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: 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 \(r=1\)
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 where \(w=e^{\frac{2\pi i}{n}}\) and \(i=\sqrt{-1}\).
$$ C_{r}= \begin{pmatrix} c_{0} & c_{1} & c_{2} & \cdots& c_{n-2} & c_{n-1} \\ rc_{n-1} & c_{0} & c_{1} & \cdots& c_{n-3} & c_{n-2} \\ rc_{n-2} & rc_{n-1} & c_{0} & \cdots& c_{n-4} & c_{n-3} \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ rc_{1} & rc_{2} & rc_{3} & \cdots& rc_{n-1} & c_{0} \end{pmatrix}. $$
$$ C= \begin{pmatrix} c_{0} & c_{1} & c_{2} & \cdots& c_{n-2} & c_{n-1} \\ c_{n-1} & c_{0} & c_{1} & \cdots& c_{n-3} & c_{n-2} \\ c_{n-2} & c_{n-1} & c_{0} & \cdots& c_{n-4} & c_{n-3} \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ c_{1} & c_{2} & c_{3} & \cdots& c_{n-1} & c_{0} \end{pmatrix} $$
$$ \lambda_{j}=\sum_{i=0}^{n-1}c_{i} \bigl(w^{j}\bigr)^{i}, $$
(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.
Anzeige
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 \(n\geq1\): where \(F_{0}=0\), \(F_{1}=1\). In [8], the authors investigated the finite sum of the reciprocals of Fibonacci numbers, which are called harmonic Fibonacci numbers. Then they gave a combinatoric identity related to harmonic Fibonacci numbers as follows: Moreover, in [8], they defined hyperharmonic Fibonacci numbers for \(n,r\geq1\)
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.
$$ F_{n+1}=F_{n}+F_{n-1}, $$
$$ \mathbb{F}_{n}=\sum_{k=1}^{n} \frac{1}{F_{k}}, $$
$$ \sum_{k=0}^{n-1}F_{k-1} \mathbb{F}_{k}=F_{n}\mathbb{F}_{n}-n. $$
(2)
$$ \mathbb{F}_{n}^{(r)}=\sum_{k=1}^{n} \mathbb{F}_{k}^{(r-1)}, $$
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 where \(\lambda_{i}(A^{H}A)\) is an eigenvalue of \(A^{H}A\) and \(A^{H}\) is the conjugate transpose of matrix A.
$$ \Vert A\Vert _{2}=\sqrt{\max_{1\leq i\leq n} \lambda_{i}\bigl(A^{H}A\bigr)}, $$
Anzeige
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
where
\(A\circ B\)
is the Hadamard product of
A
and
B.
$$ \Vert A\circ B\Vert _{2}\leq \Vert A\Vert _{2} \Vert B\Vert _{2}, $$
Lemma 2
[9]
Let
A
and
B
be two
\(n\times m\)
matrices. We have
where
$$ \Vert A\circ B\Vert _{2}\leq r_{1}(A)c_{1}(B), $$
$$\begin{aligned}& r_{1}(A)=\max_{1\leq i\leq m}\sqrt{\sum _{j=1}^{n}\vert a_{ij}\vert ^{2}},\\& c_{1}(B)=\max_{1\leq j\leq n}\sqrt{\sum _{i=1}^{m}\vert b_{ij}\vert ^{2}}. \end{aligned}$$
Definition 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
where
\(x^{\underline{m}}=x(x-1)(x-2)\cdots(x-m+1)\).
