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On the spectral norms of r-circulant matrices with the biperiodic Fibonacci and Lucas numbers
Journal of Inequalities and Applications volume 2017, Article number: 192 (2017)
Abstract
In this paper, we present new upper and lower bounds for the spectral norms of the r-circulant matrices \(Q=C_{r} ( (\frac{b}{a} )^{\frac{\xi (1)}{2}}q_{0}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}q_{1}, (\frac {b}{a} )^{\frac{\xi(3)}{2}}q_{2}, \dots, (\frac{b}{a} )^{\frac{\xi(n)}{2}}q_{n-1} )\) and \(L=C_{r} ( (\frac {b}{a} )^{\frac{\xi(0)}{2}}l_{0}, (\frac{b}{a} )^{\frac{\xi (1)}{2}}l_{1}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}l_{2}, \dots, (\frac{b}{a} )^{\frac{\xi(n-1)}{2}}l_{n-1} ) \) whose entries are the biperiodic Fibonacci and biperiodic Lucas numbers, respectively. Finally, we obtain lower and upper bounds for the spectral norms of Kronecker and Hadamard products of Q and L matrices.
1 Introduction
For \(n\in\mathbb{N}_{0}\), the Fibonacci and Lucas numbers are defined by \(F_{n+2} = F_{n+1} + F_{n}\) and \(L_{n+2} = L_{n+1} + L_{n}\) with the initial conditions \(F_{0}=0\), \(F_{1}=1\) and \(L_{0}=2\), \(L_{1}=1\), respectively. In recent years, there are several applications and generalizations of Fibonacci and Lucas numbers [1–12]. For example, Falcon and Plaza introduced the k-Fibonacci sequence by studying the recursive application of two geometrical transformations used in the well-known 4-triangle longest-edge (4TLE) partition [6]. Edson and Yayenie [3] presented a new generalization of the Fibonacci sequence: for \(n\in\mathbb{N}_{0}\),
They also obtained an extended Binet formula for this sequence:
Afterward, Bilgici [4] defined generalized the Lucas sequence by the following recurrence relation: for \(n\in\mathbb{N}_{0}\),
and The Binet formula for this sequence is
In Eqs. (2) and (4), \(\alpha= \frac{ab + \sqrt {a^{2}b^{2}+4ab}}{2}\) and \(\beta= \frac{ab - \sqrt{a^{2}b^{2}+4ab}}{2}\) are the roots of the characteristic equation of \(x^{2}-abx-ab=0\), and \(\xi(n) = n- 2\lfloor\frac{n}{2}\rfloor\).
In recent years, there have been several studies on the norms, determinants, and inverses of circulant and r-circulant matrices whose entries are special integer sequences [13–27]. For example, Shen and Cen [18] found upper and lower bounds for the spectral norms of r-circulant matrices in the forms \(A=C_{r}(F_{0}, F_{1}, F_{2}, \dots, F_{n-1})\) and \(B=C_{r}(L_{0}, L_{1}, L_{2}, \dots, L_{n-1})\). They also obtained some bounds for the spectral norms of Kronecker and Hadamard products of A and B. Afterward, Shen and Cen [19] gave the upper and lower bounds for the spectral norms of the matrices \(A=C_{r}(F_{k,0}, F_{k,1}, F_{k,2}, \dots , F_{k,n-1})\) and \(B=C_{r}(L_{k,0}, L_{k,1}, L_{k,2}, \dots, L_{k,n-1})\). They also presented some bounds for the spectral norms of Hadamard and Kronecker products of these matrices. Bahşi [16] studied the norms of r-circulant matrices \(H_{r} = \operatorname{Circr}(H_{0}^{(k)},H_{1}^{(k)},H_{2}^{(k)},\dots,H_{n-1}^{(k)})\) and \(\widehat{H_{r}} = \operatorname{Circr}(H_{k}^{(0)},H_{k}^{(1)},H_{k}^{(2)},\dots ,H_{k}^{(n-1)})\), where \(H_{n}^{(k)}\) denotes the nth hyperharmonic number of order r.
Inspired by these studies, in this paper, we compute spectral norms of r-circulant matrices whose entries are the biperiodic Fibonacci and biperiodic Lucas numbers. This study consists of three sections. The first one is the introduction. In the second section, we give some new theorems, corollaries, and some important results. We give a concise conclusion in the last section.
