# On the Spectrum and Spectral Norms ofr-Circulant Matrices with Generalized k-Horadam Numbers Entries

Authors: Lele Liu
Journal: International Journal of Computational Mathematics
 (2014)

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## Abstract

This work is concerned with the spectrum and spectral norms ofr-circulant matrices with generalized k-Horadam numbers entries. By using Abel transformation and some identities we obtain an explicit formula for the eigenvalues of them. In addition, a sufficient condition for an r-circulant matrix to be normal is presented. Based on the results we obtain the precise value for spectral norms of normal r-circulant matrix with generalized k-Horadam numbers, which generalize and improve the known results.


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## Body

## 1. Introduction

There is no doubt that ther-circulant matrices have been one of the most interesting research areas in computation mathematics. It is well known that these matrices have a wide range of applications in signal processing, digital image disposal, coding theory, linear forecast, and design of self-regress.There are many works concerning estimates for spectral norms ofr-circulant matrices with special entries. For example, Solak [1] established lower and upper bounds for the spectral norms of circulant matrices with Fibonacci and Lucas numbers entries. subsequently, Ipek [2] investigated some improved estimations for spectral norms of these matrices. Bani-Domi and Kittaneh [3] established two general norm equalities for circulant and skew circulant operator matrices. Shen and Cen [4] gave the bounds of the spectral norms of r-circulant matrices whose entries are Fibonacci and Lucas numbers. In [5] they defined r-circulant matrices involving k-Lucas and k-Fibonacci numbers and also investigated the upper and lower bounds for the spectral norms of these matrices.Recently, Yazlik and Taskara [6] define a generalization {Hk,n} of the special second-order sequences such as Fibonacci, Lucas, k-Fibonacci, k-Lucas, generalized k-Fibonacci and k-Lucas, Horadam, Pell, Jacobsthal, and Jacobsthal-Lucas sequences. For any integer number k⩾1, the generalized k-Horadam sequence {Hk,n} is defined by the following recursive relation:
(1)Hk,n+2=f(k)Hk,n+1+g(k)Hk,n,Hk,0=a,Hk,1=b,
where f(k) and g(k) are scaler-value polynomials, f2(k)+4g(k)>0. The following are some particular cases.(i)
Iff(k)=k, g(k)=1 and a=0, b=1, the k-Fibonacci sequence is obtained:
(2)Fk,n+2=kFk,n+1+Fk,n,Fk,0=0,Fk,1=1.(ii)
Iff(k)=k, g(k)=1 and a=2, b=k, the k-Lucas sequence is obtained:
(3)Lk,n+2=kLk,n+1+Lk,n,Fk,0=0,Fk,1=k.(iii)
Iff(k)=1, g(k)=1 and a=0, b=1, the Fibonacci sequence is obtained:
(4)Fn+2=Fn+1+Fn,F0=0,F1=1.(iv)
Iff(k)=1, g(k)=1 and a=2, b=1, the Lucas sequence is obtained:
(5)Ln+2=Ln+1+Ln,L0=2,L1=1.(v)
Iff(k)=1, g(k)=2 and a=0, b=1, the Jacobsthal sequence is obtained:
(6)Jn+2=Jn+1+2Jn,J0=0,J1=1.In [7], the authors present new upper and lower bounds for the spectral norm of an r-circulant matrix Cr(Hk,0,Hk,1,…,Hk,n-1), and they study the spectral norm of circulant matrix with generalized k-Horadam numbers in [8]. In this paper, we first give an explicit formula for the eigenvalues of r-circulant matrix with generalized k-Horadam numbers entries using different methods in [7]. Afterwards, we present a sufficient condition for an r-circulant matrix to be normal. Based on the results, the precise value for spectral norms of normal r-circulant matrix whose entries are generalized k-Horadam numbers is obtained, which generalize and improve the main results in [1, 2, 4, 5].


