Open Access

On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers

Journal of Inequalities and Applications20152015:253

https://doi.org/10.1186/s13660-015-0778-1

Received: 12 June 2015

Accepted: 5 August 2015

Published: 20 August 2015

Abstract

In this paper, we study norms of circulant and r-circulant matrices involving harmonic Fibonacci and hyperharmonic Fibonacci numbers. We obtain inequalities by using matrix norms.

Keywords

hyperharmonic Fibonacci number r-Circulant matrixmatrix norm

MSC

15A6015B0511B39

1 Introduction

The circulant and r-circulant matrices have a connection to signal processing, probability, numerical analysis, coding theory, and many other areas. An \(n\times n\) matrix \(C_{r}\) is called an r-circulant matrix defined as follows:
$$ C_{r}= \begin{pmatrix} c_{0} & c_{1} & c_{2} & \cdots& c_{n-2} & c_{n-1} \\ rc_{n-1} & c_{0} & c_{1} & \cdots& c_{n-3} & c_{n-2} \\ rc_{n-2} & rc_{n-1} & c_{0} & \cdots& c_{n-4} & c_{n-3} \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ rc_{1} & rc_{2} & rc_{3} & \cdots& rc_{n-1} & c_{0} \end{pmatrix}. $$
Since the matrix \(C_{r}\) is determined by its row elements and r, we denote \(C_{r}=\operatorname{Circ}(c_{0},c_{1},c_{2}, \ldots,c_{n-1})\). In particular for \(r=1\)
$$ C= \begin{pmatrix} c_{0} & c_{1} & c_{2} & \cdots& c_{n-2} & c_{n-1} \\ c_{n-1} & c_{0} & c_{1} & \cdots& c_{n-3} & c_{n-2} \\ c_{n-2} & c_{n-1} & c_{0} & \cdots& c_{n-4} & c_{n-3} \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ c_{1} & c_{2} & c_{3} & \cdots& c_{n-1} & c_{0} \end{pmatrix} $$
is called a circulant matrix and we denote it for brevity by \(C=\operatorname{Circ}(c_{0},c_{1},c_{2},\ldots,c_{n-1})\). The eigenvalues of C are
$$ \lambda_{j}=\sum_{i=0}^{n-1}c_{i} \bigl(w^{j}\bigr)^{i}, $$
(1)
where \(w=e^{\frac{2\pi i}{n}}\) and \(i=\sqrt{-1}\).

Many authors have investigated the norms of circulant and r-circulant matrices. In [1], Solak studied the lower and upper bounds for the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries. In [2], Kocer et al. obtained norms of circulant and semicirculant matrices with Horadams numbers. In [3], Zhou et al. gave the spectral norms of circulant-type matrices involving binomial coefficients and harmonic numbers. In [4], Zhou calculated spectral norms for circulant matrices with binomial coefficients combined with Fibonacci and Lucas number entries. In [5], Shen and Cen have given upper and lower bounds for the spectral norms of r-circulant matrices with classical Fibonacci and Lucas number entries. In [6], Bahşı and Solak computed the spectral norms of circulant and r-circulant matrices with the hyper-Fibonacci and hyper-Lucas numbers. In [7], Jiang and Zhou studied spectral norms of even-order r-circulant matrices.

Motivated by the above papers, we compute the spectral norms and Euclidean norm of circulant and r-circulant matrices with the harmonic and hyperharmonic Fibonacci entries. The scheme of this paper is as follows. In Section 2, we present some definitions, preliminaries, and lemmas related to our study. In Section 3, we calculate spectral norms of circulant matrix with harmonic Fibonacci entries. Moreover, we obtain the Euclidean norms of r-circulant matrices and give lower and upper bounds for the spectral norms of r-circulant matrices with harmonic and hyperharmonic Fibonacci entries.

