On the Norms of Circulant and $r-$Circulant Matrices With the Hyperharmonic Fibonacci Numbers

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.


Introduction
The circulant and r−circulant matrices have connection to signal processing, probability, numerical analysis, coding theory and many other areas. An n × 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 = Circ(c 0 , c 1 , c 2 , . . . , c n−1 ). In particular for r = 1 is called a circulant matrix and we denote it shortly by C = Circ(c 0 , c 1 , c 2 , . . . , c n−1 ). The eigenvalues of C are where w = e 2πi n and i = √ −1. Many authors investigate on 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 and et al. obtained norms of circulant and semicirculant matrices with Horadams numbers. In [7], Zhou and et all. gave 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 numbers entries. In [3], Shen and Cen have given upper and lower bounds for the spectral norms of r− circulant matrices with classical Fibonacci and Lucas numbers entries. In [5], Bahşi and Solak computed the spectral norms of circulant and r−circulant matrices with the hyper-Fibonacci and hyper-Lucas numbers. In [6], 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 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.

Preliminaries
The Fibonacci numbers F n are defined by the following recurrence relation for n ≥ 1, where F 0 = 0, F 1 = 1. In [8], authors investigated finite sum of the reciprocals of Fibonacci numbers which is called harmonic Fibonacci numbers. Then they gave a combinatoric identity related to harmonic Fibonacci numbers as follows: Moreover in [8], they defined hyperharmonic numbers for n, r ≥ 1 Fn and 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 × n matrix. The Euclidean norm of matrix A is Then the following inequalities hold for between Euclidean norm and spectral norm as follow;

Lemma 1. [9] Let
A and B be two m × n matrices. Then we have where A • B is the Hadamard product of A and B.

Lemma 2. [9] Let
A and B be two n × m matrices. We have where Definition 3. [10] Difference operator of f (x) is defined as . .

Corollary 1. We have
Proof. The proof is trivial from the Definition 1 and the relation between Euclidean norm and spectral norm in (2.2).
Proof. From the definition of Euclidean norm we have, (s + 1) 2n + s(|r| Proof. It is clear that the proof can be completed if we take k = 1 in Theorem 4 Corollary 3. [8] Let C 1 = Circ(F 0 , F 1 , . . . , F n−1 ) be n × n matrix. The Euclidean norm is 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.
Proof. Since the matrix we have i) In [8], for the sum of the squares of hyperharmonic Fibonacci numbers, we have Since |r| ≥ 1 and by (3.4), we have On the other hand, let the matrices A and B be as That is C Hence, from the (3.4) and Lemma 1, we have Thus, we have ii) Since |r| < 1 and from the (3.4), we have On the other hand, let the matrices A and B be as   ii) If |r| < 1, then |r| √ n F (2) n−1 .
Proof. It is easily seen that the proof can be completed if we take k = 1 in Theorem 5