$$ \sum x^{\underline{m}}\delta_{x}=\frac{x^{\underline{m+1}}}{m+1}, $$
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
be an
\(n\times n\)
circulant matrix. Then the spectral norm of the matrix
C
is
$$ C=\operatorname{Circ}(F_{-1}\mathbb{F}_{0},F_{0} \mathbb{F}_{1},\ldots ,F_{n-2}\mathbb{F}_{n-1}) $$
(6)
$$ \Vert C\Vert _{2}=F_{n}\mathbb{F}_{n}-n. $$
Proof
Since C is a circulant matrix, from (1), for all \(t=0,1,\ldots,s-1\), Then, for \(t=0\), and from (2), \(\lambda_{0}(C)=F_{n}\mathbb{F}_{n}-n\). Hence, for \(1\leq m\leq n-1\), we have Since C is a normal matrix, we have From (7), (8), (9), and (2), we have □
$$ \lambda_{t}(C)=\sum_{i=0}^{s-1}F_{i-1} \mathbb{F}_{i}\bigl(w^{t}\bigr)^{i}. $$
$$ \lambda_{0}(C)=\sum_{i=0}^{s-1}F_{i-1} \mathbb{F}_{i} $$
(7)
$$ \vert \lambda_{m}\vert =\Biggl\vert \sum _{i=0}^{s-1}F_{i-1}\mathbb{F}_{i} \bigl(w^{t}\bigr)^{i}\Biggr\vert \leq\Biggl\vert \sum _{i=0}^{s-1}F_{i-1} \mathbb{F}_{i}\Biggr\vert \bigl\vert \bigl(w^{t} \bigr)^{i}\bigr\vert \leq\sum_{i=0}^{s-1}F_{i-1} \mathbb {F}_{i}. $$
(8)
$$ \Vert C\Vert _{2}=\max_{0\leq m\leq n-1}\vert \lambda _{m}\vert . $$
(9)
$$ \Vert C\Vert _{2}=F_{n}\mathbb{F}_{n}-n. $$
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
Theorem 4
Let
be an
\(n\times n\)
r-circulant matrix. The Euclidean norm of
\(C_{r}^{(k)}\)
is
$$ C_{r}^{(k)}=\operatorname{Circ}\bigl(\mathbb{F}_{0}^{(k)}, \mathbb {F}_{1}^{(k)},\ldots, \mathbb{F}_{n-1}^{(k)} \bigr) $$
(10)
$$\begin{aligned} \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E}={}& \Biggl[ \frac{n}{2} \bigl( n+1+(n-1)\vert r\vert ^{2} \bigr) \bigl( \mathbb{F} _{n}^{(k)} \bigr) ^{2}\\ &{}-\frac{1}{2}\sum_{s=0}^{n-1}(s+1) \bigl( 2n+s\bigl(\vert r\vert ^{2}-1\bigr) \bigr) \bigl( \mathbb{F}_{s+1}^{(k-1)}+2 \mathbb{F}_{s}^{(k)} \bigr)\mathbb{F}_{s+1}^{(k-1)} \Biggr] ^{\frac{1}{2}}. \end{aligned}$$
Proof
From the definition of the Euclidean norm we have 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 □
$$\begin{aligned} \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E} =& \Biggl[ \sum_{s=0}^{n-1}(n-s) \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}+\sum _{s=0}^{n-1}s\vert r\vert ^{2} \bigl( \mathbb {F}_{s}^{(k)} \bigr) ^{2} \Biggr] ^{\frac{1}{2}} \\ =& \Biggl[ \sum_{s=0}^{n-1}\bigl(n+s\bigl( \vert r\vert ^{2}-1\bigr)\bigr) \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2} \Biggr] ^{\frac{1}{2}}. \end{aligned}$$
$$\begin{aligned} \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E}={}& \Biggl[ \frac{n}{2} \bigl(n+1+(n-1)\vert r\vert ^{2} \bigr) \bigl( \mathbb {F}_{n}^{(k)} \bigr)^{2}\\ &{}-\frac{1}{2}\sum_{s=0}^{n-1}(s+1) \bigl(2n+s\bigl(\vert r\vert ^{2}-1\bigr) \bigr) \bigl(\mathbb {F}_{s+1}^{(k-1)}+2\mathbb{F}_{s}^{(k)}\bigr) \mathbb{F}_{s+1}^{(k-1)} \Biggr] ^{\frac{1}{2}}. \end{aligned}$$
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.
