Research ArticleOn the Spectrum and Spectral Norms of 𝑟-Circulant Matriceswith Generalized 𝑘-Horadam Numbers Entries
Lele Liu
College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
Correspondence should be addressed to Lele Liu; [email protected]
Received 17 May 2014; Accepted 25 August 2014; Published 31 August 2014
Academic Editor: Chengpeng Bi
Copyright © 2014 Lele Liu. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
This work is concernedwith the spectrum and spectral norms of 𝑟-circulantmatrices with generalized 𝑘-Horadamnumbers entries.By using Abel transformation and some identities we obtain an explicit formula for the eigenvalues of them. In addition, a sufficientcondition for an 𝑟-circulant matrix to be normal is presented. Based on the results we obtain the precise value for spectral normsof normal 𝑟-circulant matrix with generalized 𝑘-Horadam numbers, which generalize and improve the known results.
1. Introduction
There is no doubt that the 𝑟-circulant matrices have beenone of the most interesting research areas in computationmathematics. It is well known that these matrices have awide range of applications in signal processing, digital imagedisposal, coding theory, linear forecast, and design of self-regress.
There are many works concerning estimates for spec-tral norms of 𝑟-circulant matrices with special entries. Forexample, Solak [1] established lower and upper bounds forthe spectral norms of circulant matrices with Fibonacciand Lucas numbers entries. subsequently, Ipek [2] inves-tigated some improved estimations for spectral norms ofthese matrices. Bani-Domi and Kittaneh [3] established twogeneral norm equalities for circulant and skew circulantoperator matrices. Shen and Cen [4] gave the bounds ofthe spectral norms of 𝑟-circulant matrices whose entries areFibonacci and Lucas numbers. In [5] they defined 𝑟-circulantmatrices involving 𝑘-Lucas and 𝑘-Fibonacci numbers andalso investigated the upper and lower bounds for the spectralnorms of these matrices.
Recently, Yazlik and Taskara [6] define a generaliza-tion {𝐻𝑘,𝑛} of the special second-order sequences suchas Fibonacci, Lucas, 𝑘-Fibonacci, 𝑘-Lucas, generalized 𝑘-Fibonacci and 𝑘-Lucas, Horadam, Pell, Jacobsthal, andJacobsthal-Lucas sequences. For any integer number 𝑘 ⩾ 1,
the generalized 𝑘-Horadam sequence {𝐻𝑘,𝑛} is defined by thefollowing recursive relation:
𝐻𝑘,𝑛+2 = 𝑓 (𝑘)𝐻𝑘,𝑛+1 + 𝑔 (𝑘)𝐻𝑘,𝑛,
𝐻𝑘,0 = 𝑎, 𝐻𝑘,1 = 𝑏,
(1)
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-Fibonaccisequence is obtained:
𝐹𝑘,𝑛+2 = 𝑘𝐹𝑘,𝑛+1 + 𝐹𝑘,𝑛, 𝐹𝑘,0 = 0, 𝐹𝑘,1 = 1. (2)
(ii) If 𝑓(𝑘) = 𝑘, 𝑔(𝑘) = 1 and 𝑎 = 2, 𝑏 = 𝑘, the 𝑘-Lucassequence is obtained:
𝐿𝑘,𝑛+2 = 𝑘𝐿𝑘,𝑛+1 + 𝐿𝑘,𝑛, 𝐹𝑘,0 = 0, 𝐹𝑘,1 = 𝑘. (3)
(iii) If 𝑓(𝑘) = 1, 𝑔(𝑘) = 1 and 𝑎 = 0, 𝑏 = 1, the Fibonaccisequence is obtained:
𝐹𝑛+2 = 𝐹𝑛+1 + 𝐹𝑛, 𝐹0 = 0, 𝐹1 = 1. (4)
(iv) If 𝑓(𝑘) = 1, 𝑔(𝑘) = 1 and 𝑎 = 2, 𝑏 = 1, the Lucassequence is obtained:
𝐿𝑛+2 = 𝐿𝑛+1 + 𝐿𝑛, 𝐿0 = 2, 𝐿1 = 1. (5)
Hindawi Publishing CorporationInternational Journal of Computational MathematicsVolume 2014, Article ID 795175, 6 pageshttp://dx.doi.org/10.1155/2014/795175
2 International Journal of Computational Mathematics
(v) If 𝑓(𝑘) = 1, 𝑔(𝑘) = 2 and 𝑎 = 0, 𝑏 = 1, the Jacobsthalsequence is obtained:
𝐽𝑛+2 = 𝐽𝑛+1 + 2𝐽𝑛, 𝐽0 = 0, 𝐽1 = 1. (6)
In [7], the authors present new upper and lowerbounds for the spectral norm of an 𝑟-circulant matrix𝐶𝑟(𝐻𝑘,0, 𝐻𝑘,1, . . . , 𝐻𝑘,𝑛−1), and they study the spectral normofcirculantmatrixwith generalized 𝑘-Horadamnumbers in [8].In this paper, we first give an explicit formula for the eigen-values of 𝑟-circulant matrix with generalized 𝑘-Horadamnumbers entries using different methods in [7]. Afterwards,we present a sufficient condition for an 𝑟-circulant matrixto be normal. Based on the results, the precise value forspectral norms of normal 𝑟-circulant matrix whose entriesare generalized 𝑘-Horadam numbers is obtained, whichgeneralize and improve the main results in [1, 2, 4, 5].