2 Preliminaries

The Fibonacci numbers \(F_{n}\) are defined by the following recurrence relation for \(n\geq1\):
$$ F_{n+1}=F_{n}+F_{n-1}, $$
where \(F_{0}=0\), \(F_{1}=1\). In [8], the authors investigated the finite sum of the reciprocals of Fibonacci numbers,
$$ \mathbb{F}_{n}=\sum_{k=1}^{n} \frac{1}{F_{k}}, $$
which are called harmonic Fibonacci numbers. Then they gave a combinatoric identity related to harmonic Fibonacci numbers as follows:
$$ \sum_{k=0}^{n-1}F_{k-1} \mathbb{F}_{k}=F_{n}\mathbb{F}_{n}-n. $$
(2)
Moreover, in [8], they defined hyperharmonic Fibonacci numbers for \(n,r\geq1\)
$$ \mathbb{F}_{n}^{(r)}=\sum_{k=1}^{n} \mathbb{F}_{k}^{(r-1)}, $$
where \(\mathbb{F}_{n}^{(0)}=\frac{1}{F_{n}}\) and \(\mathbb{F}_{0}=0\). At this point, we give some definitions and lemmas related to our study.

Definition 1

Let \(A=(a_{ij})\) be any \(m\times n\) matrix. The Euclidean norm of matrix A is
$$ \Vert A\Vert _{E}=\sqrt{ \Biggl(\sum _{i=1}^{m}\sum_{j=1}^{n} \vert a_{ij}\vert ^{2} \Biggr)}. $$

Definition 2

Let \(A=(a_{ij})\) be any \(m\times n\) matrix. The spectral norm of matrix A is
$$ \Vert A\Vert _{2}=\sqrt{\max_{1\leq i\leq n} \lambda_{i}\bigl(A^{H}A\bigr)}, $$
where \(\lambda_{i}(A^{H}A)\) is an eigenvalue of \(A^{H}A\) and \(A^{H}\) is the conjugate transpose of matrix A.
Then the following inequalities hold for the Euclidean norm and the spectral norm:
$$\begin{aligned}& \frac{1}{\sqrt{n}}\Vert A\Vert _{E}\leq \Vert A\Vert _{2}\leq \Vert A\Vert _{E}, \end{aligned}$$
(3)
$$\begin{aligned}& \Vert A\Vert _{2}\leq \Vert A\Vert _{E}\leq\sqrt {n}\Vert A\Vert _{2}. \end{aligned}$$
(4)

Lemma 1

[9]

Let A and B be two \(m\times n\) matrices. Then we have
$$ \Vert A\circ B\Vert _{2}\leq \Vert A\Vert _{2} \Vert B\Vert _{2}, $$
where \(A\circ B\) is the Hadamard product of A and B.

Lemma 2

[9]

Let A and B be two \(n\times m\) matrices. We have
$$ \Vert A\circ B\Vert _{2}\leq r_{1}(A)c_{1}(B), $$
where
$$\begin{aligned}& r_{1}(A)=\max_{1\leq i\leq m}\sqrt{\sum _{j=1}^{n}\vert a_{ij}\vert ^{2}},\\& c_{1}(B)=\max_{1\leq j\leq n}\sqrt{\sum _{i=1}^{m}\vert b_{ij}\vert ^{2}}. \end{aligned}$$

Definition 3

[10]

The difference operator of \(f(x)\) is defined as
$$ \Delta f(x)=f(x+1)-f(x). $$

Definition 4

[10]

A function \(f(x)\) with the property that \(\Delta f(x)=g(x)\) is called the anti-difference operator of \(g(x)\).

Lemma 3

[10]

If \(\Delta f(x)=g(x)\), then
$$ \sum_{a}^{b}g(x)\delta_{x}= \sum_{x=a}^{b-1}g(x)=f(b)-f(a). $$

Lemma 4

[10]

We have
$$ \sum_{a}^{b}u(x)\Delta v(x) \delta_{x}= u(x)v(x)| _{a}^{b+1}-\sum _{a}^{b}v(x+1)\Delta u(x) \delta_{x}. $$
(5)

Lemma 5

[10]

For \(m\neq-1\) we have
$$ \sum x^{\underline{m}}\delta_{x}=\frac{x^{\underline{m+1}}}{m+1}, $$
where \(x^{\underline{m}}=x(x-1)(x-2)\cdots(x-m+1)\).