(i)
If
\(\vert r\vert \geq1\), then
$$ \frac{1}{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq\bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq \vert r\vert \sqrt{n-1} \mathbb{F}_{n-1}^{(k+1)}. $$
(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 we have
$$ C_{r}^{(k)}= \begin{pmatrix} \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \mathbb{F}_{2}^{(k)} & \cdots& \mathbb{F}_{n-2}^{(k)} & \mathbb{F}_{n-1}^{(k)} \\ r\mathbb{F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \cdots & \mathbb{F}_{n-3}^{(k)} & \mathbb{F}_{n-2}^{(k)} \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ r\mathbb{F}_{2}^{(k)} & r\mathbb {F}_{3}^{(k)} & r\mathbb{F}_{4}^{(k)} & \cdots& \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} \\ r\mathbb{F}_{1}^{(k)} & r\mathbb{F}_{2}^{(k)} & r\mathbb{F}_{3}^{(k)} & \cdots& r\mathbb{F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} \end{pmatrix} , $$
$$ \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E}=\sqrt {\sum_{s=0}^{n-1}(n-s) \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}+\sum _{s=0}^{n-1}s\vert r\vert ^{2} \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}}. $$
(i) In [8], for the sum of the squares of hyperharmonic Fibonacci numbers, we have Since \(\vert r\vert \geq1\) and by (11), we have From (3) On the other hand, let the matrices A and B be defined by and That is, \(C_{r}^{(k)}=A\circ B\). Then we obtain and Hence, from (11) and Lemma 1, we have Thus, 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)
$$ \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E}\geq \sqrt{\sum_{s=0}^{n-1}(n-s) \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}+\sum _{s=0}^{n-1}s \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}}\geq\sqrt{n\sum_{s=0}^{n-1} \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}}\geq \mathbb {F}_{n-1}^{(k+1)}. $$
$$ \frac{1}{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq\bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}. $$
$$ A= \begin{pmatrix} \mathbb{F}_{0}^{(k)} & 1 & 1 & \cdots& 1 & 1 \\ r & \mathbb {F}_{0}^{(k)} & 1 & \cdots& 1 & 1 \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ r & r & r & \cdots& \mathbb{F}_{0}^{(k)} & 1 \\ r & r & r & \cdots& r & \mathbb{F}_{0}^{(k)} \end{pmatrix} $$
$$ B= \begin{pmatrix} \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \mathbb{F}_{2}^{(k)} & \cdots& \mathbb{F}_{n-2}^{(k)} & \mathbb{F}_{n-1}^{(k)} \\ \mathbb{F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \cdots & \mathbb{F}_{n-3}^{(k)} & \mathbb{F}_{n-2}^{(k)} \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ \mathbb{F}_{2}^{(k)} & \mathbb {F}_{3}^{(k)} & \mathbb{F}_{4}^{(k)} & \cdots& \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} \\ \mathbb{F}_{1}^{(k)} & \mathbb{F}_{2}^{(k)} & \mathbb{F}_{3}^{(k)} & \cdots& \mathbb{F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} \end{pmatrix} . $$
$$ r_{1}(A)=\max_{1\leq i\leq n}\sqrt{\sum _{j=1}^{n}\vert a_{ij}\vert ^{2}}=\sqrt{\sum_{j=1}^{n} \vert a_{nj}\vert ^{2}}=\sqrt{( n-1) \vert r\vert ^{2}} $$
$$ c_{1}(B)=\max_{1\leq j\leq n}\sqrt{\sum _{i=1}^{n}\vert b_{ij}\vert ^{2}}=\sqrt{\sum_{i=1}^{n} \vert b_{in}\vert ^{2}}=\sqrt{\sum _{s=0}^{n-1} \bigl( \mathbb {F}_{s}^{(k)} \bigr) ^{2}}. $$
