{n}$$ denote the nth Lucas and Fibonacci numbers, respectively.

### Theorem 1.5

(see )

Let $$A=\operatorname{Circ}_{r}(F_{0}, F_{1}, \ldots, F_{n-1})$$ be a r-circulant matrix, in which $$|r|\geq1$$, and then

$$\|A\|_{2}\leq|r|F_{n}F_{n-1},$$

where $$r\in\mathbb{C}$$, $$\|\cdot\|_{2}$$ is the spectral norm and $$F_{n}$$ denotes the nth Fibonacci number.

### Theorem 1.6

(see )

Let $$A=\operatorname{Circ}_{r}(L_{0}, L_{1}, \ldots, L_{n-1})$$ be a r-circulant matrix and $$|r|\geq1$$, then we obtain

$$\|A\|_{2}\leq \begin{cases} \sqrt{(5|r|^{2}F_{n}F_{n-1}+4)(5F_{n}F_{n-1}+1)}, & n\textit{ odd},\\ \sqrt{[5|r|^{2}F_{n}F_{n-1}+4(1-|r|^{2})](5F_{n}F_{n-1}-3)}, & n\textit{ even}, \end{cases}$$

where $$r\in\mathbb{C}$$, $$\|\cdot\|_{2}$$ is the spectral norm, and $$L_{n}$$ and $$F_{n}$$ denote the nth Lucas and Fibonacci numbers, respectively.

## Main results

### Theorem 2.1

Let $$A=\operatorname{Circ}(F_{0}, F_{1}, \ldots, F_{n-1})$$ be a circulant matrix, then we have

$$\|A\|_{2}\leq\sqrt{(n-1)F_{n}F_{n-1}},$$

where $$\|\cdot\|_{2}$$ is the spectral norm and $$F_{n}$$ denotes the nth Fibonacci number.

### Proof

Since $$A=\operatorname{Circ}(F_{0}, F_{1}, \ldots, F_{n-1})$$ is a circulant matrix, let the matrices B and C be

$$B= \begin{pmatrix} F_{0} & 1 & \cdots& 1\\ 1 & F_{0} & \cdots& 1\\ \cdots& \cdots& \cdots& \cdots\\ 1 & 1 & \cdots& F_{0} \end{pmatrix},\qquad C= \begin{pmatrix} F_{0} & F_{1} & \cdots& F_{n-2} & F_{n-1}\\ F_{n-1} & F_{0} & \cdots& F_{n-3} & F_{n-2}\\ \cdots& \cdots& \cdots& \cdots& \cdots\\ F_{2} & F_{3} & \cdots& F_{0} & F_{1}\\ F_{1} & F_{2} & \cdots& F_{n-1} & F_{0} \end{pmatrix} ,$$

we get $$A=B\circ C$$.

For

$$r_{1}(B)=\max_{i}\sqrt{\sum _{j}|b_{ij}|^{2}}=\sqrt{n-1}$$

and

$$c_{1}(C)=\max_{j}\sqrt{\sum _{i}|c_{ij}|^{2}}=\max _{j}\sqrt{\sum_{i=1}^{n}|c_{in}|^{2}}= \sqrt{\sum_{s=0}^{n-1}F_{s}^{2}}= \sqrt{F_{n}F_{n-1}}.$$

From (4), we have

$$\|A\|_{2}\leq\sqrt{(n-1)F_{n}F_{n-1}}.$$

□

### Corollary 2.2

Let $$A=\operatorname{SCirc}(F_{0}, F_{1}, \ldots, F_{n-1})$$ be a symmetric circulant matrix, then we have

$$\|A\|_{2}\leq\sqrt{(n-1)F_{n}F_{n-1}},$$

where $$\|\cdot\|_{2}$$ is the spectral norm and $$F_{n}$$ denotes the nth Fibonacci number.

### Corollary 2.3

Let $$A=\operatorname{Circ}(F_{0}^{2}, F_{1}^{2}, \ldots, F_{n-1}^{2})$$ be a circulant matrix, then we have

$$\|A\|_{2}\leq(n-1)F_{n}F_{n-1},$$

where $$\|\cdot\|_{2}$$ is the spectral norm and $$F_{n}$$ denotes the nth Fibonacci number.

