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Research Article On the Spectrum and Spectral Norms of -Circulant Matrices with Generalized -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. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is work is concerned with the spectrum and spectral norms of -circulant matrices with generalized -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 -circulant matrix to be normal is presented. Based on the results we obtain the precise value for spectral norms of normal -circulant matrix with generalized -Horadam numbers, which generalize and improve the known results. 1. Introduction ere is no doubt that the -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. ere are many works concerning estimates for spec- tral norms of -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] inves- tigated 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 -circulant matrices whose entries are Fibonacci and Lucas numbers. In [5] they defined -circulant matrices involving -Lucas and -Fibonacci numbers and also investigated the upper and lower bounds for the spectral norms of these matrices. Recently, Yazlik and Taskara [6] define a generaliza- tion { , } of the special second-order sequences such as Fibonacci, Lucas, -Fibonacci, -Lucas, generalized - Fibonacci and -Lucas, Horadam, Pell, Jacobsthal, and Jacobsthal-Lucas sequences. For any integer number ⩾1, the generalized -Horadam sequence { , } is defined by the following recursive relation: ,+2 = () ,+1 + () , , ,0 = , ,1 = , (1) where () and () are scaler-value polynomials, 2 () + 4() > 0. e following are some particular cases. (i) If () = , () = 1 and =0, =1, the -Fibonacci sequence is obtained: ,+2 = ,+1 + , , ,0 = 0, ,1 = 1. (2) (ii) If () = , () = 1 and =2, =, the -Lucas sequence is obtained: ,+2 = ,+1 + , , ,0 = 0, ,1 = . (3) (iii) If () = 1, () = 1 and =0, =1, the Fibonacci sequence is obtained: +2 = +1 + , 0 = 0, 1 = 1. (4) (iv) If () = 1, () = 1 and =2, =1, the Lucas sequence is obtained: +2 = +1 + , 0 = 2, 1 = 1. (5) Hindawi Publishing Corporation International Journal of Computational Mathematics Volume 2014, Article ID 795175, 6 pages http://dx.doi.org/10.1155/2014/795175
Transcript
Page 1: Research Article On the Spectrum and Spectral …downloads.hindawi.com/archive/2014/795175.pdf-Circulant Matrices In this section, we consider the spectral norms of normal - circulant

Research ArticleOn the Spectrum and Spectral Norms of 119903-Circulant Matriceswith Generalized 119896-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 ahhylau163com

Received 17 May 2014 Accepted 25 August 2014 Published 31 August 2014

Academic Editor Chengpeng Bi

Copyright copy 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 119903-circulantmatrices with generalized 119896-Horadamnumbers entriesBy using Abel transformation and some identities we obtain an explicit formula for the eigenvalues of them In addition a sufficientcondition for an 119903-circulant matrix to be normal is presented Based on the results we obtain the precise value for spectral normsof normal 119903-circulant matrix with generalized 119896-Horadam numbers which generalize and improve the known results

1 Introduction

There is no doubt that the 119903-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 119903-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 119903-circulant matrices whose entries areFibonacci and Lucas numbers In [5] they defined 119903-circulantmatrices involving 119896-Lucas and 119896-Fibonacci numbers andalso investigated the upper and lower bounds for the spectralnorms of these matrices

Recently Yazlik and Taskara [6] define a generaliza-tion 119867119896119899 of the special second-order sequences suchas Fibonacci Lucas 119896-Fibonacci 119896-Lucas generalized 119896-Fibonacci and 119896-Lucas Horadam Pell Jacobsthal andJacobsthal-Lucas sequences For any integer number 119896 ⩾ 1

the generalized 119896-Horadam sequence 119867119896119899 is defined by thefollowing recursive relation

119867119896119899+2 = 119891 (119896)119867119896119899+1 + 119892 (119896)119867119896119899

1198671198960 = 119886 1198671198961 = 119887

(1)

where 119891(119896) and 119892(119896) are scaler-value polynomials 1198912(119896) +4119892(119896) gt 0 The following are some particular cases

(i) If119891(119896) = 119896119892(119896) = 1 and 119886 = 0 119887 = 1 the 119896-Fibonaccisequence is obtained

119865119896119899+2 = 119896119865119896119899+1 + 119865119896119899 1198651198960 = 0 1198651198961 = 1 (2)

(ii) If 119891(119896) = 119896 119892(119896) = 1 and 119886 = 2 119887 = 119896 the 119896-Lucassequence is obtained

119871119896119899+2 = 119896119871119896119899+1 + 119871119896119899 1198651198960 = 0 1198651198961 = 119896 (3)

(iii) If 119891(119896) = 1 119892(119896) = 1 and 119886 = 0 119887 = 1 the Fibonaccisequence is obtained

119865119899+2 = 119865119899+1 + 119865119899 1198650 = 0 1198651 = 1 (4)

(iv) If 119891(119896) = 1 119892(119896) = 1 and 119886 = 2 119887 = 1 the Lucassequence is obtained

119871119899+2 = 119871119899+1 + 119871119899 1198710 = 2 1198711 = 1 (5)

Hindawi Publishing CorporationInternational Journal of Computational MathematicsVolume 2014 Article ID 795175 6 pageshttpdxdoiorg1011552014795175

2 International Journal of Computational Mathematics

(v) If 119891(119896) = 1 119892(119896) = 2 and 119886 = 0 119887 = 1 the Jacobsthalsequence is obtained

119869119899+2 = 119869119899+1 + 2119869119899 1198690 = 0 1198691 = 1 (6)

In [7] the authors present new upper and lowerbounds for the spectral norm of an 119903-circulant matrix119862119903(1198671198960 1198671198961 119867119896119899minus1) and they study the spectral normofcirculantmatrixwith generalized 119896-Horadamnumbers in [8]In this paper we first give an explicit formula for the eigen-values of 119903-circulant matrix with generalized 119896-Horadamnumbers entries using different methods in [7] Afterwardswe present a sufficient condition for an 119903-circulant matrixto be normal Based on the results the precise value forspectral norms of normal 119903-circulant matrix whose entriesare generalized 119896-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 1198880 1198881 119888119899minus1 isin C the 119903-circulantmatrix 119862 denoted by 119862 = 119862119903(1198880 1198881 119888119899minus1) is of the form

(

1198880 1198881 1198882 sdot sdot sdot 119888119899minus1

119903119888119899minus1 1198880 1198881 sdot sdot sdot 119888119899minus2

119903119888119899minus2 119903119888119899minus1 1198880 sdot sdot sdot 119888119899minus3

d

1199031198881 1199031198882 1199031198883 sdot sdot sdot 1198880

) (7)

It is obvious that the matrix 119862119903 turns into a classical circulantmatrix for 119903 = 1

Lemma 2 (see [9]) Let 119862 = 119862119903(1198880 1198881 119888119899minus1) be an 119903-circulant matrix then the eigenvalues of 119862 are given by

120582119894 =

119899minus1

sum

119895=0

119888119895120583119895

119894 120583119894 = 119903

1119899120596119894 119894 = 0 1 119899 minus 1 (8)

where 120596 = 119890minus2120587119894119899 is the 119899th root of unity

Let us take anymatrix119860 = [119886119894119895] of order 119899 it is well knownthat the spectral norm of matrix 119860 is

1198602 = radic max0⩽119894⩽119899minus1

120582119894 (119860119867119860) (9)

where119860119867 is the conjugate transpose of119860 and 120582119894(119860119867119860) is the

eigenvalue of 119860119867119860For a normal matrix 119860 (ie 119860119860119867 = 119860

119867119860) we have the

following lemma

Lemma 3 (see [10]) Let 119860 be a normal matrix with eigenval-ues 1205820 1205821 120582119899minus1 Then the spectral norm of 119860 is

1198602 = max0⩽119894⩽119899minus1

1003816100381610038161003816120582119894

1003816100381610038161003816 (10)

The following lemma can be found in [11]

Lemma 4 (see [11] Abel transformation) Suppose that 119886119894and 119887119894 are two sequences and 119878119894 = 1198861 + 1198862 + sdot sdot sdot + 119886119894 (119894 =

1 2 ) then119899

sum

119894=1

119886119894119887119894 = 119878119899119887119899 minus

119899minus1

sum

119894=1

(119887119894+1 minus 119887119894) 119878119894 (11)

3 Spectrum of 119903-Circulant Matrix withGeneralized 119896-Horadam Numbers

We start this section by giving the following lemma

Lemma 5 Suppose that 119867119896119894119894isinN is a generalized 119896-Horadamsequence defined in (1) The following conclusions hold

(1) If 119891(119896) + 119892(119896) = 1 then119899

sum

119894=0

119867119896119894 =

119867119896119899+1 + 119892 (119896)119867119896119899 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

(12)

(2) If 119891(119896) + 119892(119896) = 1 then119899

sum

119894=0

119867119896119894 =

119892 (119896)119867119896119899 + 119899 [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

(13)

Proof (1) According to (1) we have119899

sum

119894=0

119867119896119894 = 119891 (119896)

119899

sum

119894=0

119867119896119894minus1 + 119892 (119896)

119899

sum

119894=0

119867119896119894minus2 (14)

Changing the summation index in (14) we have119899

sum

119894=0

119867119896119894

= 119891 (119896)(

119899

sum

119894=0

119867119896119894 minus 119867119896119899 + 119867119896minus1)

+119892 (119896)(

119899

sum

119894=0

119867119896119894 minus 119867119896119899minus1 minus 119867119896119899 + 119867119896minus1 + 119867119896minus2)

(15)

By direct calculation together with recursive relation (1) onecan obtain that

[119891 (119896) + 119892 (119896) minus 1]