3 Main results

Theorem 1

[8]

Let \(C_{1}=\operatorname{Circ}(\mathbb{F}_{0},\mathbb {F}_{1},\mathbb{F}_{2},\ldots, \mathbb{F}_{n-1})\) be an \(n\times n\) circulant matrix. The spectral norm of \(C_{1}\) is
$$ \Vert C_{1}\Vert _{2}=n\mathbb{F}_{n}-\sum _{k=0}^{n-1}\frac {k+1}{F_{k+1}}. $$

Theorem 2

[8]

Let \(C^{(k)}=\operatorname{Circ}(\mathbb{F}_{0}^{(k)},\mathbb {F}_{1}^{(k)},\mathbb{F}_{2}^{(k)},\ldots,\mathbb{F}_{n-1}^{(k)})\) be an \(n\times n\) circulant matrix. The spectral norm of \(C^{(k)}\) is
$$ \bigl\Vert C^{(k)}\bigr\Vert _{2}=\mathbb{F}_{n-1}^{(k+1)}. $$

Theorem 3

Let
$$ C=\operatorname{Circ}(F_{-1}\mathbb{F}_{0},F_{0} \mathbb{F}_{1},\ldots ,F_{n-2}\mathbb{F}_{n-1}) $$
(6)
be an \(n\times n\) circulant matrix. Then the spectral norm of the matrix C is
$$ \Vert C\Vert _{2}=F_{n}\mathbb{F}_{n}-n. $$

Proof

Since C is a circulant matrix, from (1), for all \(t=0,1,\ldots,s-1\),
$$ \lambda_{t}(C)=\sum_{i=0}^{s-1}F_{i-1} \mathbb{F}_{i}\bigl(w^{t}\bigr)^{i}. $$
Then, for \(t=0\),
$$ \lambda_{0}(C)=\sum_{i=0}^{s-1}F_{i-1} \mathbb{F}_{i} $$
(7)
and from (2), \(\lambda_{0}(C)=F_{n}\mathbb{F}_{n}-n\). Hence, for \(1\leq m\leq n-1\), we have
$$ \vert \lambda_{m}\vert =\Biggl\vert \sum _{i=0}^{s-1}F_{i-1}\mathbb{F}_{i} \bigl(w^{t}\bigr)^{i}\Biggr\vert \leq\Biggl\vert \sum _{i=0}^{s-1}F_{i-1} \mathbb{F}_{i}\Biggr\vert \bigl\vert \bigl(w^{t} \bigr)^{i}\bigr\vert \leq\sum_{i=0}^{s-1}F_{i-1} \mathbb {F}_{i}. $$
(8)
Since C is a normal matrix, we have
$$ \Vert C\Vert _{2}=\max_{0\leq m\leq n-1}\vert \lambda _{m}\vert . $$
(9)
From (7), (8), (9), and (2), we have
$$ \Vert C\Vert _{2}=F_{n}\mathbb{F}_{n}-n. $$
 □

Corollary 1

We have
$$ \sqrt{\sum_{k=0}^{n-1}F_{k-1}^{2} \mathbb{F}_{n}^{2}}\leq F_{n} \mathbb{F}_{n}-n \leq\sqrt{n\sum_{k=0}^{n-1}F_{k-1}^{2} \mathbb {F} _{n}^{2}}. $$

Proof

The proof is trivial from Definition 1 and the relation between the Euclidean norm and the spectral norm in (3). □

Theorem 4

Let
$$ C_{r}^{(k)}=\operatorname{Circ}\bigl(\mathbb{F}_{0}^{(k)}, \mathbb {F}_{1}^{(k)},\ldots, \mathbb{F}_{n-1}^{(k)} \bigr) $$
(10)
be an \(n\times n\) r-circulant matrix. The Euclidean norm of \(C_{r}^{(k)}\) is
$$\begin{aligned} \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E}={}& \Biggl[ \frac{n}{2} \bigl( n+1+(n-1)\vert r\vert ^{2} \bigr) \bigl( \mathbb{F} _{n}^{(k)} \bigr) ^{2}\\ &{}-\frac{1}{2}\sum_{s=0}^{n-1}(s+1) \bigl( 2n+s\bigl(\vert r\vert ^{2}-1\bigr) \bigr) \bigl( \mathbb{F}_{s+1}^{(k-1)}+2 \mathbb{F}_{s}^{(k)} \bigr)\mathbb{F}_{s+1}^{(k-1)} \Biggr] ^{\frac{1}{2}}. \end{aligned}$$