### Proof

Since $$A=\operatorname{Circ}(F_{0}^{2}, F_{1}^{2}, \ldots, F_{n-1}^{2})$$ is a circulant matrix, if the matrices $$B=\operatorname{Circ}(F_{0}, F_{1}, \ldots, F_{n-1})$$, we get $$A=B\circ B$$; thus from (5) and Theorem 2.1 we obtain

$$\|A\|_{2}\leq(n-1)F_{n}F_{n-1}.$$

□

### Theorem 2.4

Let $$A=\operatorname{Circ}(L_{0}, L_{1}, \ldots, L_{n-1})$$ be a circulant matrix, then we have

$$\|A\|_{2}\leq \begin{cases} \sqrt{5nF_{n}F_{n-1}+4n}, & n\textit{ odd},\\ \sqrt{5nF_{n}F_{n-1}}, & n\textit{ even}, \end{cases}$$

where $$\|\cdot\|_{2}$$ is the spectral norm and $$L_{n}$$ denotes the Lucas number.

### Proof

Since $$A=\operatorname{Circ}(L_{0}, L_{1}, \ldots, L_{n-1})$$ is a circulant matrix, let the following matrices be defined:

$$B= \begin{pmatrix} 1 & 1 & \cdots& 1\\ 1 & 1 & \cdots& 1\\ \cdots& \cdots& \cdots& \cdots\\ 1 & 1 & \cdots& 1 \end{pmatrix} ,\qquad C= \begin{pmatrix} L_{0} & L_{1} & \cdots& L_{n-2} & L_{n-1}\\ L_{n-1} & L_{0} & \cdots& L_{n-3} & L_{n-2}\\ \cdots& \cdots& \cdots& \cdots& \cdots\\ L_{2} & L_{3} & \cdots& L_{0} & L_{1}\\ L_{1} & L_{2} & \cdots& L_{n-1} & L_{0} \end{pmatrix},$$

then $$A=B\circ C$$.

We have

$$r_{1}(B)=\max_{i}\sqrt{\sum _{j}|b_{ij}|^{2}}=\sqrt{n}$$

and

$$c_{1}(C)=\max_{j}\sqrt{\sum _{i}|c_{ij}|^{2}}=\sqrt {\sum _{i=1}^{n}|c_{in}|^{2}}= \sqrt{\sum_{s=0}^{n-1}L_{s}^{2}}= \sqrt {\sum_{s=0}^{n-1}(F_{s}+2F_{s-1})^{2}}.$$

Here

$$\sum_{s=0}^{n-1}F_{s}^{2}=F_{n}F_{n-1}, \qquad \sum_{s=0}^{n-1}F_{s}F_{s-1}= \begin{cases} F_{n-1}^{2} ,& n\mbox{ odd},\\ F_{n-1}^{2}-1, & n\mbox{ even}, \end{cases}\qquad \sum_{s=0}^{n-1}F_{s-1}^{2}=F_{n-1}F_{n-2}+1,$$

thus

$$c_{1}(C)= \begin{cases} \sqrt{5F_{n}F_{n-1}+4},& n\mbox{ odd},\\ \sqrt{5F_{n}F_{n-1}}, & n\mbox{ even}, \end{cases}$$

and from (4) we obtain

$$\|A\|_{2}\leq \begin{cases} \sqrt{5nF_{n}F_{n-1}+4n}, & n\mbox{ odd},\\ \sqrt{5nF_{n}F_{n-1}}, & n\mbox{ even}. \end{cases}$$

□

### Corollary 2.5

Let $$A=\operatorname{SCirc}(L_{0}, L_{1}, \ldots, L_{n-1})$$ be a symmetric circulant matrix, then we have

$$\|A\|_{2}\leq \begin{cases} \sqrt{5nF_{n}F_{n-1}+4n}, & n\textit{ odd},\\ \sqrt{5nF_{n}F_{n-1}}, & n\textit{ even}, \end{cases}$$

where $$\|\cdot\|_{2}$$ is the spectral norm, and $$L_{n}$$ and $$F_{n}$$ denote the nth Lucas and Fibonacci numbers, respectively.