119899

sum

119894=0

119867119896119894

= 119867119896119899+1 + 119892 (119896)119867119896119899 + 119891 (119896) 119886 minus 119886 minus 119887

(16)

Therefore we immediately obtain (12) from 119891(119896) + 119892(119896) = 1

(2) Suppose that 119891(119896) + 119892(119896) = 1 we first illustrate that119867119896119894+1+119892(119896)119867119896119894 equiv 119892(119896)119886+119887 Let119881119894 = 119867119896119894+1+119892(119896)119867119896119894then1198810 = 119892(119896)119886+119887 Combining (1) and 119891(119896)+119892(119896) =

1 one can obtain that

119881119894+1 = 119867119896119894+2 + 119892 (119896)119867119896119894+1

= (119891 (119896)119867119896119894+1 + 119892 (119896)119867119896119894) + 119892 (119896)119867119896119894+1

= 119867119896119894+1 + 119892 (119896)119867119896119894 = 119881119894

(17)

International Journal of Computational Mathematics 3

which shows that 119881119894 is a constant sequence and therefore

119867119896119894+1 + 119892 (119896)119867119896119894 = 119881119894 = 1198810 = 119892 (119896) 119886 + 119887 (18)

Evaluating summation from 0 to 119899 we have

119899

sum

119894=0

119867119896119894+1 + 119892 (119896)

119899

sum

119894=0

119867119896119894 = (119899 + 1) [119892 (119896) 119886 + 119887] (19)

Changing the summation index in (19) gives

(

119899

sum

119894=0

119867119896119894 + 119867119896119899+1 minus 119886) + 119892 (119896)

119899

sum

119894=0

119867119896119894

= (119899 + 1) [119892 (119896) 119886 + 119887]

(20)

Therefore

[119892 (119896) + 1]

119899

sum

119894=0

119867119896119894 = 119892 (119896)119867119896119899 + 119899 [119892 (119896) 119886 + 119887] + 119886 (21)

In view of assumptions 1198912(119896) +119892(119896) gt 1 and 119891(119896) +119892(119896) = 1we know that 119892(119896)+1 = 0Thus we obtain (13) from (21)

From Lemma 5 we have the following theorem

Theorem 6 Let 119860 = 119862119903(1198671198960 1198671198961 119867119896119899minus1) be an 119903-circulant matrix with eigenvalues 1205820 1205821 120582119899minus1 then for 119894 =0 1 2 119899 minus 1 the following hold

(1) If 119891(119896) + 119892(119896) = 1 then

120582119894 = (119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1

+1199031119899

[119891 (119896) 119886 minus 119887] 120596119894minus 119886)

times(1199031119899

120596119894119891 (119896) + 119903

21198991205962119894119892 (119896) minus 1)

minus1

(22)

(2) If 119891(119896) + 119892(119896) = 1 then

120582119894 = ((119892 (119896) 119903119867119896119899minus1 + 119886) (1 minus 1199031119899

120596119894)

+ [119892 (119896) 119886 + 119887] (1199031119899

119908119894minus 119903))

times ((1 minus 1199031119899

120596119894) [119892 (119896) 119903

1119899120596119894+ 1])

minus1

(23)

Proof According to Lemma 2 we have

120582119894 =

119899minus1

sum

119894=0

119867119896119894120583119895

119894 120583119894 = 119903

1119899119908119894 (24)

Using Abel transformation (Lemma 4) we have

120582119894 = 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus

119899minus2

sum

119895=0

((120583119895+1

119894minus 120583119895

119894)

119895

sum

119904=0

119867119896119904)

= 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus (120583119894 minus 1)

119899minus2

sum

119895=0

(120583119895

119894

119895

sum

119904=0

119867119896119904)

(25)

(1) In the light of (12) and (25) one can obtain that

120582119894 = 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus

120583119894 minus 1

119891 (119896) + 119892 (119896) minus 1

times

119899minus2

sum

119895=0

120583119895

119894[119867119896119895+1 + 119892 (119896)119867119896119895 + 119891 (119896) 119886 minus 119886 minus 119887]

= 120583119899minus1

119894

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

minus

120583119894 minus 1

119891 (119896) + 119892 (119896) minus 1

times (

119899minus2

sum

119895=0

119867119896119895+1120583119895

119894+ 119892 (119896)

119899minus2

sum

119895=0

119867119896119895120583119895

119894

+ [119891 (119896) 119886 minus 119886 minus 119887]

119899minus2

sum

119895=0

120583119895

119894)

(26)

It is clear that

119899minus2

sum

119895=0

119867119896119895+1120583119895

119894=

120582119894 minus 119886

120583119894

119899minus2

sum

119895=0

119867119896119895120583119895

119894= 120582119894 minus 120583

119899minus1

119894119867119896119899minus1

(27)

4 International Journal of Computational Mathematics

Substituting (27) into (26) we obtain that

120582119894 = 120583119899minus1

119894

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

minus

120583119894 minus 1

119891 (119896) + 119892 (119896) minus 1

times (

120582119894 minus 119886

120583119894

+ 119892 (119896) (120582119894 minus 120583119899minus1

119894119867119896119899minus1)

+ [119891 (119896) 119886 minus 119886 minus 119887]

119899minus2

sum

119895=0

120583119895

119894)

=

(1 minus 120583119894) [1 + 119892 (119896) 120583119894]

120583119894 [119891 (119896) + 119892 (119896) minus 1]

120582119894

+

120583119899minus1

119894119867119896119899 + 120583

119899

119894119892 (119896)119867119896119899minus1

119891 (119896) + 119892 (119896) minus 1

+

120583119899

119894(119891 (119896) 119886 minus 119886 minus 119887) minus 119886 (1 minus 120583119894)

120583119894 [119891 (119896) + 119892 (119896) minus 1]

+

(1 minus 120583119899minus1

119894) (119891 (119896) 119886 minus 119886 minus 119887)

119891 (119896) + 119892 (119896) minus 1

(28)

Therefore we have

[119892 (119896) 1205832

119894+ 119891 (119896) 120583119894 minus 1] 120582119894

= 120583119899

119894119867119896119899 + 119892 (119896) 120583

119899+1

119894119867119896119899minus1 + 120583

119899

119894(119891 (119896) 119886 minus 119886 minus 119887)

minus 119886 (1 minus 120583119894) + (120583119894 minus 119903) (119891 (119896) 119886 minus 119886 minus 119887)

= 119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1

+ 1199031119899

[119891 (119896) 119886 minus 119887] 120596119894minus 119886

(29)

We immediately obtain formula (22) from (29)

(2) Taking into account (13) and (25) we have

120582119894 = 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus

120583119894 minus 1

119892 (119896) + 1

times

119899minus2

sum

119895=0

120583119895

119894[119892 (119896)119867119896119895 + 119895 sdot (119892 (119896) 119886 + 119887) + 119886]

= 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 +

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

times

119899minus2

sum

119895=0

119867119896119895120583119895

119894+

1 minus 120583119894

119892 (119896) + 1

119899minus2

sum

119895=0

[119895 sdot (119892 (119896) 119886 + 119887) + 119886] 120583119895

119894

=

120583119899minus1

119894[119892 (119896)119867119896119899minus1 + (119899 minus 1) (119892 (119896) 119886 + 119887) + 119886]

119892 (119896) + 1

+

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

(120582119894 minus 119867119896119899minus1120583119899minus1

119894)

+

1 minus 120583119894

119892 (119896) + 1

[

[

((119899 minus 2) (119892 (119896) 119886 + 119887) + 119886)

times

119899minus2

sum

119895=0

120583119895

119894minus (119892 (119896) 119886 + 119887)

119899minus3

sum

119895=0

119895

sum

119904=0

120583119904

119894]

]

(30)

It follows that

120582119894 =

119892 (119896) 120583119899

119894

119892 (119896) + 1

119867119896119899minus1 +

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

120582119894

+

120583119899minus1

119894[(119899 minus 1) (119892 (119896) 119886 + 119887) + 119886]

119892 (119896) + 1

+

1 minus 120583119894

119892 (119896) + 1

[

[

((119899 minus 2) (119892 (119896) 119886 + 119887) + 119886)

times

119899minus2

sum

119895=0

120583119895

119894minus

(119892 (119896) 119886 + 119887)

1 minus 120583119894

119899minus3

sum

119895=0

(1 minus 120583119895+1

119894)]

]

=

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

120582119894

+

119892 (119896) 120583119899

119894119867119896119899minus1 + 120583

119899minus1

119894(119892 (119896) 119886 + 119887) + 119886

119892 (119896) + 1

+

(119892 (119896) 119886 + 119887) (120583119894 minus 120583119899

119894)

(119892 (119896) + 1) (1 minus 120583119894)

(31)

Therefore we obtain (23) This concludes the proof

4 Spectral Norms of Normal119903-Circulant Matrices

In this section we consider the spectral norms of normal 119903-circulant matrix whose entries are generalized 119896-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 119860 = 119862119903(1198860 1198861 119886119899minus1) be an 119903-circulantmatrix If |119903| = 1 then 119860 is normal matrix

Proof It is well known that

119860 =

119899minus1

sum

119894=0

119886119894119875119894 119875 = (

0 119868119899minus1

119903 0) (32)

International Journal of Computational Mathematics 5

If |119903| = 1 then

119875119875119867= (

0 119868119899minus1

119903 0)(

0 119903

119868119899minus1 0) = 119868119899 (33)

That is 119875119867 = 119875minus1 According to (32) we obtain that

119860119860119867= (

119899minus1

sum

119894=0

119886119894119875119894)(

119899minus1

sum

119895=0

119886119895(119875119867)