2. Preliminaries
In this section, we present some known lemmas and resultsthat will be used in the following study.
Definition 1. For any given 𝑐0, 𝑐1, . . . , 𝑐𝑛−1 ∈ C, the 𝑟-circulantmatrix 𝐶, denoted by 𝐶 = 𝐶𝑟(𝑐0, 𝑐1, . . . , 𝑐𝑛−1), is of the form
(
𝑐0 𝑐1 𝑐2 ⋅ ⋅ ⋅ 𝑐𝑛−1
𝑟𝑐𝑛−1 𝑐0 𝑐1 ⋅ ⋅ ⋅ 𝑐𝑛−2
𝑟𝑐𝑛−2 𝑟𝑐𝑛−1 𝑐0 ⋅ ⋅ ⋅ 𝑐𝑛−3
.
.
.
.
.
.
.
.
. d...
𝑟𝑐1 𝑟𝑐2 𝑟𝑐3 ⋅ ⋅ ⋅ 𝑐0
). (7)
It is obvious that the matrix 𝐶𝑟 turns into a classical circulantmatrix for 𝑟 = 1.
Lemma 2 (see [9]). Let 𝐶 = 𝐶𝑟(𝑐0, 𝑐1, . . . , 𝑐𝑛−1) be an 𝑟-circulant matrix; then the eigenvalues of 𝐶 are given by
𝜆𝑖 =
𝑛−1
∑
𝑗=0
𝑐𝑗𝜇𝑗
𝑖, 𝜇𝑖 = 𝑟
1/𝑛𝜔𝑖, 𝑖 = 0, 1, . . . , 𝑛 − 1, (8)
where 𝜔 = 𝑒−2𝜋𝑖/𝑛 is the 𝑛th root of unity.
Let us take anymatrix𝐴 = [𝑎𝑖𝑗] of order 𝑛; it is well knownthat the spectral norm of matrix 𝐴 is
‖𝐴‖2 = √ max0⩽𝑖⩽𝑛−1
𝜆𝑖 (𝐴𝐻𝐴), (9)
where𝐴𝐻 is the conjugate transpose of𝐴 and 𝜆𝑖(𝐴𝐻𝐴) is the
eigenvalue of 𝐴𝐻𝐴.For a normal matrix 𝐴 (i.e., 𝐴𝐴𝐻 = 𝐴𝐻𝐴), we have the
following lemma.
Lemma 3 (see [10]). Let 𝐴 be a normal matrix with eigenval-ues 𝜆0, 𝜆1, . . . , 𝜆𝑛−1. Then the spectral norm of 𝐴 is
‖𝐴‖2 = max0⩽𝑖⩽𝑛−1
𝜆𝑖
. (10)
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
𝑛
∑
𝑖=1
𝑎𝑖𝑏𝑖 = 𝑆𝑛𝑏𝑛 −
𝑛−1
∑
𝑖=1
(𝑏𝑖+1 − 𝑏𝑖) 𝑆𝑖. (11)
3. Spectrum of 𝑟-Circulant Matrix withGeneralized 𝑘-Horadam Numbers
We start this section by giving the following lemma.