### Corollary 2.6

Let $$A=\operatorname{Circ}(L_{0}^{2}, L_{1}^{2}, \ldots, L_{n-1}^{2})$$ be circulant matrices, then

$$\|A\|_{2}\leq \begin{cases} 5nF_{n}F_{n-1}+4n, & n\textit{ odd},\\ 5nF_{n}F_{n-1}, & n\textit{ even}, \end{cases}$$

where $$\|\cdot\|_{2}$$ is the spectral norm, and $$L_{n}$$ and $$F_{n}$$ denote the nth Lucas and Fibonacci numbers, respectively.

### Proof

Since $$A=\operatorname{Circ}(L_{0}^{2}, L_{1}^{2}, \ldots, L_{n-1}^{2})$$ is a circulant matrix, if the matrices $$B=\operatorname{Circ}(L_{0}, L_{1}, \ldots, L_{n-1})$$, we get $$A=B\circ B$$; thus from (5) and Theorem 2.4, we obtain

$$\|A\|_{2}\leq \begin{cases} 5nF_{n}F_{n-1}+4n, & n\mbox{ odd},\\ 5nF_{n}F_{n-1}, & n\mbox{ even}. \end{cases}$$

□

### Corollary 2.7

Let $$A=\operatorname{Circ}(F_{0}L_{0}, F_{1}L_{1}, \ldots, F_{n-1}L_{n-1})$$ be circulant matrices, then

$$\|A\|_{2}\leq \begin{cases} \sqrt{(n-1)nF_{n}F_{n-1}(5F_{n}F_{n-1}+4)}, & n\textit{ odd},\\ \sqrt{5(n-1)n}F_{n}F_{n-1}, & n\textit{ even}, \end{cases}$$

where $$\|\cdot\|_{2}$$ is the spectral norm, and $$L_{n}$$ and $$F_{n}$$ denote the nth Lucas and Fibonacci numbers, respectively.

### Proof

Since $$A=\operatorname{Circ}(F_{0}L_{0}, F_{1}L_{1}, \ldots, F_{n-1}L_{n-1})$$ is a circulant matrix, if the matrices $$B=\operatorname{Circ}(F_{0}, F_{1}, \ldots, F_{n-1})$$ and $$C=\operatorname{Circ}(L_{0}, L_{1}, \ldots, L_{n-1})$$, we get $$A=B\circ C$$; thus from (5), Theorems 2.1, and 2.4, we obtain

$$\|A\|_{2}\leq \begin{cases} \sqrt{(n-1)nF_{n}F_{n-1}(5F_{n}F_{n-1}+4)}, & n\mbox{ odd},\\ \sqrt{5(n-1)n}F_{n}F_{n-1}, & n\mbox{ even}. \end{cases}$$

□

### Theorem 2.8

Let $$A=\operatorname{Circ}_{r}(F_{0}, F_{1}, \ldots, F_{n-1})$$ be a r-circulant matrix, in which $$|r|\geq1$$, and then

$$\|A\|_{2}\leq\sqrt{(n-1)|r|^{2}F_{n}F_{n-1}},$$

where $$r\in\mathbb{C}$$, $$\|\cdot\|_{2}$$ is the spectral norm and $$F_{n}$$ denotes the nth Fibonacci number.

### Proof

Since $$A=\operatorname{Circ}_{r}(F_{0}, F_{1}, \ldots, F_{n-1})$$ is a r-circulant matrix, let B and C, respectively, be

$$B= \begin{pmatrix} F_{0} & 1 & 1 & \cdots& 1\\ r & F_{0} & 1 & \cdots& 1\\ r & r & F_{0} & \cdots& 1\\ \cdots& \cdots& \cdots& \cdots& \cdots\\ r & r & r & \cdots& F_{0} \end{pmatrix} ,\qquad C= \begin{pmatrix} F_{0} & F_{1} & F_{2} & \cdots& F_{n-1}\\ F_{n-1} & F_{0} & F_{1} & \cdots& F_{n-2}\\ F_{n-2} & F_{n-1} & F_{0} & \cdots& F_{n-3}\\ \cdots& \cdots& \cdots& \cdots& \cdots\\ F_{1} & F_{2} & F_{3} & \cdots& F_{0} \end{pmatrix},$$

then $$A=B\circ C$$.