119895

)

= (

119899minus1

sum

119894=0

119886119894119875119894)(

119899minus1

sum

119895=0

119886119895(119875minus1)

119895

) =

119899minus1

sum

119894=0

119899minus1

sum

119895=0

119886119894119886119895119875119894minus119895

119860119867119860 = (

119899minus1

sum

119895=0

119886119895(119875119867)

119895

)(

119899minus1

sum

119894=0

119886119894119875119894)

= (

119899minus1

sum

119895=0

119886119895119875minus119895)(

119899minus1

sum

119894=0

119886119895119875119894) =

119899minus1

sum

119894=0

119899minus1

sum

119895=0

119886119894119886119895119875119894minus119895

(34)

Therefore 119860119860119867 = 119860119867119860 which shows that 119860 is normal

According to Theorem 6 and Lemma 7 we have thefollowing theorem

Theorem 8 Suppose that119860 = 119862119903(1198671198960 1198671198961 119867119896119899minus1) is an119903-circulant matrix If |119903| = 1 and119867119896119894 ⩾ 0 119894 = 0 1 2 119899 minus 1then the spectral norm of 119860 is

1198602 =

max0⩽119894⩽119899minus1

1003816100381610038161003816100381610038161003816100381610038161003816

119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1 + 119903

1119899[119891 (119896) 119886 minus 119887] 120596

119894minus 119886

1199031119899

120596119894119891 (119896) + 119903

21198991205962119894119892 (119896) minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

119891 (119896) + 119892 (119896) = 1

max0⩽119894⩽119899minus1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119892 (119896) 119903119867119896119899minus1 + 119886) (1 minus 1199031119899

120596119894) + [119892 (119896) 119886 + 119887] (119903

1119899119908119894minus 119903)

(1 minus 1199031119899

120596119894) [119892 (119896) 119903

1119899120596119894+ 1]

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (119896) + 119892 (119896) = 1

(35)

The following theorem simplifies and generalizes theresults of Theorem 22 in [12]

Theorem 9 Let 119860 = Circ(1198671198960 1198671198961 119867119896119899minus1) be a circu-lant matrix then

1198602 =

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

119891 (119896) + 119892 (119896) = 1

119892 (119896)119867119896119899minus1 + (119899 minus 1) [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

119891 (119896) + 119892 (119896) = 1

(36)

Proof Suppose that 119903 = 1 it follows from Lemma 7 that 119860 isnormal Notice that

1003816100381610038161003816120582119894

1003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=0

119867119896119894120583119895

119894

1003816100381610038161003816100381610038161003816100381610038161003816

⩽

119899minus1

sum

119894=0

119867119896119894

1003816100381610038161003816120583119894

1003816100381610038161003816

119895= 1205820 (37)

It follows from Lemma 3 that 1198602 = 1205820 According toTheorem 6 if 119891(119896) + 119892(119896) = 1 and 119903 = 1 we obtain that

1198602 = 1205820 =

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

(38)

Similarly if 119891(119896) + 119892(119896) = 1 it follows that

1198602 = 1205820 =

119892 (119896)119867119896119899minus1 + (119899 minus 1) [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

(39)

This completes the proof

Taking into account formulae (4)ndash(6) we have the follow-ing corollary

Corollary 10 Let 1198601 = Circ(1198650 1198651 119865119899minus1) be a circulantmatrix then

10038171003817100381710038171198601

10038171003817100381710038172

= 119865119899+1 minus 1 (40)

Corollary 11 Let 1198602 = Circ(1198710 1198711 119871119899minus1) be a circulantmatrix then

10038171003817100381710038171198602

10038171003817100381710038172

= 119865119899+2 + 119865119899 minus 1 (41)

Corollary 12 Let 1198603 = Circ(1198690 1198691 119869119899minus1) be a circulantmatrix then

10038171003817100381710038171198603

10038171003817100381710038172

=

119869119899+1 minus 1

2

(42)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005

[2] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011

6 International Journal of Computational Mathematics

[3] W Bani-Domi and F Kittaneh ldquoNorm equalities and inequali-ties for operator matricesrdquo Linear Algebra and Its Applicationsvol 429 no 1 pp 57ndash67 2008

[4] S Shen and J Cen ldquoOn the bounds for the norms of 119903-circulant matrices with the Fibonacci and Lucas numbersrdquoApplied Mathematics and Computation vol 216 no 10 pp2891ndash2897 2010

[5] S Shen and J Cen ldquoOn the spectral norms of r-circulant matri-ces with the k-Fibonacci and k-Lucas numbersrdquo InternationalJournal of Contemporary Mathematical Sciences vol 5 no 9ndash12 pp 569ndash578 2010

[6] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012

[7] Y Yazlik and N Taskara ldquoOn the norms of an r-circulantmatrix with the generalized k-Horadam numbersrdquo Journal ofInequalities and Applications vol 2013 article 394 2013

[8] Y Yazlik and N Taskara ldquoSpectral norm eigenvalues anddeterminant of circulant matrix involving the generalized k-Horadam numbersrdquo Ars Combinatoria vol 104 pp 505ndash5122012

[9] Z Jiang and Z Zhou ldquoNonsingularity of 119903-circulant matricesrdquoApplied Mathematics A Journal of Chinese Universities vol 10no 2 pp 222ndash226 1995

[10] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press 1985

[11] W Rudin Principles of Mathematical Analysis McGraw-Hill3rd edition 1976

[12] E G Kocer T Mansour and N Tuglu ldquoNorms of circulantand semicirculant matrices with Horadamrsquos numbersrdquo ArsCombinatoria vol 85 pp 353ndash359 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Research Article On the Spectrum and Spectral …downloads.hindawi.com/archive/2014/795175.pdf-Circulant Matrices In this section, we consider the spectral norms of normal - circulant

2 International Journal of Computational Mathematics

(v) If 119891(119896) = 1 119892(119896) = 2 and 119886 = 0 119887 = 1 the Jacobsthalsequence is obtained

119869119899+2 = 119869119899+1 + 2119869119899 1198690 = 0 1198691 = 1 (6)

In [7] the authors present new upper and lowerbounds for the spectral norm of an 119903-circulant matrix119862119903(1198671198960 1198671198961 119867119896119899minus1) and they study the spectral normofcirculantmatrixwith generalized 119896-Horadamnumbers in [8]In this paper we first give an explicit formula for the eigen-values of 119903-circulant matrix with generalized 119896-Horadamnumbers entries using different methods in [7] Afterwardswe present a sufficient condition for an 119903-circulant matrixto be normal Based on the results the precise value forspectral norms of normal 119903-circulant matrix whose entriesare generalized 119896-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 1198880 1198881 119888119899minus1 isin C the 119903-circulantmatrix 119862 denoted by 119862 = 119862119903(1198880 1198881 119888119899minus1) is of the form

(

1198880 1198881 1198882 sdot sdot sdot 119888119899minus1

119903119888119899minus1 1198880 1198881 sdot sdot sdot 119888119899minus2

119903119888119899minus2 119903119888119899minus1 1198880 sdot sdot sdot 119888119899minus3

d

1199031198881 1199031198882 1199031198883 sdot sdot sdot 1198880

) (7)

It is obvious that the matrix 119862119903 turns into a classical circulantmatrix for 119903 = 1

Lemma 2 (see [9]) Let 119862 = 119862119903(1198880 1198881 119888119899minus1) be an 119903-circulant matrix then the eigenvalues of 119862 are given by

120582119894 =

119899minus1

sum

119895=0

119888119895120583119895

119894 120583119894 = 119903

1119899120596119894 119894 = 0 1 119899 minus 1 (8)

where 120596 = 119890minus2120587119894119899 is the 119899th root of unity

Let us take anymatrix119860 = [119886119894119895] of order 119899 it is well knownthat the spectral norm of matrix 119860 is

1198602 = radic max0⩽119894⩽119899minus1

120582119894 (119860119867119860) (9)

where119860119867 is the conjugate transpose of119860 and 120582119894(119860119867119860) is the

eigenvalue of 119860119867119860For a normal matrix 119860 (ie 119860119860119867 = 119860

119867119860) we have the

following lemma

Lemma 3 (see [10]) Let 119860 be a normal matrix with eigenval-ues 1205820 1205821 120582119899minus1 Then the spectral norm of 119860 is

1198602 = max0⩽119894⩽119899minus1

1003816100381610038161003816120582119894

1003816100381610038161003816 (10)

The following lemma can be found in [11]

Lemma 4 (see [11] Abel transformation) Suppose that 119886119894and 119887119894 are two sequences and 119878119894 = 1198861 + 1198862 + sdot sdot sdot + 119886119894 (119894 =

1 2 ) then119899

sum

119894=1

119886119894119887119894 = 119878119899119887119899 minus

119899minus1

sum

119894=1

(119887119894+1 minus 119887119894) 119878119894 (11)

3 Spectrum of 119903-Circulant Matrix withGeneralized 119896-Horadam Numbers

We start this section by giving the following lemma

Lemma 5 Suppose that 119867119896119894119894isinN is a generalized 119896-Horadamsequence defined in (1) The following conclusions hold

(1) If 119891(119896) + 119892(119896) = 1 then119899

sum

119894=0

119867119896119894 =

119867119896119899+1 + 119892 (119896)119867119896119899 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

(12)

(2) If 119891(119896) + 119892(119896) = 1 then119899

sum

119894=0

119867119896119894 =

119892 (119896)119867119896119899 + 119899 [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

(13)

Proof (1) According to (1) we have119899

sum

119894=0

119867119896119894 = 119891 (119896)