Definition 1.1
For any given \(c_{0},c_{1},c_{2},\dots,c_{n-1} \in\mathbb{C}\), the r-circulant matrix \(C_{r} = (c_{ij})_{n\times n}\) is defined by
It is clear that, for \(r=1\), \(C_{r}\) turns into a classical circulant matrix. Let us take any \(A=[a_{ij}] \in M_{n,n}(\mathbb{C})\). The Frobenius norm of the matrix A is defined by
Also, the spectral norm of the matrix A is given by
where \(\lambda_{i} (A^{H} A)\) are the eigenvalues of \(A^{H} A\) such that \(A^{H}\) is the conjugate transpose of A. Then, the well-known inequality [28] is given by
Lemma 1.2
[28]
For any matrices \(A,B\in M_{m,n}(\mathbb{C})\), we have
where \(A\circ B\) is the Hadamard product of A and B.
Lemma 1.3
[28]
For any matrices \(A\in M_{m,n}(\mathbb{C})\) and \(B\in M_{p,q}(\mathbb {C})\), we have
where \(A\otimes B\) is the Kronecker product of A and B.
Lemma 1.4
[29]
For any matrices \(A=[a_{ij}] \in M_{n,n}(\mathbb{C})\) and \(B=[b_{ij}] \in M_{n,n}(\mathbb{C})\), we have
where \(A \circ B\) is the Hadamard product, \(r_{1}(A) = \max_{1\leq i \leq n}\sqrt{\sum_{j=1}^{n}|a_{ij}|^{2}}\), and \(c_{1}(B) = \max_{1\leq j \leq n}\sqrt{\sum_{i=1}^{n}|b_{ij}|^{2}}\).
Theorem 1.5
[5]
For any positive integer n, we have
2 Main results
In this section, we first give the sum of squares of biperiodic Lucas numbers.
Theorem 2.1
For any positive integer m, we have
Proof
Using the Binet formula of the biperiodic Lucas numbers, we have
Therefore, for any \(k\geq1\),
Using the properties \(ab(\alpha+1)=\alpha^{2}\) and \(ab(\beta+1)=\beta ^{2}\), we get
Observe that
Therefore,
□
Theorem 2.2
Let \(Q=C_{r} ( (\frac{b}{a} )^{\frac{\xi(1)}{2}}q_{0}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}q_{1}, (\frac{b}{a} )^{\frac{\xi(3)}{2}}q_{2}, \dots, (\frac{b}{a} )^{\frac{\xi (n)}{2}}q_{n-1} ) \) be an r-circulant matrix. Then, for \(r \in \mathbb{C}\), we have:
-
if \(|r| \geq1\), then
$$\sqrt{\frac{q_{n} q_{n-1}}{a}} \leq\|Q\|_{2} \leq|r|\frac{q_{n} q_{n-1}}{a}; $$ -
if \(|r| < 1\), then
$$|r|\sqrt{\frac{q_{n} q_{n-1}}{a}} \leq\|Q\|_{2} \leq\sqrt{(n-1) \frac{q_{n} q_{n-1}}{a}}. $$
Proof
The matrix Q is of the form
Then we have
Hence, for \(|r|\geq1\), using Eq. (8), we obtain
that is,
From (6) we have
Now, for \(|r|\geq1\), we give an bound for the spectral norm of the matrix Q. Let the matrices B and C be
and
so that \(Q = B \circ C\). Then we obtain
By Lemma 1.4 we have
Thus,
On the other hand, for \(|r|<1\), we have
that is,
Thus, we obtain
Now, for \(|r|< 1\), we give an upper bound for the spectral norm of the matrix Q. Let the matrices D and E be
and
so that \(Q=D \circ E\). Then we obtain
By Lemma 1.4 we have
Thus,
□
Theorem 2.3
Let \(L=C_{r} ( (\frac{b}{a} )^{\frac{\xi(0)}{2}}l_{0}, (\frac{b}{a} )^{\frac{\xi(1)}{2}}l_{1}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}l_{2}, \dots, (\frac{b}{a} )^{\frac{\xi (n-1)}{2}}l_{n-1} ) \) be an r-circulant matrix. Then, for \(r \in\mathbb{C}\), we have:
-
if \(|r| \geq1\), then
$$\sqrt{\frac{l_{n} l_{n-1}}{a}+2} \leq\|L\|_{2} \leq|r| \biggl( \frac{l_{n} l_{n-1}}{a}+2 \biggr); $$ -
if \(|r| < 1\), then
$$|r|\sqrt{\frac{l_{n} l_{n-1}}{a}+2} \leq\|L\|_{2} \leq\sqrt{n \biggl( \frac {l_{n} l_{n-1}}{a}+2 \biggr)}. $$
Proof
The matrix L is of the form
Then we have
Hence, for \(|r|\geq1\), using Eq. (9), we obtain
that is,
From (6) we have
Now, for \(|r|\geq1\), we give an upper bound for the spectral norm of the matrix L. Let the matrices F and H be
and
so that \(L = F \circ H\). Then we obtain
By Lemma 1.4 we have
Thus,
On the other hand, for \(|r|<1\), we have
that is,
Thus, we obtain
Now, for \(|r|< 1\), we give an upper bound for the spectral norm of the matrix L. Let the matrices G and K be
and
so that \(L=G \circ K\). Then we obtain
By Lemma 1.4 we have
Thus,
□
Corollary 2.1
Let \(Q=C_{r} ( (\frac{b}{a} )^{\frac{\xi(1)}{2}}q_{0}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}q_{1}, (\frac{b}{a} )^{\frac{\xi(3)}{2}}q_{2}, \dots, (\frac{b}{a} )^{\frac{\xi (n)}{2}}q_{n-1} )\) and \(L= C_{r} ( (\frac{b}{a} )^{\frac{\xi(0)}{2}}l_{0}, (\frac{b}{a} )^{\frac{\xi (1)}{2}}l_{1}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}l_{2}, \dots, (\frac{b}{a} )^{\frac{\xi(n-1)}{2}}l_{n-1} ) \) be r-circulant matrices, where \(r\in\mathbb{C}\).