Proof

From the definition of the Euclidean norm we have
$$\begin{aligned} \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E} =& \Biggl[ \sum_{s=0}^{n-1}(n-s) \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}+\sum _{s=0}^{n-1}s\vert r\vert ^{2} \bigl( \mathbb {F}_{s}^{(k)} \bigr) ^{2} \Biggr] ^{\frac{1}{2}} \\ =& \Biggl[ \sum_{s=0}^{n-1}\bigl(n+s\bigl( \vert r\vert ^{2}-1\bigr)\bigr) \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2} \Biggr] ^{\frac{1}{2}}. \end{aligned}$$
Now we will use the property of the difference operator in Lemma 4. Let \(u(s)= ( \mathbb{F}_{s}^{(k)} ) ^{2}\) and \(\Delta v(s)=n+s(\vert r\vert ^{2}-1)\). Then using the definition of the hyperharmonic Fibonacci numbers we obtain \(\Delta u(s)=\mathbb{F} _{s+1}^{(k-1)}(\mathbb{F}_{s+1}^{(k-1)}+2\mathbb {F}_{s}^{(k)})\) and \(v(s)=ns+\frac{s^{\underline{2}}}{2}(\vert r\vert ^{2}-1)\). By using (5), we have
$$\begin{aligned} \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E}={}& \Biggl[ \frac{n}{2} \bigl(n+1+(n-1)\vert r\vert ^{2} \bigr) \bigl( \mathbb {F}_{n}^{(k)} \bigr)^{2}\\ &{}-\frac{1}{2}\sum_{s=0}^{n-1}(s+1) \bigl(2n+s\bigl(\vert r\vert ^{2}-1\bigr) \bigr) \bigl(\mathbb {F}_{s+1}^{(k-1)}+2\mathbb{F}_{s}^{(k)}\bigr) \mathbb{F}_{s+1}^{(k-1)} \Biggr] ^{\frac{1}{2}}. \end{aligned}$$
 □

Corollary 2

Let \(C_{r}=\operatorname{Circ}(\mathbb{F}_{0},\mathbb{F}_{1},\ldots ,\mathbb{F}_{n-1})\) be an \(n\times n\) r-circulant matrix. The Euclidean norm of \(C_{r}\) is
$$\begin{aligned} \Vert C_{r}\Vert _{E}={}& \Biggl[ \biggl( n^{2}+\frac {n^{\underline{2}}}{2}\bigl(\vert r\vert ^{2}-1\bigr) \biggr) \mathbb {F}_{n}^{2}\\ &{}-\sum _{s=0}^{n-1} \biggl(n(s+1)+\frac {(s+1)^{{\underline{2}}}}{2}\bigl( \vert r\vert ^{2}-1\bigr) \biggr) \biggl( 2\mathbb{F}_{s}+ \frac{1}{F_{s+1}} \biggr) \frac{1}{F_{s+1}} \Biggr] ^{\frac{1}{2}}. \end{aligned}$$

Proof

It is clear that the proof can be completed if we take \(k=1\) in Theorem 4. □

Corollary 3

[8]

Let \(C_{1}=\operatorname{Circ}(\mathbb{F}_{0},\mathbb {F}_{1},\ldots,\mathbb{F}_{n-1})\) be an \(n\times n\) matrix. The Euclidean norm is
$$ \Vert C_{1}\Vert _{E}= \Biggl[ n^{2} \mathbb{F}_{n}^{2}-n\sum_{k=0}^{n-1} \frac{k+1}{F_{k+1}} \biggl( 2\mathbb{F}_{k}+\frac {1}{F_{k+1}} \biggr) \Biggr] ^{\frac{1}{2}}. $$

Proof

It is easily seen that the proof can be completed if we take \(k=r=1\) in Theorem 4. □

Now we give upper and lower bounds for the spectral norms of r-circulant matrices.