119899

sum

119894=0

119867119896119894minus1 + 119892 (119896)

119899

sum

119894=0

119867119896119894minus2 (14)

Changing the summation index in (14) we have119899

sum

119894=0

119867119896119894

= 119891 (119896)(

119899

sum

119894=0

119867119896119894 minus 119867119896119899 + 119867119896minus1)

+119892 (119896)(

119899

sum

119894=0

119867119896119894 minus 119867119896119899minus1 minus 119867119896119899 + 119867119896minus1 + 119867119896minus2)

(15)

By direct calculation together with recursive relation (1) onecan obtain that

[119891 (119896) + 119892 (119896) minus 1]

119899

sum

119894=0

119867119896119894

= 119867119896119899+1 + 119892 (119896)119867119896119899 + 119891 (119896) 119886 minus 119886 minus 119887

(16)

Therefore we immediately obtain (12) from 119891(119896) + 119892(119896) = 1

(2) Suppose that 119891(119896) + 119892(119896) = 1 we first illustrate that119867119896119894+1+119892(119896)119867119896119894 equiv 119892(119896)119886+119887 Let119881119894 = 119867119896119894+1+119892(119896)119867119896119894then1198810 = 119892(119896)119886+119887 Combining (1) and 119891(119896)+119892(119896) =

1 one can obtain that

119881119894+1 = 119867119896119894+2 + 119892 (119896)119867119896119894+1

= (119891 (119896)119867119896119894+1 + 119892 (119896)119867119896119894) + 119892 (119896)119867119896119894+1

= 119867119896119894+1 + 119892 (119896)119867119896119894 = 119881119894

(17)

International Journal of Computational Mathematics 3

which shows that 119881119894 is a constant sequence and therefore

119867119896119894+1 + 119892 (119896)119867119896119894 = 119881119894 = 1198810 = 119892 (119896) 119886 + 119887 (18)

Evaluating summation from 0 to 119899 we have

119899

sum

119894=0

119867119896119894+1 + 119892 (119896)

119899

sum

119894=0

119867119896119894 = (119899 + 1) [119892 (119896) 119886 + 119887] (19)

Changing the summation index in (19) gives

(

119899

sum

119894=0

119867119896119894 + 119867119896119899+1 minus 119886) + 119892 (119896)

119899

sum

119894=0

119867119896119894

= (119899 + 1) [119892 (119896) 119886 + 119887]

(20)

Therefore

[119892 (119896) + 1]

119899

sum

119894=0

119867119896119894 = 119892 (119896)119867119896119899 + 119899 [119892 (119896) 119886 + 119887] + 119886 (21)

In view of assumptions 1198912(119896) +119892(119896) gt 1 and 119891(119896) +119892(119896) = 1we know that 119892(119896)+1 = 0Thus we obtain (13) from (21)

From Lemma 5 we have the following theorem

Theorem 6 Let 119860 = 119862119903(1198671198960 1198671198961 119867119896119899minus1) be an 119903-circulant matrix with eigenvalues 1205820 1205821 120582119899minus1 then for 119894 =0 1 2 119899 minus 1 the following hold

(1) If 119891(119896) + 119892(119896) = 1 then

120582119894 = (119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1

+1199031119899

[119891 (119896) 119886 minus 119887] 120596119894minus 119886)

times(1199031119899

120596119894119891 (119896) + 119903

21198991205962119894119892 (119896) minus 1)

minus1

(22)

(2) If 119891(119896) + 119892(119896) = 1 then

120582119894 = ((119892 (119896) 119903119867119896119899minus1 + 119886) (1 minus 1199031119899

120596119894)

+ [119892 (119896) 119886 + 119887] (1199031119899

119908119894minus 119903))

times ((1 minus 1199031119899

120596119894) [119892 (119896) 119903

1119899120596119894+ 1])

minus1

(23)

Proof According to Lemma 2 we have

120582119894 =

119899minus1

sum

119894=0

119867119896119894120583119895

119894 120583119894 = 119903

1119899119908119894 (24)

Using Abel transformation (Lemma 4) we have

120582119894 = 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus

119899minus2

sum

119895=0

((120583119895+1

119894minus 120583119895

119894)

119895

sum

119904=0

119867119896119904)

= 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus (120583119894 minus 1)

119899minus2

sum

119895=0

(120583119895

119894

119895

sum

119904=0

119867119896119904)

(25)

(1) In the light of (12) and (25) one can obtain that

120582119894 = 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus

120583119894 minus 1

119891 (119896) + 119892 (119896) minus 1

times

119899minus2

sum

119895=0

120583119895

119894[119867119896119895+1 + 119892 (119896)119867119896119895 + 119891 (119896) 119886 minus 119886 minus 119887]

= 120583119899minus1

119894

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

minus

120583119894 minus 1

119891 (119896) + 119892 (119896) minus 1

times (

119899minus2

sum

119895=0

119867119896119895+1120583119895

119894+ 119892 (119896)

119899minus2

sum

119895=0

119867119896119895120583119895

119894

+ [119891 (119896) 119886 minus 119886 minus 119887]

119899minus2

sum

119895=0

120583119895

119894)

(26)

It is clear that

119899minus2

sum

119895=0

119867119896119895+1120583119895

119894=

120582119894 minus 119886

120583119894

119899minus2

sum

119895=0

119867119896119895120583119895

119894= 120582119894 minus 120583

119899minus1

119894119867119896119899minus1

(27)

4 International Journal of Computational Mathematics

Substituting (27) into (26) we obtain that

120582119894 = 120583119899minus1

119894

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

minus

120583119894 minus 1

119891 (119896) + 119892 (119896) minus 1

times (

120582119894 minus 119886

120583119894

+ 119892 (119896) (120582119894 minus 120583119899minus1

119894119867119896119899minus1)

+ [119891 (119896) 119886 minus 119886 minus 119887]

119899minus2

sum

119895=0

120583119895

119894)

=

(1 minus 120583119894) [1 + 119892 (119896) 120583119894]

120583119894 [119891 (119896) + 119892 (119896) minus 1]

120582119894

+

120583119899minus1

119894119867119896119899 + 120583

119899

119894119892 (119896)119867119896119899minus1

119891 (119896) + 119892 (119896) minus 1

+

120583119899

119894(119891 (119896) 119886 minus 119886 minus 119887) minus 119886 (1 minus 120583119894)

120583119894 [119891 (119896) + 119892 (119896) minus 1]

+

(1 minus 120583119899minus1

119894) (119891 (119896) 119886 minus 119886 minus 119887)

119891 (119896) + 119892 (119896) minus 1

(28)

Therefore we have

[119892 (119896) 1205832

119894+ 119891 (119896) 120583119894 minus 1] 120582119894

= 120583119899

119894119867119896119899 + 119892 (119896) 120583

119899+1

119894119867119896119899minus1 + 120583

119899

119894(119891 (119896) 119886 minus 119886 minus 119887)

minus 119886 (1 minus 120583119894) + (120583119894 minus 119903) (119891 (119896) 119886 minus 119886 minus 119887)

= 119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1

+ 1199031119899

[119891 (119896) 119886 minus 119887] 120596119894minus 119886

(29)

We immediately obtain formula (22) from (29)

(2) Taking into account (13) and (25) we have

120582119894 = 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus

120583119894 minus 1

119892 (119896) + 1

times

119899minus2

sum

119895=0

120583119895

119894[119892 (119896)119867119896119895 + 119895 sdot (119892 (119896) 119886 + 119887) + 119886]

= 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 +

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

times

119899minus2

sum

119895=0

119867119896119895120583119895

119894+

1 minus 120583119894

119892 (119896) + 1

119899minus2

sum

119895=0

[119895 sdot (119892 (119896) 119886 + 119887) + 119886] 120583119895

119894

=

120583119899minus1

119894[119892 (119896)119867119896119899minus1 + (119899 minus 1) (119892 (119896) 119886 + 119887) + 119886]

119892 (119896) + 1

+

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

(120582119894 minus 119867119896119899minus1120583119899minus1

119894)

+

1 minus 120583119894

119892 (119896) + 1

[

[

((119899 minus 2) (119892 (119896) 119886 + 119887) + 119886)

times

119899minus2

sum

119895=0

120583119895

119894minus (119892 (119896) 119886 + 119887)

119899minus3

sum

119895=0

119895

sum

119904=0

120583119904

119894]

]

(30)

It follows that

120582119894 =

119892 (119896) 120583119899

119894

119892 (119896) + 1

119867119896119899minus1 +

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

120582119894

+

120583119899minus1

119894[(119899 minus 1) (119892 (119896) 119886 + 119887) + 119886]

119892 (119896) + 1

+

1 minus 120583119894

119892 (119896) + 1

[

[

((119899 minus 2) (119892 (119896) 119886 + 119887) + 119886)

times

119899minus2

sum

119895=0

120583119895

119894minus

(119892 (119896) 119886 + 119887)

1 minus 120583119894

119899minus3

sum

119895=0

(1 minus 120583119895+1

119894)]

]

=

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

120582119894

+

119892 (119896) 120583119899

119894119867119896119899minus1 + 120583

119899minus1

119894(119892 (119896) 119886 + 119887) + 119886

119892 (119896) + 1

+

(119892 (119896) 119886 + 119887) (120583119894 minus 120583119899

119894)

(119892 (119896) + 1) (1 minus 120583119894)

(31)

Therefore we obtain (23) This concludes the proof

4 Spectral Norms of Normal119903-Circulant Matrices

In this section we consider the spectral norms of normal 119903-circulant matrix whose entries are generalized 119896-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 119860 = 119862119903(1198860 1198861 119886119899minus1) be an 119903-circulantmatrix If |119903| = 1 then 119860 is normal matrix