Lemma 5. Suppose that {𝐻𝑘,𝑖}𝑖∈N is a generalized 𝑘-Horadamsequence defined in (1). The following conclusions hold.
(1) If 𝑓(𝑘) + 𝑔(𝑘) ̸= 1, then𝑛
∑
𝑖=0
𝐻𝑘,𝑖 =
𝐻𝑘,𝑛+1 + 𝑔 (𝑘)𝐻𝑘,𝑛 + 𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏
𝑓 (𝑘) + 𝑔 (𝑘) − 1
. (12)
(2) If 𝑓(𝑘) + 𝑔(𝑘) = 1, then𝑛
∑
𝑖=0
𝐻𝑘,𝑖 =
𝑔 (𝑘)𝐻𝑘,𝑛 + 𝑛 [𝑔 (𝑘) 𝑎 + 𝑏] + 𝑎
𝑔 (𝑘) + 1
. (13)
Proof. (1) According to (1), we have𝑛
∑
𝑖=0
𝐻𝑘,𝑖 = 𝑓 (𝑘)
𝑛
∑
𝑖=0
𝐻𝑘,𝑖−1 + 𝑔 (𝑘)
𝑛
∑
𝑖=0
𝐻𝑘,𝑖−2. (14)
Changing the summation index in (14), we have𝑛
∑
𝑖=0
𝐻𝑘,𝑖
= 𝑓 (𝑘)(
𝑛
∑
𝑖=0
𝐻𝑘,𝑖 − 𝐻𝑘,𝑛 + 𝐻𝑘,−1)
+𝑔 (𝑘)(
𝑛
∑
𝑖=0
𝐻𝑘,𝑖 − 𝐻𝑘,𝑛−1 − 𝐻𝑘,𝑛 + 𝐻𝑘,−1 + 𝐻𝑘,−2) .
(15)
By direct calculation, together with recursive relation (1), onecan obtain that
[𝑓 (𝑘) + 𝑔 (𝑘) − 1]
𝑛
∑
𝑖=0
𝐻𝑘,𝑖
= 𝐻𝑘,𝑛+1 + 𝑔 (𝑘)𝐻𝑘,𝑛 + 𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏.
(16)
Therefore we immediately obtain (12) from 𝑓(𝑘) + 𝑔(𝑘) ̸= 1.
(2) Suppose that 𝑓(𝑘) + 𝑔(𝑘) = 1; we first illustrate that𝐻𝑘,𝑖+1+𝑔(𝑘)𝐻𝑘,𝑖 ≡ 𝑔(𝑘)𝑎+𝑏. Let𝑉𝑖 = 𝐻𝑘,𝑖+1+𝑔(𝑘)𝐻𝑘,𝑖;then𝑉0 = 𝑔(𝑘)𝑎+𝑏. Combining (1) and 𝑓(𝑘)+𝑔(𝑘) =1, one can obtain that
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,
(17)
International Journal of Computational Mathematics 3
which shows that {𝑉𝑖} is a constant sequence, and therefore
𝐻𝑘,𝑖+1 + 𝑔 (𝑘)𝐻𝑘,𝑖 = 𝑉𝑖 = 𝑉0 = 𝑔 (𝑘) 𝑎 + 𝑏. (18)
Evaluating summation from 0 to 𝑛, we have
𝑛
∑
𝑖=0
𝐻𝑘,𝑖+1 + 𝑔 (𝑘)
𝑛
∑
𝑖=0
𝐻𝑘,𝑖 = (𝑛 + 1) [𝑔 (𝑘) 𝑎 + 𝑏] . (19)
Changing the summation index in (19) gives
(
𝑛
∑
𝑖=0
𝐻𝑘,𝑖 + 𝐻𝑘,𝑛+1 − 𝑎) + 𝑔 (𝑘)
𝑛
∑
𝑖=0
𝐻𝑘,𝑖
= (𝑛 + 1) [𝑔 (𝑘) 𝑎 + 𝑏] .
(20)
Therefore
[𝑔 (𝑘) + 1]
𝑛
∑
𝑖=0
𝐻𝑘,𝑖 = 𝑔 (𝑘)𝐻𝑘,𝑛 + 𝑛 [𝑔 (𝑘) 𝑎 + 𝑏] + 𝑎. (21)
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 Lemma 5 we have the following theorem.