## 2. Preliminaries

In this section, we present some known lemmas and results that will be used in the following study.Definition 1.
For any givenc0,c1,…,cn-1∈C, the r-circulant matrix C, denoted by C=Cr(c0,c1,…,cn-1), is of the form
(7)(c0c1c2⋯cn-1rcn-1c0c1⋯cn-2rcn-2rcn-1c0⋯cn-3⋮⋮⋮⋱⋮rc1rc2rc3⋯c0).
It is obvious that the matrix Cr turns into a classical circulant matrix for r=1.Lemma 2 (see [9]).
LetC=Cr(c0,c1,…,cn-1) be an r-circulant matrix; then the eigenvalues of C are given by
(8)λi=∑j=0n-1‍cjμij,μi=r1/nωi,i=0,1,…,n-1,
where ω=e-2πi/n is the nth root of unity.Let us take any matrixA=[aij] of order n; it is well known that the spectral norm of matrix A is
(9)∥A∥2=max⁡0⩽i⩽n-1λi(AHA),
where AH is the conjugate transpose of A and λi(AHA) is the eigenvalue of AHA.For a normal matrixA (i.e., AAH=AHA), we have the following lemma.Lemma 3 (see [10]).
LetA be a normal matrix with eigenvalues λ0,λ1,…,λn-1. Then the spectral norm of A is
(10)∥A∥2=max⁡0⩽i⩽n-1|λi|.The following lemma can be found in [11].Lemma 4 (see [11], Abel transformation).
Suppose that{ai} and {bi} are two sequences, and Si=a1+a2+⋯+ai(i=1,2,…); then
(11)∑i=1n‍aibi=Snbn-∑i=1n-1‍(bi+1-bi)Si.