Proof It is well known that

119860 =

119899minus1

sum

119894=0

119886119894119875119894 119875 = (

0 119868119899minus1

119903 0) (32)

International Journal of Computational Mathematics 5

If |119903| = 1 then

119875119875119867= (

0 119868119899minus1

119903 0)(

0 119903

119868119899minus1 0) = 119868119899 (33)

That is 119875119867 = 119875minus1 According to (32) we obtain that

119860119860119867= (

119899minus1

sum

119894=0

119886119894119875119894)(

119899minus1

sum

119895=0

119886119895(119875119867)

119895

)

= (

119899minus1

sum

119894=0

119886119894119875119894)(

119899minus1

sum

119895=0

119886119895(119875minus1)

119895

) =

119899minus1

sum

119894=0

119899minus1

sum

119895=0

119886119894119886119895119875119894minus119895

119860119867119860 = (

119899minus1

sum

119895=0

119886119895(119875119867)

119895

)(

119899minus1

sum

119894=0

119886119894119875119894)

= (

119899minus1

sum

119895=0

119886119895119875minus119895)(

119899minus1

sum

119894=0

119886119895119875119894) =

119899minus1

sum

119894=0

119899minus1

sum

119895=0

119886119894119886119895119875119894minus119895

(34)

Therefore 119860119860119867 = 119860119867119860 which shows that 119860 is normal

According to Theorem 6 and Lemma 7 we have thefollowing theorem

Theorem 8 Suppose that119860 = 119862119903(1198671198960 1198671198961 119867119896119899minus1) is an119903-circulant matrix If |119903| = 1 and119867119896119894 ⩾ 0 119894 = 0 1 2 119899 minus 1then the spectral norm of 119860 is

1198602 =

max0⩽119894⩽119899minus1

1003816100381610038161003816100381610038161003816100381610038161003816

119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1 + 119903

1119899[119891 (119896) 119886 minus 119887] 120596

119894minus 119886

1199031119899

120596119894119891 (119896) + 119903

21198991205962119894119892 (119896) minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

119891 (119896) + 119892 (119896) = 1

max0⩽119894⩽119899minus1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119892 (119896) 119903119867119896119899minus1 + 119886) (1 minus 1199031119899

120596119894) + [119892 (119896) 119886 + 119887] (119903

1119899119908119894minus 119903)

(1 minus 1199031119899

120596119894) [119892 (119896) 119903

1119899120596119894+ 1]

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (119896) + 119892 (119896) = 1

(35)

The following theorem simplifies and generalizes theresults of Theorem 22 in [12]

Theorem 9 Let 119860 = Circ(1198671198960 1198671198961 119867119896119899minus1) be a circu-lant matrix then

1198602 =

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

119891 (119896) + 119892 (119896) = 1

119892 (119896)119867119896119899minus1 + (119899 minus 1) [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

119891 (119896) + 119892 (119896) = 1

(36)

Proof Suppose that 119903 = 1 it follows from Lemma 7 that 119860 isnormal Notice that

1003816100381610038161003816120582119894

1003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=0

119867119896119894120583119895

119894

1003816100381610038161003816100381610038161003816100381610038161003816

⩽

119899minus1

sum

119894=0

119867119896119894

1003816100381610038161003816120583119894

1003816100381610038161003816

119895= 1205820 (37)

It follows from Lemma 3 that 1198602 = 1205820 According toTheorem 6 if 119891(119896) + 119892(119896) = 1 and 119903 = 1 we obtain that

1198602 = 1205820 =

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

(38)

Similarly if 119891(119896) + 119892(119896) = 1 it follows that

1198602 = 1205820 =

119892 (119896)119867119896119899minus1 + (119899 minus 1) [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

(39)

This completes the proof

Taking into account formulae (4)ndash(6) we have the follow-ing corollary

Corollary 10 Let 1198601 = Circ(1198650 1198651 119865119899minus1) be a circulantmatrix then

10038171003817100381710038171198601

10038171003817100381710038172

= 119865119899+1 minus 1 (40)

Corollary 11 Let 1198602 = Circ(1198710 1198711 119871119899minus1) be a circulantmatrix then

10038171003817100381710038171198602

10038171003817100381710038172

= 119865119899+2 + 119865119899 minus 1 (41)

Corollary 12 Let 1198603 = Circ(1198690 1198691 119869119899minus1) be a circulantmatrix then

10038171003817100381710038171198603

10038171003817100381710038172

=

119869119899+1 minus 1

2

(42)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005

[2] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011

6 International Journal of Computational Mathematics

[3] W Bani-Domi and F Kittaneh ldquoNorm equalities and inequali-ties for operator matricesrdquo Linear Algebra and Its Applicationsvol 429 no 1 pp 57ndash67 2008

[4] S Shen and J Cen ldquoOn the bounds for the norms of 119903-circulant matrices with the Fibonacci and Lucas numbersrdquoApplied Mathematics and Computation vol 216 no 10 pp2891ndash2897 2010

[5] S Shen and J Cen ldquoOn the spectral norms of r-circulant matri-ces with the k-Fibonacci and k-Lucas numbersrdquo InternationalJournal of Contemporary Mathematical Sciences vol 5 no 9ndash12 pp 569ndash578 2010

[6] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012

[7] Y Yazlik and N Taskara ldquoOn the norms of an r-circulantmatrix with the generalized k-Horadam numbersrdquo Journal ofInequalities and Applications vol 2013 article 394 2013

[8] Y Yazlik and N Taskara ldquoSpectral norm eigenvalues anddeterminant of circulant matrix involving the generalized k-Horadam numbersrdquo Ars Combinatoria vol 104 pp 505ndash5122012

[9] Z Jiang and Z Zhou ldquoNonsingularity of 119903-circulant matricesrdquoApplied Mathematics A Journal of Chinese Universities vol 10no 2 pp 222ndash226 1995

[10] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press 1985

[11] W Rudin Principles of Mathematical Analysis McGraw-Hill3rd edition 1976

[12] E G Kocer T Mansour and N Tuglu ldquoNorms of circulantand semicirculant matrices with Horadamrsquos numbersrdquo ArsCombinatoria vol 85 pp 353ndash359 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Research Article On the Spectrum and Spectral …downloads.hindawi.com/archive/2014/795175.pdf-Circulant Matrices In this section, we consider the spectral norms of normal - circulant

International Journal of Computational Mathematics 3

which shows that 119881119894 is a constant sequence and therefore

119867119896119894+1 + 119892 (119896)119867119896119894 = 119881119894 = 1198810 = 119892 (119896) 119886 + 119887 (18)

Evaluating summation from 0 to 119899 we have

119899

sum

119894=0

119867119896119894+1 + 119892 (119896)

119899

sum

119894=0

119867119896119894 = (119899 + 1) [119892 (119896) 119886 + 119887] (19)

Changing the summation index in (19) gives

(

119899

sum

119894=0

119867119896119894 + 119867119896119899+1 minus 119886) + 119892 (119896)

119899

sum

119894=0

119867119896119894

= (119899 + 1) [119892 (119896) 119886 + 119887]

(20)

Therefore

[119892 (119896) + 1]

119899

sum

119894=0

119867119896119894 = 119892 (119896)119867119896119899 + 119899 [119892 (119896) 119886 + 119887] + 119886 (21)

In view of assumptions 1198912(119896) +119892(119896) gt 1 and 119891(119896) +119892(119896) = 1we know that 119892(119896)+1 = 0Thus we obtain (13) from (21)

From Lemma 5 we have the following theorem

Theorem 6 Let 119860 = 119862119903(1198671198960 1198671198961 119867119896119899minus1) be an 119903-circulant matrix with eigenvalues 1205820 1205821 120582119899minus1 then for 119894 =0 1 2 119899 minus 1 the following hold

(1) If 119891(119896) + 119892(119896) = 1 then

120582119894 = (119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1

+1199031119899

[119891 (119896) 119886 minus 119887] 120596119894minus 119886)

times(1199031119899

120596119894119891 (119896) + 119903

21198991205962119894119892 (119896) minus 1)

minus1

(22)

(2) If 119891(119896) + 119892(119896) = 1 then

120582119894 = ((119892 (119896) 119903119867119896119899minus1 + 119886) (1 minus 1199031119899

120596119894)

+ [119892 (119896) 119886 + 119887] (1199031119899

119908119894minus 119903))

times ((1 minus 1199031119899

120596119894) [119892 (119896) 119903

1119899120596119894+ 1])

minus1

(23)

Proof According to Lemma 2 we have

120582119894 =

119899minus1

sum

119894=0

119867119896119894120583119895

119894 120583119894 = 119903

1119899119908119894 (24)

Using Abel transformation (Lemma 4) we have

120582119894 = 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus

119899minus2

sum

119895=0

((120583119895+1

119894minus 120583119895

119894)

119895

sum

119904=0

119867119896119904)

= 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus (120583119894 minus 1)

119899minus2

sum

119895=0

(120583119895

119894

119895

sum

119904=0

119867119896119904)

(25)

(1) In the light of (12) and (25) one can obtain that

120582119894 = 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus

120583119894 minus 1

119891 (119896) + 119892 (119896) minus 1

times

119899minus2

sum

119895=0

120583119895

119894[119867119896119895+1 + 119892 (119896)119867119896119895 + 119891 (119896) 119886 minus 119886 minus 119887]

= 120583119899minus1

119894

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

minus

120583119894 minus 1

119891 (119896) + 119892 (119896) minus 1

times (

119899minus2

sum

119895=0

119867119896119895+1120583119895

119894+ 119892 (119896)