Theorem 5

Let \(C_{r}^{(k)}=\operatorname{Circ}(\mathbb{F}_{0}^{(k)},\mathbb {F}_{1}^{(k)},\ldots,\mathbb{F}_{n-1}^{(k)})\) be an \(n\times n\) r-circulant matrix.
  1. (i)
    If \(\vert r\vert \geq1\), then
    $$ \frac{1}{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq\bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq \vert r\vert \sqrt{n-1} \mathbb{F}_{n-1}^{(k+1)}. $$
     
  2. (ii)
    If \(\vert r\vert <1\), then
    $$ \frac{\vert r\vert }{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq \bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq\sqrt{n-1} \mathbb{F}_{n-1}^{(k+1)}. $$
     

Proof

Since we have the matrix
$$ C_{r}^{(k)}= \begin{pmatrix} \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \mathbb{F}_{2}^{(k)} & \cdots& \mathbb{F}_{n-2}^{(k)} & \mathbb{F}_{n-1}^{(k)} \\ r\mathbb{F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \cdots & \mathbb{F}_{n-3}^{(k)} & \mathbb{F}_{n-2}^{(k)} \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ r\mathbb{F}_{2}^{(k)} & r\mathbb {F}_{3}^{(k)} & r\mathbb{F}_{4}^{(k)} & \cdots& \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} \\ r\mathbb{F}_{1}^{(k)} & r\mathbb{F}_{2}^{(k)} & r\mathbb{F}_{3}^{(k)} & \cdots& r\mathbb{F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} \end{pmatrix} , $$
we have
$$ \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E}=\sqrt {\sum_{s=0}^{n-1}(n-s) \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}+\sum _{s=0}^{n-1}s\vert r\vert ^{2} \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}}. $$
(i) In [8], for the sum of the squares of hyperharmonic Fibonacci numbers, we have
$$ \frac{1}{\sqrt{n}}\mathbb{F}_{n-1}^{(r+1)}\leq\sqrt{\sum _{k=0}^{n-1} \bigl( \mathbb{F}_{k}^{(r)} \bigr) ^{2}}\leq\mathbb {F}_{n-1}^{(r+1)}. $$
(11)
Since \(\vert r\vert \geq1\) and by (11), we have
$$ \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E}\geq \sqrt{\sum_{s=0}^{n-1}(n-s) \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}+\sum _{s=0}^{n-1}s \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}}\geq\sqrt{n\sum_{s=0}^{n-1} \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}}\geq \mathbb {F}_{n-1}^{(k+1)}. $$
From (3)
$$ \frac{1}{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq\bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}. $$
On the other hand, let the matrices A and B be defined by
$$ A= \begin{pmatrix} \mathbb{F}_{0}^{(k)} & 1 & 1 & \cdots& 1 & 1 \\ r & \mathbb {F}_{0}^{(k)} & 1 & \cdots& 1 & 1 \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ r & r & r & \cdots& \mathbb{F}_{0}^{(k)} & 1 \\ r & r & r & \cdots& r & \mathbb{F}_{0}^{(k)} \end{pmatrix} $$
and
$$ B= \begin{pmatrix} \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \mathbb{F}_{2}^{(k)} & \cdots& \mathbb{F}_{n-2}^{(k)} & \mathbb{F}_{n-1}^{(k)} \\ \mathbb{F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \cdots & \mathbb{F}_{n-3}^{(k)} & \mathbb{F}_{n-2}^{(k)} \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ \mathbb{F}_{2}^{(k)} & \mathbb {F}_{3}^{(k)} & \mathbb{F}_{4}^{(k)} & \cdots& \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} \\ \mathbb{F}_{1}^{(k)} & \mathbb{F}_{2}^{(k)} & \mathbb{F}_{3}^{(k)} & \cdots& \mathbb{F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} \end{pmatrix} . $$