-
(i)
If \(|r|\geq1\), then
$$ \|Q\circ L\|_{2} \leq|r|^{2} \frac{q_{n}q_{n-1}}{a} \biggl( \frac{l_{n} l_{n-1}}{a} + 2 \biggr). $$ -
(ii)
If \(|r| < 1\), then
$$ \|Q\circ L\|_{2} \leq\sqrt{n(n-1)\frac{q_{n}q_{n-1}}{a} \biggl( \frac{l_{n} l_{n-1}}{a} +2 \biggr)}. $$
Proof
Since \(\|Q\circ L\|_{2} \leq\|Q\|_{2} \|L\|_{2}\), the proof is trivial by Theorems 2.2 and 2.3. □
Corollary 2.2
Let \(Q=C_{r} ( (\frac{b}{a} )^{\frac{\xi(1)}{2}}q_{0}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}q_{1}, (\frac{b}{a} )^{\frac{\xi(3)}{2}}q_{2}, \dots, (\frac{b}{a} )^{\frac{\xi (n)}{2}}q_{n-1} )\) and \(L= C_{r} ( (\frac{b}{a} )^{\frac{\xi(0)}{2}}l_{0}, (\frac{b}{a} )^{\frac{\xi (1)}{2}}l_{1}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}l_{2}, \dots, (\frac{b}{a} )^{\frac{\xi(n-1)}{2}}l_{n-1} ) \) be r-circulant matrices, where \(r\in\mathbb{C}\).
-
(i)
If \(|r|\geq1\), then
$$ \|Q \otimes L\|_{2} \geq\sqrt{\frac{q_{n}q_{n-1}}{a} \biggl( \frac{l_{n} l_{n-1}}{a} + 2 \biggr)} $$and
$$ \|Q\otimes L\|_{2} \leq|r|^{2} \frac{q_{n}q_{n-1}}{a} \biggl( \frac{l_{n} l_{n-1}}{a} + 2 \biggr)l. $$ -
(ii)
If \(|r|< 1\), then
$$ \|Q \otimes L\|_{2} \geq|r|^{2} \sqrt{\frac{q_{n} q_{n-1}}{a} \biggl( \frac {l_{n} l_{n-1}}{a} + 2 \biggr)} $$and
$$ \|Q\otimes L\|_{2} \leq\sqrt{n(n-1)\frac{q_{n}q_{n-1}}{a} \biggl( \frac {l_{n} l_{n-1}}{a} +2 \biggr)}. $$
Proof
Since \(\|Q\otimes L\|_{2} = \|Q\|_{2}\|L\|_{2}\), the proof is trivial by Theorems 2.2 and 2.3. □
3 Conclusion
In this paper, we obtain new upper and lower bounds for the spectral norms of the r-circulant matrices Q and L whose entries are the biperiodic Fibonacci and biperiodic Lucas numbers. This study can be reduced to various studies for the specific values of a and b in the literature. For example, if \(a=b=r=1\), \(a=b=1\), and \(a=b=k\) in Q and L, our results reduce to the studies [13, 18], and [19], respectively. Since this study is a generalization of these studies, it contributes to the literature by providing essential information on the spectral norms of r-circulant matrices.
Change history
22 February 2018
In the publication of this article (Köme and Yazlik in J. Inequal. Appl. 2017(1):192), there are a few errors.
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The authors are grateful to the anonymous referees who have contributed to improve the quality of the paper. The authors declare that they have not received any financial support for this research.
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Köme, C., Yazlik, Y. On the spectral norms of r-circulant matrices with the biperiodic Fibonacci and Lucas numbers. J Inequal Appl 2017, 192 (2017). https://doi.org/10.1186/s13660-017-1466-0
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DOI: https://doi.org/10.1186/s13660-017-1466-0