Theorem 6. Let 𝐴 = 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) If 𝑓(𝑘) + 𝑔(𝑘) ̸= 1, then
𝜆𝑖 = (𝑟𝐻𝑘,𝑛 + 𝑔 (𝑘) 𝑟1+(1/𝑛)
𝜔𝑖𝐻𝑘,𝑛−1
+𝑟1/𝑛
[𝑓 (𝑘) 𝑎 − 𝑏] 𝜔𝑖− 𝑎)
×(𝑟1/𝑛
𝜔𝑖𝑓 (𝑘) + 𝑟
2/𝑛𝜔2𝑖𝑔 (𝑘) − 1)
−1
.
(22)
(2) If 𝑓(𝑘) + 𝑔(𝑘) = 1, then
𝜆𝑖 = ((𝑔 (𝑘) 𝑟𝐻𝑘,𝑛−1 + 𝑎) (1 − 𝑟1/𝑛
𝜔𝑖)
+ [𝑔 (𝑘) 𝑎 + 𝑏] (𝑟1/𝑛
𝑤𝑖− 𝑟))
× ((1 − 𝑟1/𝑛
𝜔𝑖) [𝑔 (𝑘) 𝑟
1/𝑛𝜔𝑖+ 1])
−1
.
(23)
Proof. According to Lemma 2, we have
𝜆𝑖 =
𝑛−1
∑
𝑖=0
𝐻𝑘,𝑖𝜇𝑗
𝑖, 𝜇𝑖 = 𝑟
1/𝑛𝑤𝑖. (24)
Using Abel transformation (Lemma 4), we have
𝜆𝑖 = 𝜇𝑛−1
𝑖
𝑛−1
∑
𝑗=0
𝐻𝑘,𝑗 −
𝑛−2
∑
𝑗=0
((𝜇𝑗+1
𝑖− 𝜇𝑗
𝑖)
𝑗
∑
𝑠=0
𝐻𝑘,𝑠)
= 𝜇𝑛−1
𝑖
𝑛−1
∑
𝑗=0
𝐻𝑘,𝑗 − (𝜇𝑖 − 1)
𝑛−2
∑
𝑗=0
(𝜇𝑗
𝑖
𝑗
∑
𝑠=0
𝐻𝑘,𝑠) .
(25)
(1) In the light of (12) and (25), one can obtain that
𝜆𝑖 = 𝜇𝑛−1
𝑖
𝑛−1
∑
𝑗=0
𝐻𝑘,𝑗 −
𝜇𝑖 − 1
𝑓 (𝑘) + 𝑔 (𝑘) − 1
×
𝑛−2
∑
𝑗=0
𝜇𝑗
𝑖[𝐻𝑘,𝑗+1 + 𝑔 (𝑘)𝐻𝑘,𝑗 + 𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏]
= 𝜇𝑛−1
𝑖
𝐻𝑘,𝑛 + 𝑔 (𝑘)𝐻𝑘,𝑛−1 + 𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏
𝑓 (𝑘) + 𝑔 (𝑘) − 1
−
𝜇𝑖 − 1
𝑓 (𝑘) + 𝑔 (𝑘) − 1
× (
𝑛−2
∑
𝑗=0
𝐻𝑘,𝑗+1𝜇𝑗
𝑖+ 𝑔 (𝑘)
𝑛−2
∑
𝑗=0
𝐻𝑘,𝑗𝜇𝑗
𝑖
+ [𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏]
𝑛−2
∑
𝑗=0
𝜇𝑗
𝑖) .
(26)
It is clear that
𝑛−2
∑
𝑗=0
𝐻𝑘,𝑗+1𝜇𝑗
𝑖=
𝜆𝑖 − 𝑎
𝜇𝑖
,
𝑛−2
∑
𝑗=0
𝐻𝑘,𝑗𝜇𝑗
𝑖= 𝜆𝑖 − 𝜇
𝑛−1
𝑖𝐻𝑘,𝑛−1.