## 3. Spectrum ofr-Circulant Matrix with Generalized k-Horadam Numbers

We start this section by giving the following lemma.Lemma 5.
Suppose that{Hk,i}i∈N is a generalized k-Horadam sequence defined in (1). The following conclusions hold.(1)
Iff(k)+g(k)≠1, then
(12)∑i=0n‍Hk,i=Hk,n+1+g(k)Hk,n+f(k)a-a-bf(k)+g(k)-1.(2)
Iff(k)+g(k)=1, then
(13)∑i=0n‍Hk,i=g(k)Hk,n+n[g(k)a+b]+ag(k)+1.Proof.
(1) According to (1), we have
(14)∑i=0n‍Hk,i=f(k)∑i=0n‍Hk,i-1+g(k)∑i=0n‍Hk,i-2.
Changing the summation index in (14), we have
(15)∑i=0n‍Hk,i=f(k)(∑i=0n‍Hk,i-Hk,n+Hk,-1)+g(k)(∑i=0n‍Hk,i-Hk,n-1-Hk,n+Hk,-1+Hk,-2).
By direct calculation, together with recursive relation (1), one can obtain that
(16)[f(k)+g(k)-1]∑i=0n‍Hk,i=Hk,n+1+g(k)Hk,n+f(k)a-a-b.
Therefore we immediately obtain (12) from f(k)+g(k)≠1.(2)
Suppose thatf(k)+g(k)=1; we first illustrate that Hk,i+1+g(k)Hk,i≡g(k)a+b. Let Vi=Hk,i+1+g(k)Hk,i; then V0=g(k)a+b. Combining (1) and f(k)+g(k)=1, one can obtain that
(17)Vi+1=Hk,i+2+g(k)Hk,i+1=(f(k)Hk,i+1+g(k)Hk,i)+g(k)Hk,i+1=Hk,i+1+g(k)Hk,i=Vi, which shows that {Vi} is a constant sequence, and therefore
(18)Hk,i+1+g(k)Hk,i=Vi=V0=g(k)a+b.
Evaluating summation from 0 to n, we have
(19)∑i=0n‍Hk,i+1+g(k)∑i=0n‍Hk,i=(n+1)[g(k)a+b].
Changing the summation index in (19) gives
(20)(∑i=0n‍Hk,i+Hk,n+1-a)+g(k)∑i=0n‍Hk,i=(n+1)[g(k)a+b].
Therefore
(21)[g(k)+1]∑i=0n‍Hk,i=g(k)Hk,n+n[g(k)a+b]+a.
In view of assumptions f2(k)+g(k)>1 and f(k)+g(k)=1, we know that g(k)+1≠0. Thus we obtain (13) from (21).From Lemma5 we have the following theorem.Theorem 6.
LetA=Cr(Hk,0,Hk,1,…,Hk,n-1) be an r-circulant matrix with eigenvalues λ0,λ1,…,λn-1; then for i=0,1,2,…,n-1 the following hold.(1)
Iff(k)+g(k)≠1, then
(22)λi=(rHk,n+g(k)r1+(1/n)ωiHk,n-1+r1/n[f(k)a-b]ωi-a)×(r1/nωif(k)+r2/nω2ig(k)-1)-1.(2)
Iff(k)+g(k)=1, then
(23)λi=((g(k)rHk,n-1+a)(1-r1/nωi)+[g(k)a+b](r1/nwi-r))×((1-r1/nωi)[g(k)r1/nωi+1])-1.Proof.
According to Lemma2, we have
(24)λi=∑i=0n-1‍Hk,iμij,μi=r1/nwi.
Using Abel transformation (Lemma 4), we have
(25)λi=μin-1∑j=0n-1‍Hk,j-∑j=0n-2‍((μij+1-μij)∑s=0j‍Hk,s)=μin-1∑j=0n-1‍Hk,j-(μi-1)∑j=0n-2‍(μij∑s=0j‍Hk,s).(1)
In the light of (12) and (25), one can obtain that
(26)λi=μin-1∑j=0n-1‍Hk,j-μi-1f(k)+g(k)-1×∑j=0n-2‍μij[Hk,j+1+g(k)Hk,j+f(k)a-a-b]=μin-1Hk,n+g(k)Hk,n-1+f(k)a-a-bf(k)+g(k)-1-μi-1f(k)+g(k)-1×(∑j=0n-2‍Hk,j+1μij+g(k)∑j=0n-2‍Hk,jμij+[f(k)a-a-b]∑j=0n-2‍μij). It is clear that
(27)∑j=0n-2‍Hk,j+1μij=λi-aμi,∑j=0n-2‍Hk,jμij=λi-μin-1Hk,n-1.
Substituting (27) into (26), we obtain that
(28)λi=μin-1Hk,n+g(k)Hk,n-1+f(k)a-a-bf(k)+g(k)-1-μi-1f(k)+g(k)-1×(∑j=0n-2‍μijλi-aμi+g(k)(λi-μin-1Hk,n-1)+[f(k)a-a-b]∑j=0n-2‍μij)=(1-μi)[1+g(k)μi]μi[f(k)+g(k)-1]λi+μin-1Hk,n+μing(k)Hk,n-1f(k)+g(k)-1+μin(f(k)a-a-b)-a(1-μi)μi[f(k)+g(k)-1]+(1-μin-1)(f(k)a-a-b)f(k)+g(k)-1.
Therefore we have
(29)[g(k)μi2+f(k)μi-1]λi=μinHk,n+g(k)μin+1Hk,n-1+μin(f(k)a-a-b)-a(1-μi)+(μi-r)(f(k)a-a-b)=rHk,n+g(k)r1+(1/n)ωiHk,n-1+r1/n[f(k)a-b]ωi-a.
We immediately obtain formula (22) from (29).(2)
Taking into account (13) and (25), we have
(30)λi=μin-1∑j=0n-1‍Hk,j-μi-1g(k)+1×∑j=0n-2‍μij[g(k)Hk,j+j·(g(k)a+b)+a]=μin-1∑j=0n-1‍Hk,j+g(k)(1-μi)g(k)+1×∑j=0n-2‍Hk,jμij+1-μig(k)+1∑j=0n-2‍[j·(g(k)a+b)+a]μij=μin-1[g(k)Hk,n-1+(n-1)(g(k)a+b)+a]g(k)+1+g(k)(1-μi)g(k)+1(λi-Hk,n-1μin-1)+1-μig(k)+1[∑j=0n-3‍((n-2)(g(k)a+b)+a)×∑j=0n-2‍μij-(g(k)a+b)∑j=0n-3‍∑s=0j‍μis]. It follows that
(31)λi=g(k)μing(k)+1Hk,n-1+g(k)(1-μi)g(k)+1λi+μin-1[(n-1)(g(k)a+b)+a]g(k)+1+1-μig(k)+1[∑j=0n-3‍((n-2)(g(k)a+b)+a)×∑j=0n-2‍μij-(g(k)a+b)1-μi∑j=0n-3‍(1-μij+1)]=g(k)(1-μi)g(k)+1λi+g(k)μinHk,n-1+μin-1(g(k)a+b)+ag(k)+1+(g(k)a+b)(μi-μin)(g(k)+1)(1-μi).
Therefore we obtain (23). This concludes the proof.