119899minus2

sum

119895=0

119867119896119895120583119895

119894

+ [119891 (119896) 119886 minus 119886 minus 119887]

119899minus2

sum

119895=0

120583119895

119894)

(26)

It is clear that

119899minus2

sum

119895=0

119867119896119895+1120583119895

119894=

120582119894 minus 119886

120583119894

119899minus2

sum

119895=0

119867119896119895120583119895

119894= 120582119894 minus 120583

119899minus1

119894119867119896119899minus1

(27)

4 International Journal of Computational Mathematics

Substituting (27) into (26) we obtain that

120582119894 = 120583119899minus1

119894

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

minus

120583119894 minus 1

119891 (119896) + 119892 (119896) minus 1

times (

120582119894 minus 119886

120583119894

+ 119892 (119896) (120582119894 minus 120583119899minus1

119894119867119896119899minus1)

+ [119891 (119896) 119886 minus 119886 minus 119887]

119899minus2

sum

119895=0

120583119895

119894)

=

(1 minus 120583119894) [1 + 119892 (119896) 120583119894]

120583119894 [119891 (119896) + 119892 (119896) minus 1]

120582119894

+

120583119899minus1

119894119867119896119899 + 120583

119899

119894119892 (119896)119867119896119899minus1

119891 (119896) + 119892 (119896) minus 1

+

120583119899

119894(119891 (119896) 119886 minus 119886 minus 119887) minus 119886 (1 minus 120583119894)

120583119894 [119891 (119896) + 119892 (119896) minus 1]

+

(1 minus 120583119899minus1

119894) (119891 (119896) 119886 minus 119886 minus 119887)

119891 (119896) + 119892 (119896) minus 1

(28)

Therefore we have

[119892 (119896) 1205832

119894+ 119891 (119896) 120583119894 minus 1] 120582119894

= 120583119899

119894119867119896119899 + 119892 (119896) 120583

119899+1

119894119867119896119899minus1 + 120583

119899

119894(119891 (119896) 119886 minus 119886 minus 119887)

minus 119886 (1 minus 120583119894) + (120583119894 minus 119903) (119891 (119896) 119886 minus 119886 minus 119887)

= 119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1

+ 1199031119899

[119891 (119896) 119886 minus 119887] 120596119894minus 119886

(29)

We immediately obtain formula (22) from (29)

(2) Taking into account (13) and (25) we have

120582119894 = 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus

120583119894 minus 1

119892 (119896) + 1

times

119899minus2

sum

119895=0

120583119895

119894[119892 (119896)119867119896119895 + 119895 sdot (119892 (119896) 119886 + 119887) + 119886]

= 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 +

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

times

119899minus2

sum

119895=0

119867119896119895120583119895

119894+

1 minus 120583119894

119892 (119896) + 1

119899minus2

sum

119895=0

[119895 sdot (119892 (119896) 119886 + 119887) + 119886] 120583119895

119894

=

120583119899minus1

119894[119892 (119896)119867119896119899minus1 + (119899 minus 1) (119892 (119896) 119886 + 119887) + 119886]

119892 (119896) + 1

+

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

(120582119894 minus 119867119896119899minus1120583119899minus1

119894)

+

1 minus 120583119894

119892 (119896) + 1

[

[

((119899 minus 2) (119892 (119896) 119886 + 119887) + 119886)

times

119899minus2

sum

119895=0

120583119895

119894minus (119892 (119896) 119886 + 119887)

119899minus3

sum

119895=0

119895

sum

119904=0

120583119904

119894]

]

(30)

It follows that

120582119894 =

119892 (119896) 120583119899

119894

119892 (119896) + 1

119867119896119899minus1 +

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

120582119894

+

120583119899minus1

119894[(119899 minus 1) (119892 (119896) 119886 + 119887) + 119886]

119892 (119896) + 1

+

1 minus 120583119894

119892 (119896) + 1

[

[

((119899 minus 2) (119892 (119896) 119886 + 119887) + 119886)

times

119899minus2

sum

119895=0

120583119895

119894minus

(119892 (119896) 119886 + 119887)

1 minus 120583119894

119899minus3

sum

119895=0

(1 minus 120583119895+1

119894)]

]

=

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

120582119894

+

119892 (119896) 120583119899

119894119867119896119899minus1 + 120583

119899minus1

119894(119892 (119896) 119886 + 119887) + 119886

119892 (119896) + 1

+

(119892 (119896) 119886 + 119887) (120583119894 minus 120583119899

119894)

(119892 (119896) + 1) (1 minus 120583119894)

(31)

Therefore we obtain (23) This concludes the proof

4 Spectral Norms of Normal119903-Circulant Matrices

In this section we consider the spectral norms of normal 119903-circulant matrix whose entries are generalized 119896-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 119860 = 119862119903(1198860 1198861 119886119899minus1) be an 119903-circulantmatrix If |119903| = 1 then 119860 is normal matrix

Proof It is well known that

119860 =

119899minus1

sum

119894=0

119886119894119875119894 119875 = (

0 119868119899minus1

119903 0) (32)

International Journal of Computational Mathematics 5

If |119903| = 1 then

119875119875119867= (

0 119868119899minus1

119903 0)(

0 119903

119868119899minus1 0) = 119868119899 (33)

That is 119875119867 = 119875minus1 According to (32) we obtain that

119860119860119867= (

119899minus1

sum

119894=0

119886119894119875119894)(

119899minus1

sum

119895=0

119886119895(119875119867)

119895

)

= (

119899minus1

sum

119894=0

119886119894119875119894)(

119899minus1

sum

119895=0

119886119895(119875minus1)

119895

) =

119899minus1

sum

119894=0

119899minus1

sum

119895=0

119886119894119886119895119875119894minus119895

119860119867119860 = (

119899minus1

sum

119895=0

119886119895(119875119867)

119895

)(

119899minus1

sum

119894=0

119886119894119875119894)

= (

119899minus1

sum

119895=0

119886119895119875minus119895)(

119899minus1

sum

119894=0

119886119895119875119894) =

119899minus1

sum

119894=0

119899minus1

sum

119895=0

119886119894119886119895119875119894minus119895

(34)

Therefore 119860119860119867 = 119860119867119860 which shows that 119860 is normal

According to Theorem 6 and Lemma 7 we have thefollowing theorem

Theorem 8 Suppose that119860 = 119862119903(1198671198960 1198671198961 119867119896119899minus1) is an119903-circulant matrix If |119903| = 1 and119867119896119894 ⩾ 0 119894 = 0 1 2 119899 minus 1then the spectral norm of 119860 is

1198602 =

max0⩽119894⩽119899minus1

1003816100381610038161003816100381610038161003816100381610038161003816

119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1 + 119903

1119899[119891 (119896) 119886 minus 119887] 120596

119894minus 119886

1199031119899

120596119894119891 (119896) + 119903

21198991205962119894119892 (119896) minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

119891 (119896) + 119892 (119896) = 1

max0⩽119894⩽119899minus1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119892 (119896) 119903119867119896119899minus1 + 119886) (1 minus 1199031119899

120596119894) + [119892 (119896) 119886 + 119887] (119903

1119899119908119894minus 119903)

(1 minus 1199031119899

120596119894) [119892 (119896) 119903

1119899120596119894+ 1]

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (119896) + 119892 (119896) = 1

(35)

The following theorem simplifies and generalizes theresults of Theorem 22 in [12]

Theorem 9 Let 119860 = Circ(1198671198960 1198671198961 119867119896119899minus1) be a circu-lant matrix then

1198602 =

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

119891 (119896) + 119892 (119896) = 1

119892 (119896)119867119896119899minus1 + (119899 minus 1) [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

119891 (119896) + 119892 (119896) = 1

(36)

Proof Suppose that 119903 = 1 it follows from Lemma 7 that 119860 isnormal Notice that

1003816100381610038161003816120582119894

1003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=0

119867119896119894120583119895

119894

1003816100381610038161003816100381610038161003816100381610038161003816

⩽

119899minus1

sum

119894=0

119867119896119894

1003816100381610038161003816120583119894

1003816100381610038161003816

119895= 1205820 (37)

It follows from Lemma 3 that 1198602 = 1205820 According toTheorem 6 if 119891(119896) + 119892(119896) = 1 and 119903 = 1 we obtain that

1198602 = 1205820 =

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

(38)

Similarly if 119891(119896) + 119892(119896) = 1 it follows that

1198602 = 1205820 =

119892 (119896)119867119896119899minus1 + (119899 minus 1) [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

(39)

This completes the proof

Taking into account formulae (4)ndash(6) we have the follow-ing corollary

Corollary 10 Let 1198601 = Circ(1198650 1198651 119865119899minus1) be a circulantmatrix then

10038171003817100381710038171198601

10038171003817100381710038172

= 119865119899+1 minus 1 (40)

Corollary 11 Let 1198602 = Circ(1198710 1198711 119871119899minus1) be a circulantmatrix then

10038171003817100381710038171198602

10038171003817100381710038172

= 119865119899+2 + 119865119899 minus 1 (41)

Corollary 12 Let 1198603 = Circ(1198690 1198691 119869119899minus1) be a circulantmatrix then

10038171003817100381710038171198603

10038171003817100381710038172

=

119869119899+1 minus 1

2

(42)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005

[2] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011

6 International Journal of Computational Mathematics

[3] W Bani-Domi and F Kittaneh ldquoNorm equalities and inequali-ties for operator matricesrdquo Linear Algebra and Its Applicationsvol 429 no 1 pp 57ndash67 2008