(27)
4 International Journal of Computational Mathematics
Substituting (27) into (26), we obtain that
𝜆𝑖 = 𝜇𝑛−1
𝑖
𝐻𝑘,𝑛 + 𝑔 (𝑘)𝐻𝑘,𝑛−1 + 𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏
𝑓 (𝑘) + 𝑔 (𝑘) − 1
−
𝜇𝑖 − 1
𝑓 (𝑘) + 𝑔 (𝑘) − 1
× (
𝜆𝑖 − 𝑎
𝜇𝑖
+ 𝑔 (𝑘) (𝜆𝑖 − 𝜇𝑛−1
𝑖𝐻𝑘,𝑛−1)
+ [𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏]
𝑛−2
∑
𝑗=0
𝜇𝑗
𝑖)
=
(1 − 𝜇𝑖) [1 + 𝑔 (𝑘) 𝜇𝑖]
𝜇𝑖 [𝑓 (𝑘) + 𝑔 (𝑘) − 1]
𝜆𝑖
+
𝜇𝑛−1
𝑖𝐻𝑘,𝑛 + 𝜇
𝑛
𝑖𝑔 (𝑘)𝐻𝑘,𝑛−1
𝑓 (𝑘) + 𝑔 (𝑘) − 1
+
𝜇𝑛
𝑖(𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏) − 𝑎 (1 − 𝜇𝑖)
𝜇𝑖 [𝑓 (𝑘) + 𝑔 (𝑘) − 1]
+
(1 − 𝜇𝑛−1
𝑖) (𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏)
𝑓 (𝑘) + 𝑔 (𝑘) − 1
.
(28)
Therefore we have
[𝑔 (𝑘) 𝜇2
𝑖+ 𝑓 (𝑘) 𝜇𝑖 − 1] 𝜆𝑖
= 𝜇𝑛
𝑖𝐻𝑘,𝑛 + 𝑔 (𝑘) 𝜇
𝑛+1
𝑖𝐻𝑘,𝑛−1 + 𝜇
𝑛
𝑖(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.
(29)
We immediately obtain formula (22) from (29).
(2) Taking into account (13) and (25), we have
𝜆𝑖 = 𝜇𝑛−1
𝑖
𝑛−1
∑
𝑗=0
𝐻𝑘,𝑗 −
𝜇𝑖 − 1
𝑔 (𝑘) + 1
×
𝑛−2
∑
𝑗=0
𝜇𝑗
𝑖[𝑔 (𝑘)𝐻𝑘,𝑗 + 𝑗 ⋅ (𝑔 (𝑘) 𝑎 + 𝑏) + 𝑎]
= 𝜇𝑛−1
𝑖
𝑛−1
∑
𝑗=0
𝐻𝑘,𝑗 +
𝑔 (𝑘) (1 − 𝜇𝑖)
𝑔 (𝑘) + 1
×
𝑛−2
∑
𝑗=0
𝐻𝑘,𝑗𝜇𝑗
𝑖+
1 − 𝜇𝑖
𝑔 (𝑘) + 1
𝑛−2
∑
𝑗=0
[𝑗 ⋅ (𝑔 (𝑘) 𝑎 + 𝑏) + 𝑎] 𝜇𝑗
𝑖
=
𝜇𝑛−1
𝑖[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−1𝜇𝑛−1
𝑖)
+
1 − 𝜇𝑖
𝑔 (𝑘) + 1
[
[
((𝑛 − 2) (𝑔 (𝑘) 𝑎 + 𝑏) + 𝑎)
×
𝑛−2
∑
𝑗=0
𝜇𝑗
𝑖− (𝑔 (𝑘) 𝑎 + 𝑏)
𝑛−3
∑
𝑗=0
𝑗
∑
𝑠=0
𝜇𝑠
𝑖]
]
.