## 4. Spectral Norms of Normalr-Circulant Matrices

In this section, we consider the spectral norms of normalr-circulant matrix whose entries are generalized k-Horadam numbers. Our results generalize and improve the results in [1, 2, 4, 5]. The following lemma can be found in [9], and we give a concise proof.Lemma 7.
LetA=Cr(a0,a1,…,an-1) be an r-circulant matrix. If |r|=1, then A is normal matrix.Proof.
It is well known that(32)A=∑i=0n-1‍aiPi,P=(0In-1r0).
If |r|=1, then
(33)PPH=(0In-1r0)(0r¯In-10)=In.
That is,PH=P-1. According to (32), we obtain that
(34)AAH=(∑i=0n-1aiPi)(∑j=0n-1aj(PH)j)=(∑i=0n-1aiPi)(∑j=0n-1aj(P-1)j)=∑i=0n-1∑j=0n-1aiajPi-j,AHA=(∑j=0n-1aj(PH)j)(∑i=0n-1aiPi)=(∑j=0n-1ajP-j)(∑i=0n-1ajPi)=∑i=0n-1∑j=0n-1aiajPi-j.
Therefore AAH=AHA, which shows that A is normal.According to Theorem6 and Lemma 7, we have the following theorem.Theorem 8.
Suppose thatA=Cr(Hk,0,Hk,1,…,Hk,n-1) is an r-circulant matrix. If |r|=1 and Hk,i⩾0,i=0,1,2,…,n-1, then the spectral norm of A is (35)∥A∥2={max⁡0⩽i⩽n-1|rHk,n+g(k)r1+(1/n)ωiHk,n-1+r1/n[f(k)a-b]ωi-ar1/nωif(k)+r2/nω2ig(k)-1|,f(k)+g(k)≠1,max⁡0⩽i⩽n-1|(g(k)rHk,n-1+a)(1-r1/nωi)+[g(k)a+b](r1/nwi-r)(1-r1/nωi)[g(k)r1/nωi+1]|,f(k)+g(k)=1.The following theorem simplifies and generalizes the results of Theorem 2.2 in [12].Theorem 9.
LetA=Circ(Hk,0,Hk,1,…,Hk,n-1) be a circulant matrix; then
(36)∥A∥2={Hk,n+g(k)Hk,n-1+f(k)a-a-bf(k)+g(k)-1,f(k)+g(k)≠1,g(k)Hk,n-1+(n-1)[g(k)a+b]+ag(k)+1,f(k)+g(k)=1.Proof.
Suppose thatr=1; it follows from Lemma 7 that A is normal. Notice that
(37)|λi|=|∑i=0n-1Hk,iμij|⩽∑i=0n-1Hk,i|μi|j=λ0.
It follows from Lemma 3 that ∥A∥2=λ0. According to Theorem 6, if f(k)+g(k)≠1 and r=1, we obtain that
(38)∥A∥2=λ0=Hk,n+g(k)Hk,n-1+f(k)a-a-bf(k)+g(k)-1.
Similarly, if f(k)+g(k)=1, it follows that
(39)∥A∥2=λ0=g(k)Hk,n-1+(n-1)[g(k)a+b]+ag(k)+1.
This completes the proof.Taking into account formulae (4)–(6), we have the following corollary.Corollary 10.
LetA1=Circ(F0,F1,…,Fn-1) be a circulant matrix; then
(40)∥A1∥2=Fn+1-1.Corollary 11.
LetA2=Circ(L0,L1,…,Ln-1) be a circulant matrix; then
(41)∥A2∥2=Fn+2+Fn-1.Corollary 12.
LetA3=Circ(J0,J1,…,Jn-1) be a circulant matrix; then
(42)∥A3∥2=Jn+1-12.



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