On the spectral norms of r-circulant matrices with the biperiodic Fibonacci and Lucas numbers

In this paper, we present new upper and lower bounds for the spectral norms of the r-circulant matrices \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q=C_{r} ( (\frac{b}{a} )^{\frac{\xi (1)}{2}}q_{0}, (\frac{b}{a} )^{\frac{\xi(2)}{2}}q_{1}, (\frac {b}{a} )^{\frac{\xi(3)}{2}}q_{2}, \dots, (\frac{b}{a} )^{\frac{\xi(n)}{2}}q_{n-1} )$\end{document}Q=Cr((ba)ξ(1)2q0,(ba)ξ(2)2q1,(ba)ξ(3)2q2,…,(ba)ξ(n)2qn−1) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$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} ) $\end{document}L=Cr((ba)ξ(0)2l0,(ba)ξ(1)2l1,(ba)ξ(2)2l2,…,(ba)ξ(n−1)2ln−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.


Abstract
In this paper, we present new upper and lower bounds for the spectral norms of the r-circulant matrices Q = C r (( b a ) ξ (1) They also obtained an extended Binet formula for this sequence: Afterward, Bilgici [] defined generalized the Lucas sequence by the following recurrence relation: for n ∈ N  , l  = , l  = a, l n+ = ⎧ ⎨ ⎩ bl n+ + l n if n is even, and The Binet formula for this sequence is In Eqs. () and (), α = ab+ are the roots of the characteristic equation of x abxab = , and ξ (n) = n - n  . 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 [-]. For example, Shen and Cen [] found upper and lower bounds for the spectral norms of r-circulant matrices in the forms A = C r (F  , F  , F  , . . . , F n- ) and B = C r (L  , L  , L  , . . . , L n- ). They also obtained some bounds for the spectral norms of Kronecker and Hadamard products of A and B. Afterward, Shen and Cen [] gave the upper and lower bounds for the spectral norms of the matrices A = C r (F k, , F k, , F k, , . . . , F k,n- ) and B = C r (L k, , L k, , L k, , . . . , L k,n- ). They also presented some bounds for the spectral norms of Hadamard and Kronecker products of these matrices. Bahşi [] studied the norms of r-circulant matrices n 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 . For any given c  , c  , c  , . . . , c n- ∈ C, the r-circulant matrix C r = (c ij ) n×n is defined by It is clear that, for r = , C r turns into a classical circulant matrix. Let us take any A = [a ij ] ∈ M n,n (C). The Frobenius norm of the matrix A is defined by Also, the spectral norm of the matrix A is given by where A • B is the Hadamard product of A and B.
where A ⊗ B is the Kronecker product of A and B.

Main results
In this section, we first give the sum of squares of biperiodic Lucas numbers.

Theorem . For any positive integer m, we have
Proof Using the Binet formula of the biperiodic Lucas numbers, we have Using the properties ab(α + ) = α  and ab(β Proof The matrix Q is of the form Then we have Hence, for |r| ≥ , using Eq. (), we obtain that is, From () we have Now, for |r| ≥ , we give an bound for the spectral norm of the matrix Q. Let the matrices B and C be By Lemma . we have Thus, q n q n- a ≤ Q  ≤ |r| q n q n- a .
On the other hand, for |r| < , we have Thus, we obtain Q  ≥ |r| q n q n- a .
Now, for |r| < , we give an upper bound for the spectral norm of the matrix Q. Let the matrices D and E be By Lemma . we have Thus,  l n- ) be an r-circulant matrix. Then, for r ∈ C, we have: if |r| ≥ , then l n l n- a +  ≤ L  ≤ |r| l n l n- a +  ; if |r| < , then |r| l n l n- a +  ≤ L  ≤ n l n l n- a +  .
Proof The matrix L is of the form Then we have Hence, for |r| ≥ , using Eq. (), we obtain Now, for |r| ≥ , we give an upper bound for the spectral norm of the matrix L. Let the matrices F and H be and On the other hand, for |r| < , we have Thus, we obtain L  ≥ |r| l n l n- a + . Now, for |r| < , we give an upper bound for the spectral norm of the matrix L. Let the matrices G and K be By Lemma . we have Thus, |r| l n l n- a +  ≤ L  ≤ n l n l n- a +  . (ii) If |r| < , then Q ⊗ L  ≥ |r|  q n q n- a l n l n- a +  and Q ⊗ L  ≤ n(n -) q n q n- a l n l n- a +  .
Proof Since Q ⊗ L  = Q  L  , the proof is trivial by Theorems . and ..

Conclusion
In this paper, we obtain new upper and lower bounds for the spectral norms of the rcirculant 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 = , a = b = , and a = b = k in Q and L, our results reduce to the studies [, ], and [], 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.