That is, \(C_{r}^{(k)}=A\circ B\). Then we obtain
$$ r_{1}(A)=\max_{1\leq i\leq n}\sqrt{\sum _{j=1}^{n}\vert a_{ij}\vert ^{2}}=\sqrt{\sum_{j=1}^{n} \vert a_{nj}\vert ^{2}}=\sqrt{( n-1) \vert r\vert ^{2}} $$
and
$$ c_{1}(B)=\max_{1\leq j\leq n}\sqrt{\sum _{i=1}^{n}\vert b_{ij}\vert ^{2}}=\sqrt{\sum_{i=1}^{n} \vert b_{in}\vert ^{2}}=\sqrt{\sum _{s=0}^{n-1} \bigl( \mathbb {F}_{s}^{(k)} \bigr) ^{2}}. $$
Hence, from (11) and Lemma 1, we have
$$ \bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq \vert r\vert \sqrt{ n-1} \mathbb{F}_{n-1}^{(k+1)}. $$
Thus, we have
$$ \frac{1}{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq\bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq \vert r\vert \sqrt{ n-1} \mathbb{F}_{n-1}^{(k+1)} . $$
(ii) From \(\vert r\vert <1\) and from (11), we have
$$\begin{aligned} \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E} =& \sqrt{\sum_{s=0}^{n-1}(n-s) \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}+\sum _{s=0}^{n-1}s\vert r\vert ^{2} \bigl( \mathbb {F}_{s}^{(k)} \bigr) ^{2}} \\ \geq&\sqrt{\sum_{s=0}^{n-1}(n-s) \vert r\vert ^{2} \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}+\sum_{s=0}^{n-1}s\vert r\vert ^{2} \bigl(\mathbb {F}_{s}^{(k)} \bigr) ^{2}} \\ =&\vert r\vert \sqrt{n\sum_{s=0}^{n-1} \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}} \\ \geq&\vert r\vert \mathbb{F}_{n-1}^{(k+1)}. \end{aligned}$$
From (3),
$$ \frac{\vert r\vert }{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq \bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}. $$
On the other hand, let the matrices A and B be defined by
$$ A= \begin{pmatrix} \mathbb{F}_{0}^{(k)} & 1 & 1 & \cdots& 1 & 1 \\ r & \mathbb {F}_{0}^{(k)} & 1 & \cdots& 1 & 1 \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ r & r & r & \cdots& \mathbb{F}_{0}^{(k)} & 1 \\ r & r & r & \cdots& r & \mathbb{F}_{0}^{(k)} \end{pmatrix} $$
and
$$ B= \begin{pmatrix} \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \mathbb{F}_{2}^{(k)} & \cdots& \mathbb{F}_{n-2}^{(k)} & \mathbb{F}_{n-1}^{(k)} \\ \mathbb {F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \cdots& \mathbb{F}_{n-3}^{(k)} & \mathbb{F}_{n-2}^{(k)} \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ \mathbb{F}_{2}^{(k)} & \mathbb {F}_{3}^{(k)} & \mathbb{F}_{4}^{(k)} & \cdots& \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} \\ \mathbb{F}_{1}^{(k)} & \mathbb{F}_{2}^{(k)} & \mathbb{F}_{3}^{(k)} & \cdots& \mathbb{F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} \end{pmatrix} . $$
Thus \(C_{r}^{(k)}=A\circ B\). Then we obtain
$$ r_{1}(A)=\max_{1\leq i\leq n}\sqrt{\sum _{j=1}^{n}\vert a_{ij}\vert ^{2}}=\sqrt{ \bigl( \mathbb{F}_{0}^{(k)} \bigr) ^{2}+n-1}=\sqrt{n-1} $$
and
$$ c_{1}(B)=\max_{1\leq j\leq n}\sqrt{\sum _{i=1}^{n}\vert b_{ij}\vert ^{2}}=\sqrt{\sum_{s=0}^{n-1} \bigl( \mathbb {F}_{s}^{(k)} \bigr) ^{2}}. $$
Therefore, from (11) and Lemma 1, we have
$$ \bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq \sqrt{n-1} \mathbb{F}_{n-1}^{(k+1)}. $$
Thus, we have
$$ \frac{\vert r\vert }{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq \bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq\sqrt{n-1} \mathbb{F}_{n-1}^{(k+1)}. $$
 □