(30)
It follows that
𝜆𝑖 =
𝑔 (𝑘) 𝜇𝑛
𝑖
𝑔 (𝑘) + 1
𝐻𝑘,𝑛−1 +
𝑔 (𝑘) (1 − 𝜇𝑖)
𝑔 (𝑘) + 1
𝜆𝑖
+
𝜇𝑛−1
𝑖[(𝑛 − 1) (𝑔 (𝑘) 𝑎 + 𝑏) + 𝑎]
𝑔 (𝑘) + 1
+
1 − 𝜇𝑖
𝑔 (𝑘) + 1
[
[
((𝑛 − 2) (𝑔 (𝑘) 𝑎 + 𝑏) + 𝑎)
×
𝑛−2
∑
𝑗=0
𝜇𝑗
𝑖−
(𝑔 (𝑘) 𝑎 + 𝑏)
1 − 𝜇𝑖
𝑛−3
∑
𝑗=0
(1 − 𝜇𝑗+1
𝑖)]
]
=
𝑔 (𝑘) (1 − 𝜇𝑖)
𝑔 (𝑘) + 1
𝜆𝑖
+
𝑔 (𝑘) 𝜇𝑛
𝑖𝐻𝑘,𝑛−1 + 𝜇
𝑛−1
𝑖(𝑔 (𝑘) 𝑎 + 𝑏) + 𝑎
𝑔 (𝑘) + 1
+
(𝑔 (𝑘) 𝑎 + 𝑏) (𝜇𝑖 − 𝜇𝑛
𝑖)
(𝑔 (𝑘) + 1) (1 − 𝜇𝑖)
.
(31)
Therefore we obtain (23). This concludes the proof.
4. Spectral Norms of Normal𝑟-Circulant Matrices
In this section, we consider the spectral norms of normal 𝑟-circulant matrix whose entries are generalized 𝑘-Horadamnumbers. Our results generalize and improve the results in[1, 2, 4, 5]. The following lemma can be found in [9], and wegive a concise proof.
Lemma 7. Let 𝐴 = 𝐶𝑟(𝑎0, 𝑎1, . . . , 𝑎𝑛−1) be an 𝑟-circulantmatrix. If |𝑟| = 1, then 𝐴 is normal matrix.
Proof. It is well known that
𝐴 =
𝑛−1
∑
𝑖=0
𝑎𝑖𝑃𝑖, 𝑃 = (
0 𝐼𝑛−1
𝑟 0) . (32)
International Journal of Computational Mathematics 5
If |𝑟| = 1, then
𝑃𝑃𝐻= (
0 𝐼𝑛−1
𝑟 0)(
0 𝑟
𝐼𝑛−1 0) = 𝐼𝑛. (33)
That is, 𝑃𝐻 = 𝑃−1. According to (32), we obtain that
𝐴𝐴𝐻= (
𝑛−1
∑
𝑖=0
𝑎𝑖𝑃𝑖)(
𝑛−1
∑
𝑗=0
𝑎𝑗(𝑃𝐻)
𝑗
)
= (
𝑛−1
∑
𝑖=0
𝑎𝑖𝑃𝑖)(
𝑛−1
∑
𝑗=0
𝑎𝑗(𝑃−1)
𝑗
) =
𝑛−1
∑
𝑖=0
𝑛−1
∑
𝑗=0
𝑎𝑖𝑎𝑗𝑃𝑖−𝑗
,
𝐴𝐻𝐴 = (
𝑛−1
∑
𝑗=0
𝑎𝑗(𝑃𝐻)
𝑗
)(
𝑛−1
∑
𝑖=0
𝑎𝑖𝑃𝑖)
= (
𝑛−1
∑
𝑗=0
𝑎𝑗𝑃−𝑗)(
𝑛−1
∑
𝑖=0
𝑎𝑗𝑃𝑖) =
𝑛−1
∑
𝑖=0
𝑛−1
∑
𝑗=0
𝑎𝑖𝑎𝑗𝑃𝑖−𝑗
.
(34)
Therefore 𝐴𝐴𝐻 = 𝐴𝐻𝐴, which shows that 𝐴 is normal.
According to Theorem 6 and Lemma 7, we have thefollowing theorem.