For

$$r_{1}(B)=\max_{i}\sqrt{\sum _{j}|b_{ij}|^{2}}=\sqrt{(n-1)|r|^{2}}$$

and

$$c_{1}(C)=\max_{j}\sqrt{\sum _{i}|c_{ij}|^{2}}=\sqrt {\sum _{i=1}^{n}|c_{in}|^{2}}= \sqrt{\sum_{s=0}^{n-1}F_{s}^{2}}= \sqrt {F_{n}F_{n-1}},$$

from (4), we have

$$\|A\|_{2}\leq\sqrt{(n-1)|r|^{2}F_{n}F_{n-1}}.$$

□

### Corollary 2.9

Let $$A=\operatorname{SCirc}_{r}(F_{0}, F_{1}, \ldots, F_{n-1})$$ be a symmetric r-circulant matrix, in which $$|r|\geq1$$, and then

$$\|A\|_{2}\leq\sqrt{(n-1)|r|^{2}F_{n}F_{n-1}},$$

where $$r\in\mathbb{C}$$, $$\|\cdot\|_{2}$$ is the spectral norm and $$F_{n}$$ denotes the nth Fibonacci number.

### Corollary 2.10

Let $$A=\operatorname{Circ}_{r}(F_{0}^{2}, F_{1}^{2}, \ldots, F_{n-1}^{2})$$ be a r-circulant matrix, while $$|r|\geq1$$, then we obtain

$$\|A\|_{2}\leq(n-1)|r|F_{n}F_{n-1},$$

where $$r\in\mathbb{C}$$, $$\|\cdot\|_{2}$$ is the spectral norm and $$F_{n}$$ denotes the Fibonacci number.

### Proof

Since $$A=\operatorname{Circ}_{r}(F_{0}^{2}, F_{1}^{2}, \ldots, F_{n-1}^{2})$$ is a r-circulant matrix, if the matrices $$B=\operatorname{Circ}_{r}(F_{0}, F_{1}, \ldots, F_{n-1})$$ and $$C=\operatorname{Circ}(F_{0}, F_{1}, \ldots, F_{n-1})$$, we get $$A=B\circ C$$; thus from (5), Theorems 2.1, and 2.8, we obtain

$$\|A\|_{2}\leq(n-1)|r|F_{n}F_{n-1}.$$

□

### Corollary 2.11

Let $$A=\operatorname{Circ}_{r}(F_{0}L_{0}, F_{1}L_{1}, \ldots, F_{n-1}L_{n-1})$$ be a r-circulant matrix, while $$|r|\geq1$$, then we obtain

$$\|A\|_{2} \leq \begin{cases} \sqrt{(n-1)n|r|^{2}F_{n}F_{n-1}(5F_{n}F_{n-1}+4)}, & n\textit{ odd},\\ F_{n}F_{n-1} \sqrt{5|r|^{2}(n-1)n}, & n\textit{ even}, \end{cases}$$

where $$r\in\mathbb{C}$$, $$\|\cdot\|_{2}$$ is the spectral norm, and $$L_{n}$$ and $$F_{n}$$ denote the nth Lucas and Fibonacci numbers, respectively.

### Proof

Since $$A=\operatorname{Circ}_{r}(F_{0}L_{0}, F_{1}L_{1}, \ldots, F_{n-1}L_{n-1})$$ is a r-circulant matrix, if the matrices $$B=\operatorname{Circ}_{r}(F_{0}, F_{1}, \ldots, F_{n-1})$$ and $$C=\operatorname{Circ}(L_{0}, L_{1}, \ldots, L_{n-1})$$, we get $$A=B\circ C$$; thus from (5), Theorems 2.4, and 2.8, we obtain

$$\|A\|_{2}\leq \begin{cases} \sqrt{(n-1)n|r|^{2}F_{n}F_{n-1}(5F_{n}F_{n-1}+4)}, & n\mbox{ odd},\\ F_{n}F_{n-1} \sqrt{5|r|^{2}(n-1)n},& n\mbox{ even}. \end{cases}$$