[4] S Shen and J Cen ldquoOn the bounds for the norms of 119903-circulant matrices with the Fibonacci and Lucas numbersrdquoApplied Mathematics and Computation vol 216 no 10 pp2891ndash2897 2010

[5] S Shen and J Cen ldquoOn the spectral norms of r-circulant matri-ces with the k-Fibonacci and k-Lucas numbersrdquo InternationalJournal of Contemporary Mathematical Sciences vol 5 no 9ndash12 pp 569ndash578 2010

[6] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012

[7] Y Yazlik and N Taskara ldquoOn the norms of an r-circulantmatrix with the generalized k-Horadam numbersrdquo Journal ofInequalities and Applications vol 2013 article 394 2013

[8] Y Yazlik and N Taskara ldquoSpectral norm eigenvalues anddeterminant of circulant matrix involving the generalized k-Horadam numbersrdquo Ars Combinatoria vol 104 pp 505ndash5122012

[9] Z Jiang and Z Zhou ldquoNonsingularity of 119903-circulant matricesrdquoApplied Mathematics A Journal of Chinese Universities vol 10no 2 pp 222ndash226 1995

[10] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press 1985

[11] W Rudin Principles of Mathematical Analysis McGraw-Hill3rd edition 1976

[12] E G Kocer T Mansour and N Tuglu ldquoNorms of circulantand semicirculant matrices with Horadamrsquos numbersrdquo ArsCombinatoria vol 85 pp 353ndash359 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Research Article On the Spectrum and Spectral …downloads.hindawi.com/archive/2014/795175.pdf-Circulant Matrices In this section, we consider the spectral norms of normal - circulant

4 International Journal of Computational Mathematics

Substituting (27) into (26) we obtain that

120582119894 = 120583119899minus1

119894

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

minus

120583119894 minus 1

119891 (119896) + 119892 (119896) minus 1

times (

120582119894 minus 119886

120583119894

+ 119892 (119896) (120582119894 minus 120583119899minus1

119894119867119896119899minus1)

+ [119891 (119896) 119886 minus 119886 minus 119887]

119899minus2

sum

119895=0

120583119895

119894)

=

(1 minus 120583119894) [1 + 119892 (119896) 120583119894]

120583119894 [119891 (119896) + 119892 (119896) minus 1]

120582119894

+

120583119899minus1

119894119867119896119899 + 120583

119899

119894119892 (119896)119867119896119899minus1

119891 (119896) + 119892 (119896) minus 1

+

120583119899

119894(119891 (119896) 119886 minus 119886 minus 119887) minus 119886 (1 minus 120583119894)

120583119894 [119891 (119896) + 119892 (119896) minus 1]

+

(1 minus 120583119899minus1

119894) (119891 (119896) 119886 minus 119886 minus 119887)

119891 (119896) + 119892 (119896) minus 1

(28)

Therefore we have

[119892 (119896) 1205832

119894+ 119891 (119896) 120583119894 minus 1] 120582119894

= 120583119899

119894119867119896119899 + 119892 (119896) 120583

119899+1

119894119867119896119899minus1 + 120583

119899

119894(119891 (119896) 119886 minus 119886 minus 119887)

minus 119886 (1 minus 120583119894) + (120583119894 minus 119903) (119891 (119896) 119886 minus 119886 minus 119887)

= 119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1

+ 1199031119899

[119891 (119896) 119886 minus 119887] 120596119894minus 119886

(29)

We immediately obtain formula (22) from (29)

(2) Taking into account (13) and (25) we have

120582119894 = 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 minus

120583119894 minus 1

119892 (119896) + 1

times

119899minus2

sum

119895=0

120583119895

119894[119892 (119896)119867119896119895 + 119895 sdot (119892 (119896) 119886 + 119887) + 119886]

= 120583119899minus1

119894

119899minus1

sum

119895=0

119867119896119895 +

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

times

119899minus2

sum

119895=0

119867119896119895120583119895

119894+

1 minus 120583119894

119892 (119896) + 1

119899minus2

sum

119895=0

[119895 sdot (119892 (119896) 119886 + 119887) + 119886] 120583119895

119894

=

120583119899minus1

119894[119892 (119896)119867119896119899minus1 + (119899 minus 1) (119892 (119896) 119886 + 119887) + 119886]

119892 (119896) + 1

+

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

(120582119894 minus 119867119896119899minus1120583119899minus1

119894)

+

1 minus 120583119894

119892 (119896) + 1

[

[

((119899 minus 2) (119892 (119896) 119886 + 119887) + 119886)

times

119899minus2

sum

119895=0

120583119895

119894minus (119892 (119896) 119886 + 119887)

119899minus3

sum

119895=0

119895

sum

119904=0

120583119904

119894]

]

(30)

It follows that

120582119894 =

119892 (119896) 120583119899

119894

119892 (119896) + 1

119867119896119899minus1 +

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

120582119894

+

120583119899minus1

119894[(119899 minus 1) (119892 (119896) 119886 + 119887) + 119886]

119892 (119896) + 1

+

1 minus 120583119894

119892 (119896) + 1

[

[

((119899 minus 2) (119892 (119896) 119886 + 119887) + 119886)

times

119899minus2

sum

119895=0

120583119895

119894minus

(119892 (119896) 119886 + 119887)

1 minus 120583119894

119899minus3

sum

119895=0

(1 minus 120583119895+1

119894)]

]

=

119892 (119896) (1 minus 120583119894)

119892 (119896) + 1

120582119894

+

119892 (119896) 120583119899

119894119867119896119899minus1 + 120583

119899minus1

119894(119892 (119896) 119886 + 119887) + 119886

119892 (119896) + 1

+

(119892 (119896) 119886 + 119887) (120583119894 minus 120583119899

119894)

(119892 (119896) + 1) (1 minus 120583119894)

(31)

Therefore we obtain (23) This concludes the proof

4 Spectral Norms of Normal119903-Circulant Matrices

In this section we consider the spectral norms of normal 119903-circulant matrix whose entries are generalized 119896-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 119860 = 119862119903(1198860 1198861 119886119899minus1) be an 119903-circulantmatrix If |119903| = 1 then 119860 is normal matrix

Proof It is well known that

119860 =

119899minus1

sum

119894=0

119886119894119875119894 119875 = (

0 119868119899minus1

119903 0) (32)

International Journal of Computational Mathematics 5

If |119903| = 1 then

119875119875119867= (

0 119868119899minus1

119903 0)(

0 119903

119868119899minus1 0) = 119868119899 (33)

That is 119875119867 = 119875minus1 According to (32) we obtain that

119860119860119867= (

119899minus1

sum

119894=0

119886119894119875119894)(

119899minus1

sum

119895=0

119886119895(119875119867)

119895

)

= (

119899minus1

sum

119894=0

119886119894119875119894)(

119899minus1

sum

119895=0

119886119895(119875minus1)

119895

) =

119899minus1

sum

119894=0

119899minus1

sum

119895=0

119886119894119886119895119875119894minus119895

119860119867119860 = (

119899minus1

sum

119895=0

119886119895(119875119867)

119895

)(

119899minus1

sum

119894=0

119886119894119875119894)

= (

119899minus1

sum

119895=0

119886119895119875minus119895)(

119899minus1

sum

119894=0

119886119895119875119894) =

119899minus1

sum

119894=0

119899minus1

sum

119895=0

119886119894119886119895119875119894minus119895

(34)

Therefore 119860119860119867 = 119860119867119860 which shows that 119860 is normal

According to Theorem 6 and Lemma 7 we have thefollowing theorem

Theorem 8 Suppose that119860 = 119862119903(1198671198960 1198671198961 119867119896119899minus1) is an119903-circulant matrix If |119903| = 1 and119867119896119894 ⩾ 0 119894 = 0 1 2 119899 minus 1then the spectral norm of 119860 is

1198602 =

max0⩽119894⩽119899minus1

1003816100381610038161003816100381610038161003816100381610038161003816

119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1 + 119903

1119899[119891 (119896) 119886 minus 119887] 120596

119894minus 119886

1199031119899

120596119894119891 (119896) + 119903

21198991205962119894119892 (119896) minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

119891 (119896) + 119892 (119896) = 1

max0⩽119894⩽119899minus1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119892 (119896) 119903119867119896119899minus1 + 119886) (1 minus 1199031119899

120596119894) + [119892 (119896) 119886 + 119887] (119903

1119899119908119894minus 119903)

(1 minus 1199031119899

120596119894) [119892 (119896) 119903

1119899120596119894+ 1]

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (119896) + 119892 (119896) = 1

(35)

The following theorem simplifies and generalizes theresults of Theorem 22 in [12]

Theorem 9 Let 119860 = Circ(1198671198960 1198671198961 119867119896119899minus1) be a circu-lant matrix then

1198602 =

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

119891 (119896) + 119892 (119896) = 1

119892 (119896)119867119896119899minus1 + (119899 minus 1) [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

119891 (119896) + 119892 (119896) = 1

(36)

Proof Suppose that 119903 = 1 it follows from Lemma 7 that 119860 isnormal Notice that

1003816100381610038161003816120582119894

1003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=0

119867119896119894120583119895

119894

1003816100381610038161003816100381610038161003816100381610038161003816

⩽

119899minus1

sum

119894=0

119867119896119894

1003816100381610038161003816120583119894

1003816100381610038161003816

119895= 1205820 (37)

It follows from Lemma 3 that 1198602 = 1205820 According toTheorem 6 if 119891(119896) + 119892(119896) = 1 and 119903 = 1 we obtain that

1198602 = 1205820 =

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

(38)