Theorem 8. Suppose that𝐴 = 𝐶𝑟(𝐻𝑘,0, 𝐻𝑘,1, . . . , 𝐻𝑘,𝑛−1) is an𝑟-circulant matrix. If |𝑟| = 1 and𝐻𝑘,𝑖 ⩾ 0, 𝑖 = 0, 1, 2, . . . , 𝑛 − 1,then the spectral norm of 𝐴 is
‖𝐴‖2 =
{{{{{{{{
{{{{{{{{
{
max0⩽𝑖⩽𝑛−1
𝑟𝐻𝑘,𝑛 + 𝑔 (𝑘) 𝑟1+(1/𝑛)
𝜔𝑖𝐻𝑘,𝑛−1 + 𝑟
1/𝑛[𝑓 (𝑘) 𝑎 − 𝑏] 𝜔
𝑖− 𝑎
𝑟1/𝑛
𝜔𝑖𝑓 (𝑘) + 𝑟
2/𝑛𝜔2𝑖𝑔 (𝑘) − 1
, 𝑓 (𝑘) + 𝑔 (𝑘) ̸= 1,
max0⩽𝑖⩽𝑛−1
(𝑔 (𝑘) 𝑟𝐻𝑘,𝑛−1 + 𝑎) (1 − 𝑟1/𝑛
𝜔𝑖) + [𝑔 (𝑘) 𝑎 + 𝑏] (𝑟
1/𝑛𝑤𝑖− 𝑟)
(1 − 𝑟1/𝑛
𝜔𝑖) [𝑔 (𝑘) 𝑟
1/𝑛𝜔𝑖+ 1]
, 𝑓 (𝑘) + 𝑔 (𝑘) = 1.
(35)
The following theorem simplifies and generalizes theresults of Theorem 2.2 in [12].
Theorem 9. Let 𝐴 = Circ(𝐻𝑘,0, 𝐻𝑘,1, . . . , 𝐻𝑘,𝑛−1) be a circu-lant matrix; then
‖𝐴‖2 =
{{{{{{{{{{{
{{{{{{{{{{{
{
𝐻𝑘,𝑛 + 𝑔 (𝑘)𝐻𝑘,𝑛−1 + 𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏
𝑓 (𝑘) + 𝑔 (𝑘) − 1
,
𝑓 (𝑘) + 𝑔 (𝑘) ̸= 1,
𝑔 (𝑘)𝐻𝑘,𝑛−1 + (𝑛 − 1) [𝑔 (𝑘) 𝑎 + 𝑏] + 𝑎
𝑔 (𝑘) + 1
,
𝑓 (𝑘) + 𝑔 (𝑘) = 1.
(36)
Proof. Suppose that 𝑟 = 1; it follows from Lemma 7 that 𝐴 isnormal. Notice that
𝜆𝑖
=
𝑛−1
∑
𝑖=0
𝐻𝑘,𝑖𝜇𝑗
𝑖
⩽
𝑛−1
∑
𝑖=0
𝐻𝑘,𝑖
𝜇𝑖
𝑗= 𝜆0. (37)
It follows from Lemma 3 that ‖𝐴‖2 = 𝜆0. According toTheorem 6, if 𝑓(𝑘) + 𝑔(𝑘) ̸= 1 and 𝑟 = 1, we obtain that
‖𝐴‖2 = 𝜆0 =
𝐻𝑘,𝑛 + 𝑔 (𝑘)𝐻𝑘,𝑛−1 + 𝑓 (𝑘) 𝑎 − 𝑎 − 𝑏
𝑓 (𝑘) + 𝑔 (𝑘) − 1
. (38)
Similarly, if 𝑓(𝑘) + 𝑔(𝑘) = 1, it follows that
‖𝐴‖2 = 𝜆0 =
𝑔 (𝑘)𝐻𝑘,𝑛−1 + (𝑛 − 1) [𝑔 (𝑘) 𝑎 + 𝑏] + 𝑎
𝑔 (𝑘) + 1
. (39)
This completes the proof.
Taking into account formulae (4)–(6), we have the follow-ing corollary.
Corollary 10. Let 𝐴1 = Circ(𝐹0, 𝐹1, . . . , 𝐹𝑛−1) be a circulantmatrix; then
𝐴1
2
= 𝐹𝑛+1 − 1. (40)
Corollary 11. Let 𝐴2 = Circ(𝐿0, 𝐿1, . . . , 𝐿𝑛−1) be a circulantmatrix; then
𝐴2
2
= 𝐹𝑛+2 + 𝐹𝑛 − 1. (41)
Corollary 12. Let 𝐴3 = Circ(𝐽0, 𝐽1, . . . , 𝐽𝑛−1) be a circulantmatrix; then
𝐴3
2
=
𝐽𝑛+1 − 1
2
. (42)
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper.
References
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6 International Journal of Computational Mathematics
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