□

### Theorem 2.12

Let $$A=\operatorname{Circ}_{r}(L_{0}, L_{1}, \ldots, L_{n-1})$$ be a r-circulant matrix and $$|r|\geq1$$, then we obtain

$$\|A\|_{2}\leq \begin{cases} \sqrt{(n-1)|r|^{2}+1}\times\sqrt{5F_{n}F_{n-1}+4}, & n\textit{ odd},\\ \sqrt{(n-1)|r|^{2}+1}\times\sqrt{5F_{n}F_{n-1}}, & n\textit{ even}, \end{cases}$$

where $$r\in\mathbb{C}$$, $$\|\cdot\|_{2}$$ is the spectral norm, and $$L_{n}$$ and $$F_{n}$$ denote the nth Lucas and Fibonacci numbers, respectively.

### Proof

Since $$A=\operatorname{Circ}_{r}(L_{0}, L_{1}, \ldots, L_{n-1})$$ is a r-circulant matrix, let B and C, respectively, be

$$B= \begin{pmatrix} 1 & 1 & 1 & \cdots& 1\\ r & 1 & 1 & \cdots& 1\\ r & r & 1 & \cdots& 1\\ \cdots& \cdots& \cdots& \cdots& \cdots\\ r & r & r & \cdots& 1 \end{pmatrix} ,\qquad C= \begin{pmatrix} L_{0} & L_{1} & \cdots& L_{n-2}& L_{n-1}\\ L_{n-1} & L_{0} & \cdots& L_{n-3} & L_{n-2}\\ \cdots& \cdots& \cdots& \cdots& \cdots\\ L_{2} & L_{3} & \cdots& L_{0} & L_{1}\\ L_{1} & L_{2} & \cdots& L_{n-1} & L_{0} \end{pmatrix} ,$$

and then $$A=B\circ C$$.

We have

$$r_{1}(B)=\max_{i}\sqrt{\sum _{j}|b_{ij}|^{2}}=\sqrt {(n-1)|r|^{2}+1}$$

and

$$c_{1}(C)=\max_{j}\sqrt{\sum _{i}|c_{ij}|^{2}}=\sqrt {\sum _{i=1}^{n}|c_{in}|^{2}}= \sqrt{\sum_{s=0}^{n-1}L_{s}^{2}}= \sqrt {\sum_{s=0}^{n-1}(F_{s}+2F_{s-1})^{2}},$$

in which

$$\sum_{s=0}^{n-1}F_{s}^{2}=F_{n}F_{n-1}, \qquad \sum_{s=0}^{n-1}F_{s-1}F_{s}= \begin{cases} F_{n-1}^{2}, & n\mbox{ odd},\\ F_{n-1}^{2}-1, & n\mbox{ even}, \end{cases}\qquad \sum_{s=0}^{n-1}F_{s-1}^{2}=F_{n-1}F_{n-2}+1,$$

and we get

$$c_{1}(C)= \begin{cases} \sqrt{5F_{n}F_{n-1}+4}, & n\mbox{ odd},\\ \sqrt{5F_{n}F_{n-1}}, & n\mbox{ even}. \end{cases}$$

From (4), we further infer

$$\|A\|_{2}\leq \begin{cases} \sqrt{(n-1)|r|^{2}+1}\times\sqrt{5F_{n}F_{n-1}+4}, & n\mbox{ odd},\\ \sqrt{(n-1)|r|^{2}+1}\times\sqrt{5F_{n}F_{n-1}}, & n\mbox{ even}. \end{cases}$$

□

### Corollary 2.13

Let $$A=\operatorname{SCirc}_{r}(L_{0}, L_{1}, \ldots, L_{n-1})$$ be a symmetric r-circulant matrix and $$|r|\geq1$$, then we obtain

$$\|A\|_{2}\leq \begin{cases} \sqrt{(n-1)|r|^{2}+1}\times\sqrt{5F_{n}F_{n-1}+4}, & n\textit{ odd},\\ \sqrt{(n-1)|r|^{2}+1}\times\sqrt{5F_{n}F_{n-1}}, & n\textit{ even}, \end{cases}$$

where $$r\in\mathbb{C}$$, $$\|\cdot\|_{2}$$ is the spectral norm, and $$L_{n}$$ and $$F_{n}$$ denote the nth Lucas and Fibonacci numbers, respectively.