$$ \bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq \vert r\vert \sqrt{ n-1} \mathbb{F}_{n-1}^{(k+1)}. $$
$$ \frac{1}{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq\bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq \vert r\vert \sqrt{ n-1} \mathbb{F}_{n-1}^{(k+1)} . $$
(ii) From \(\vert r\vert <1\) and from (11), we have From (3), On the other hand, let the matrices A and B be defined by and Thus \(C_{r}^{(k)}=A\circ B\). Then we obtain and Therefore, from (11) and Lemma 1, we have Thus, we have □
$$\begin{aligned} \bigl\Vert C_{r}^{(k)}\bigr\Vert _{E} =& \sqrt{\sum_{s=0}^{n-1}(n-s) \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}+\sum _{s=0}^{n-1}s\vert r\vert ^{2} \bigl( \mathbb {F}_{s}^{(k)} \bigr) ^{2}} \\ \geq&\sqrt{\sum_{s=0}^{n-1}(n-s) \vert r\vert ^{2} \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}+\sum_{s=0}^{n-1}s\vert r\vert ^{2} \bigl(\mathbb {F}_{s}^{(k)} \bigr) ^{2}} \\ =&\vert r\vert \sqrt{n\sum_{s=0}^{n-1} \bigl( \mathbb{F}_{s}^{(k)} \bigr) ^{2}} \\ \geq&\vert r\vert \mathbb{F}_{n-1}^{(k+1)}. \end{aligned}$$
$$ \frac{\vert r\vert }{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq \bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}. $$
$$ A= \begin{pmatrix} \mathbb{F}_{0}^{(k)} & 1 & 1 & \cdots& 1 & 1 \\ r & \mathbb {F}_{0}^{(k)} & 1 & \cdots& 1 & 1 \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ r & r & r & \cdots& \mathbb{F}_{0}^{(k)} & 1 \\ r & r & r & \cdots& r & \mathbb{F}_{0}^{(k)} \end{pmatrix} $$
$$ B= \begin{pmatrix} \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \mathbb{F}_{2}^{(k)} & \cdots& \mathbb{F}_{n-2}^{(k)} & \mathbb{F}_{n-1}^{(k)} \\ \mathbb {F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} & \cdots& \mathbb{F}_{n-3}^{(k)} & \mathbb{F}_{n-2}^{(k)} \\ \vdots& \vdots& \vdots& & \vdots& \vdots\\ \mathbb{F}_{2}^{(k)} & \mathbb {F}_{3}^{(k)} & \mathbb{F}_{4}^{(k)} & \cdots& \mathbb{F}_{0}^{(k)} & \mathbb{F}_{1}^{(k)} \\ \mathbb{F}_{1}^{(k)} & \mathbb{F}_{2}^{(k)} & \mathbb{F}_{3}^{(k)} & \cdots& \mathbb{F}_{n-1}^{(k)} & \mathbb{F}_{0}^{(k)} \end{pmatrix} . $$
$$ r_{1}(A)=\max_{1\leq i\leq n}\sqrt{\sum _{j=1}^{n}\vert a_{ij}\vert ^{2}}=\sqrt{ \bigl( \mathbb{F}_{0}^{(k)} \bigr) ^{2}+n-1}=\sqrt{n-1} $$
$$ c_{1}(B)=\max_{1\leq j\leq n}\sqrt{\sum _{i=1}^{n}\vert b_{ij}\vert ^{2}}=\sqrt{\sum_{s=0}^{n-1} \bigl( \mathbb {F}_{s}^{(k)} \bigr) ^{2}}. $$
$$ \bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq \sqrt{n-1} \mathbb{F}_{n-1}^{(k+1)}. $$
$$ \frac{\vert r\vert }{\sqrt{n}}\mathbb{F}_{n-1}^{(k+1)}\leq \bigl\Vert C_{r}^{(k)}\bigr\Vert _{2}\leq\sqrt{n-1} \mathbb{F}_{n-1}^{(k+1)}. $$
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.
(i)
If
\(\vert r\vert \geq1\), then
$$ \frac{1}{\sqrt{n}}\mathbb{F}_{n-1}^{(2)}\leq \Vert C_{r}\Vert _{2}\leq \vert r\vert \sqrt{ n-1} \mathbb{F}_{n-1}^{(2)}. $$
(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 |
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors, NT, and CK, contributed to each part of this work equally and read and approved the final version of the manuscript.