Corollary 4

Let \(C_{r}=\operatorname{Circ}(\mathbb{F}_{0},\mathbb{F}_{1},\ldots ,\mathbb{F}_{n-1})\) be an \(n\times n\) r-circulant matrix.
  1. (i)
    If \(\vert r\vert \geq1\), then
    $$ \frac{1}{\sqrt{n}}\mathbb{F}_{n-1}^{(2)}\leq \Vert C_{r}\Vert _{2}\leq \vert r\vert \sqrt{ n-1} \mathbb{F}_{n-1}^{(2)}. $$
     
  2. (ii)
    If \(\vert r\vert <1\), then
    $$ \frac{\vert r\vert }{\sqrt{n}}\mathbb{F}_{n-1}^{(2)}\leq \Vert C_{r}\Vert _{2}\leq\sqrt{n-1}\mathbb{F}_{n-1}^{(2)}. $$
     

Proof

It is easily seen that the proof can be completed if we take \(k=1\) in Theorem 5. □

4 Numerical examples

In this section, we present some numerical examples by using Maple 11.

Example 1

Let \(C=\operatorname{Circ}(F_{-1}\mathbb{F}_{0},F_{0}\mathbb {F}_{1},\ldots,F_{n-2}\mathbb{F}_{n-1})\) be as in (6). We obtain the spectral norms of some \(n\times n\) C matrices, with the aid of Theorem 3 (see Table 1).
Table 1

Spectral norms of C

n

\(\boldsymbol {\| C\|_{2}}\)

n = 5

0.1016666667 × 102

n = 10

0.1731757972 × 103

n = 50

0.4228842484 × 1011

n = 100

0.1190154990 × 1022

n = 500

0.4684460937 × 10105

n = 1,000

0.1460426641 × 10210

Example 2

Let \(C_{r}^{(k)}=\operatorname{Circ}(\mathbb{F}_{0}^{(k)},\mathbb {F}_{1}^{(k)},\ldots, \mathbb{F}_{4}^{(k)})\) be \(5\times5\) r-circulant matrix as in (10). We obtain Euclidean norms of \(C_{r}^{(k)}\) for some values of r and k, with the aid of Theorem 4 (see Table 2).
Table 2

Euclidean norms of \(\pmb{C_{r}^{(k)}}\) for \(\pmb{n=5}\)

k / r

k  = 1

k  = 2

k  = 3

k  = 4

r = −2

\(\frac{\sqrt{9{,}935}}{6}\)

\(\frac{\sqrt{61{,}598}}{6}\)

\(\frac{ \sqrt{246{,}743}}{6}\)

\(\frac{\sqrt{755{,}966}}{6}\)

r = −0.5

\(\frac{\sqrt{7{,}415}}{12}\)

\(\frac{\sqrt{37{,}127}}{12}\)

\(\frac{\sqrt{136{,}007}}{12}\)

\(\frac{\sqrt{397{,}559}}{12}\)

r = 0.1

\(\frac{\sqrt{133{,}655}}{60}\)

\(\frac{\sqrt{593{,}351}}{60}\)

\(\frac{\sqrt{2{,}038{,}631}}{60}\)

\(\frac{\sqrt{5{,}736{,}887}}{60}\)

r = 0.9

\(\frac{\sqrt{306{,}055}}{60}\)

\(\frac{\sqrt{1{,}709{,}431}}{60}\)

\(\frac{\sqrt{6{,}577{,}111}}{60}\)

\(\frac{\sqrt{19{,}743{,}847}}{60}\)

r = 1

\(\frac{\sqrt{3{,}470}}{6}\)

\(\frac{\sqrt{19{,}745}}{6}\)