### Corollary 2.14

Let $$A=\operatorname{Circ}_{r}(L_{0}^{2}, L_{1}^{2}, \ldots, L_{n-1}^{2})$$ be a r-circulant matrix and $$|r|\geq1$$, then

$$\|A\|_{2}\leq \begin{cases} (5F_{n}F_{n-1}+4)\sqrt{n[(n-1)|r|^{2}+1]}, & n\textit{ odd},\\ 5F_{n}F_{n-1}\sqrt{n[(n-1)|r|^{2}+1]}, & n\textit{ even}, \end{cases}$$

where $$r\in\mathbb{C}$$, $$\|\cdot\|_{2}$$ is the spectral norm, and $$L_{n}$$ and $$F_{n}$$ denote the nth Lucas and Fibonacci numbers, respectively.

### Proof

Since $$A=\operatorname{Circ}_{r}(L_{0}^{2}, L_{1}^{2}, \ldots, L_{n-1}^{2})$$ is a r-circulant matrix, if the matrices $$B=\operatorname{Circ}(L_{0}, L_{1}, \ldots, L_{n-1})$$ and $$C=\operatorname{Circ}_{r}(L_{0}, L_{1}, \ldots, L_{n-1})$$, we get $$A=B\circ C$$; thus from (5), Theorems 2.4, and 2.12, we obtain

$$\|A\|_{2}\leq \begin{cases} (5F_{n}F_{n-1}+4)\sqrt{n[(n-1)|r|^{2}+1]}, & n\mbox{ odd},\\ 5F_{n}F_{n-1}\sqrt{n[(n-1)|r|^{2}+1]}, & n\mbox{ even}. \end{cases}$$

□

## Examples

### Example 1

Let $$A=\operatorname{Circ}(F_{0}, F_{1}, \ldots, F_{n-1})$$ be a circulant matrix, in which $$F_{i}$$ ($$i=0,1,\ldots,n-1$$) denotes the Fibonacci number.

From Table 1, it is easy to find that the upper bounds for the spectral norm, of Theorem 2.1 are more accurate than Theorem 1.3 when $$n\geq4$$.

### Example 2

Let $$A=\operatorname{Circ}(L_{0}, L_{1}, \ldots, L_{n-1})$$ be a circulant matrix, where $$L_{i}$$ ($$i=0,1,\ldots,n-1$$) denotes the Lucas sequence.

Let $$n\geq3$$, and it is easy to find that the upper bounds for the spectral norm of Theorem 2.4 are more accurate than Theorem 1.4 (see Table 2).

### Example 3

Let $$A=\operatorname{Circ}_{2}(F_{0}, F_{1}, \ldots, F_{n-1})$$ be a 2-circulant matrix, in which $$F_{i}$$ ($$i=0,1,\ldots,n-1$$) denotes the Fibonacci number.

Let $$n\geq4$$, and it is easy to find that the upper bounds for the spectral norm of Theorem 2.8 are more precise than Theorem 1.5 (see Table 3).

### Example 4

Let $$A=\operatorname{Circ}_{2}(L_{0}, L_{1}, \ldots, L_{n-1})$$ be a 2-circulant matrix where $$L_{i}$$ ($$i=0,1,\ldots, n-1$$) denotes the Lucas sequence.

It can be seen from Table 4 that the upper bounds for the spectral norm of Theorem 2.12 are more precise than Theorem 1.6 when $$n\geq3$$.

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## Acknowledgements

Project supported by Applied Fundamental Research Plan of Sichuan Province (No. 2013JY0178).

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Correspondence to Chengyuan He.

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The authors declare that they have no competing interests.

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All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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