\(\frac{5 \sqrt{3{,}062}}{6}\)

\(\frac{\sqrt{230{,}705}}{6}\)

r = 1.1

\(\frac{\sqrt{392{,}255}}{60}\)

\(\frac{\sqrt{2{,}267{,}471}}{60}\)

\(\frac{\sqrt{8{,}846{,}351}}{60}\)

\(\frac{\sqrt{26{,}747{,}327}}{60}\)

r = 10

\(\frac{\sqrt{216{,}815}}{6}\)

\(\frac{\sqrt{1{,}400{,}894}}{6}\)

\(\frac{\sqrt{5{,}692{,}919}}{6}\)

\(\frac{\sqrt{17{,}564{,}318}}{6}\)

Example 3

Let \(C_{r}^{(k)}=\operatorname{Circ}(\mathbb{F}_{0}^{(k)},\mathbb {F}_{1}^{(k)},\ldots,\mathbb{F}_{4}^{(k)})\) be \(5\times5\) r-circulant matrix as in (10). We obtain some lower and upper bounds for the spectral norms of \(C_{r}^{(k)}\) for some values of r and k, with the aid of Theorem 5 (see Tables 3 and 4).
Table 3

Some lower and upper bounds for the spectral norms of \(\pmb{C_{r}^{(k)}}\) for \(\pmb{n=5}\) and \(\pmb{|r|\geq1}\)

 

| r | ≥ 1

\(\boldsymbol {\frac{1}{\sqrt{n}}\mathbb {F}_{n-1}^{(k+1)}}\)

\(\boldsymbol {\|C_{r}^{(k)}\|_{2}}\)

\(\boldsymbol {|r|\sqrt{n-1}\mathbb{F}_{n-1}^{(k+1)}}\)

k = 1

r = −2

3.726779962

13.07393997

33.33333333

r = 1

3.726779962

8.333333332

16.66666667

r = 1.1

3.726779962

6.187213310

18.33333333

k = 2

r = −2

7.975309119

29.94984421

71.33333333

r = 1

7.975309119

17.83333333

35.66666667

r = 1.1

7.975309119

19.01179206

39.23333333

k = 3

r = −2

14.45990625

56.50736302

129.3333333

r = 1

14.45990625

32.33333334

64.66666667

r = 1.1

14.45990625

34.60097132

71.13333333

k = 4

r = −2

23.62778496

94.70523788

211.3333333

r = 1

23.62778496

52.83333332

105.6666667

r = 1.1

23.62778496

56.68238740

116.2333333

Table 4

Some lower and upper bounds for the spectral norms of \(\pmb{C_{r}^{(k)}}\) for \(\pmb{n=5}\) and \(\pmb{|r|<1}\)

 

| r |<1

\(\boldsymbol {\frac{| r|}{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}}\)

\(\boldsymbol {\| C_{r}^{(k)}\|_{2}}\)

\(\boldsymbol {\sqrt{n-1}\mathbb{F}_{n-1}^{(k+1)}}\)

k = 1

r = −0.5

1.863389981

6.051069352

16.66666667

r = 0.1

0.372677996

5.912862964

16.66666667

r = 0.9

3.354101966

7.874554252

16.66666667

k = 2

r = −0.5

3.987654558

13.14607243

35.66666667

r = 0.1

0.797530912

12.64953828

35.66666667

r = 0.9

7.177778206

16.74610810

35.66666667

k = 3

r = −0.5

7.229953125

24.39979647

64.66666667

r = 0.1

1.445990625

23.46328510

64.66666667

r = 0.9

13.01391563

30.26866878

64.66666667

k = 4

r = −0.5

11.81389248

40.74250279

105.6666667

r = 0.1

2.362778496

39.32798158

105.6666667

r = 0.9

21.26500646

49.37728355

105.6666667

Declarations

Acknowledgements

The authors are grateful to two anonymous referees and the associate editor for their careful reading, helpful comments, and constructive suggestions, which improved the presentation of the results.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Gazi University
(2)
Department of Mathematics, Bülent Ecevit University

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© Tuglu and Kızılateş 2015