Similarly if 119891(119896) + 119892(119896) = 1 it follows that

1198602 = 1205820 =

119892 (119896)119867119896119899minus1 + (119899 minus 1) [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

(39)

This completes the proof

Taking into account formulae (4)ndash(6) we have the follow-ing corollary

Corollary 10 Let 1198601 = Circ(1198650 1198651 119865119899minus1) be a circulantmatrix then

10038171003817100381710038171198601

10038171003817100381710038172

= 119865119899+1 minus 1 (40)

Corollary 11 Let 1198602 = Circ(1198710 1198711 119871119899minus1) be a circulantmatrix then

10038171003817100381710038171198602

10038171003817100381710038172

= 119865119899+2 + 119865119899 minus 1 (41)

Corollary 12 Let 1198603 = Circ(1198690 1198691 119869119899minus1) be a circulantmatrix then

10038171003817100381710038171198603

10038171003817100381710038172

=

119869119899+1 minus 1

2

(42)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005

[2] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011

6 International Journal of Computational Mathematics

[3] W Bani-Domi and F Kittaneh ldquoNorm equalities and inequali-ties for operator matricesrdquo Linear Algebra and Its Applicationsvol 429 no 1 pp 57ndash67 2008

[4] S Shen and J Cen ldquoOn the bounds for the norms of 119903-circulant matrices with the Fibonacci and Lucas numbersrdquoApplied Mathematics and Computation vol 216 no 10 pp2891ndash2897 2010

[5] S Shen and J Cen ldquoOn the spectral norms of r-circulant matri-ces with the k-Fibonacci and k-Lucas numbersrdquo InternationalJournal of Contemporary Mathematical Sciences vol 5 no 9ndash12 pp 569ndash578 2010

[6] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012

[7] Y Yazlik and N Taskara ldquoOn the norms of an r-circulantmatrix with the generalized k-Horadam numbersrdquo Journal ofInequalities and Applications vol 2013 article 394 2013

[8] Y Yazlik and N Taskara ldquoSpectral norm eigenvalues anddeterminant of circulant matrix involving the generalized k-Horadam numbersrdquo Ars Combinatoria vol 104 pp 505ndash5122012

[9] Z Jiang and Z Zhou ldquoNonsingularity of 119903-circulant matricesrdquoApplied Mathematics A Journal of Chinese Universities vol 10no 2 pp 222ndash226 1995

[10] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press 1985

[11] W Rudin Principles of Mathematical Analysis McGraw-Hill3rd edition 1976

[12] E G Kocer T Mansour and N Tuglu ldquoNorms of circulantand semicirculant matrices with Horadamrsquos numbersrdquo ArsCombinatoria vol 85 pp 353ndash359 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Research Article On the Spectrum and Spectral …downloads.hindawi.com/archive/2014/795175.pdf-Circulant Matrices In this section, we consider the spectral norms of normal - circulant

International Journal of Computational Mathematics 5

If |119903| = 1 then

119875119875119867= (

0 119868119899minus1

119903 0)(

0 119903

119868119899minus1 0) = 119868119899 (33)

That is 119875119867 = 119875minus1 According to (32) we obtain that

119860119860119867= (

119899minus1

sum

119894=0

119886119894119875119894)(

119899minus1

sum

119895=0

119886119895(119875119867)

119895

)

= (

119899minus1

sum

119894=0

119886119894119875119894)(

119899minus1

sum

119895=0

119886119895(119875minus1)

119895

) =

119899minus1

sum

119894=0

119899minus1

sum

119895=0

119886119894119886119895119875119894minus119895

119860119867119860 = (

119899minus1

sum

119895=0

119886119895(119875119867)

119895

)(

119899minus1

sum

119894=0

119886119894119875119894)

= (

119899minus1

sum

119895=0

119886119895119875minus119895)(

119899minus1

sum

119894=0

119886119895119875119894) =

119899minus1

sum

119894=0

119899minus1

sum

119895=0

119886119894119886119895119875119894minus119895

(34)

Therefore 119860119860119867 = 119860119867119860 which shows that 119860 is normal

According to Theorem 6 and Lemma 7 we have thefollowing theorem

Theorem 8 Suppose that119860 = 119862119903(1198671198960 1198671198961 119867119896119899minus1) is an119903-circulant matrix If |119903| = 1 and119867119896119894 ⩾ 0 119894 = 0 1 2 119899 minus 1then the spectral norm of 119860 is

1198602 =

max0⩽119894⩽119899minus1

1003816100381610038161003816100381610038161003816100381610038161003816

119903119867119896119899 + 119892 (119896) 1199031+(1119899)

120596119894119867119896119899minus1 + 119903

1119899[119891 (119896) 119886 minus 119887] 120596

119894minus 119886

1199031119899

120596119894119891 (119896) + 119903

21198991205962119894119892 (119896) minus 1

1003816100381610038161003816100381610038161003816100381610038161003816

119891 (119896) + 119892 (119896) = 1

max0⩽119894⩽119899minus1

10038161003816100381610038161003816100381610038161003816100381610038161003816

(119892 (119896) 119903119867119896119899minus1 + 119886) (1 minus 1199031119899

120596119894) + [119892 (119896) 119886 + 119887] (119903

1119899119908119894minus 119903)

(1 minus 1199031119899

120596119894) [119892 (119896) 119903

1119899120596119894+ 1]

10038161003816100381610038161003816100381610038161003816100381610038161003816

119891 (119896) + 119892 (119896) = 1

(35)

The following theorem simplifies and generalizes theresults of Theorem 22 in [12]

Theorem 9 Let 119860 = Circ(1198671198960 1198671198961 119867119896119899minus1) be a circu-lant matrix then

1198602 =

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

119891 (119896) + 119892 (119896) = 1

119892 (119896)119867119896119899minus1 + (119899 minus 1) [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

119891 (119896) + 119892 (119896) = 1

(36)

Proof Suppose that 119903 = 1 it follows from Lemma 7 that 119860 isnormal Notice that

1003816100381610038161003816120582119894

1003816100381610038161003816=

1003816100381610038161003816100381610038161003816100381610038161003816

119899minus1

sum

119894=0

119867119896119894120583119895

119894

1003816100381610038161003816100381610038161003816100381610038161003816

⩽

119899minus1

sum

119894=0

119867119896119894

1003816100381610038161003816120583119894

1003816100381610038161003816

119895= 1205820 (37)

It follows from Lemma 3 that 1198602 = 1205820 According toTheorem 6 if 119891(119896) + 119892(119896) = 1 and 119903 = 1 we obtain that

1198602 = 1205820 =

119867119896119899 + 119892 (119896)119867119896119899minus1 + 119891 (119896) 119886 minus 119886 minus 119887

119891 (119896) + 119892 (119896) minus 1

(38)

Similarly if 119891(119896) + 119892(119896) = 1 it follows that

1198602 = 1205820 =

119892 (119896)119867119896119899minus1 + (119899 minus 1) [119892 (119896) 119886 + 119887] + 119886

119892 (119896) + 1

(39)

This completes the proof

Taking into account formulae (4)ndash(6) we have the follow-ing corollary

Corollary 10 Let 1198601 = Circ(1198650 1198651 119865119899minus1) be a circulantmatrix then

10038171003817100381710038171198601

10038171003817100381710038172

= 119865119899+1 minus 1 (40)

Corollary 11 Let 1198602 = Circ(1198710 1198711 119871119899minus1) be a circulantmatrix then

10038171003817100381710038171198602

10038171003817100381710038172

= 119865119899+2 + 119865119899 minus 1 (41)

Corollary 12 Let 1198603 = Circ(1198690 1198691 119869119899minus1) be a circulantmatrix then

10038171003817100381710038171198603

10038171003817100381710038172

=

119869119899+1 minus 1

2

(42)

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] S Solak ldquoOn the norms of circulantmatrices with the Fibonacciand Lucas numbersrdquo Applied Mathematics and Computationvol 160 no 1 pp 125ndash132 2005

[2] A Ipek ldquoOn the spectral norms of circulant matrices withclassical Fibonacci and Lucas numbers entriesrdquo Applied Mathe-matics and Computation vol 217 no 12 pp 6011ndash6012 2011

6 International Journal of Computational Mathematics

[3] W Bani-Domi and F Kittaneh ldquoNorm equalities and inequali-ties for operator matricesrdquo Linear Algebra and Its Applicationsvol 429 no 1 pp 57ndash67 2008

[4] S Shen and J Cen ldquoOn the bounds for the norms of 119903-circulant matrices with the Fibonacci and Lucas numbersrdquoApplied Mathematics and Computation vol 216 no 10 pp2891ndash2897 2010

[5] S Shen and J Cen ldquoOn the spectral norms of r-circulant matri-ces with the k-Fibonacci and k-Lucas numbersrdquo InternationalJournal of Contemporary Mathematical Sciences vol 5 no 9ndash12 pp 569ndash578 2010

[6] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012

[7] Y Yazlik and N Taskara ldquoOn the norms of an r-circulantmatrix with the generalized k-Horadam numbersrdquo Journal ofInequalities and Applications vol 2013 article 394 2013

[8] Y Yazlik and N Taskara ldquoSpectral norm eigenvalues anddeterminant of circulant matrix involving the generalized k-Horadam numbersrdquo Ars Combinatoria vol 104 pp 505ndash5122012

[9] Z Jiang and Z Zhou ldquoNonsingularity of 119903-circulant matricesrdquoApplied Mathematics A Journal of Chinese Universities vol 10no 2 pp 222ndash226 1995

[10] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press 1985

[11] W Rudin Principles of Mathematical Analysis McGraw-Hill3rd edition 1976

[12] E G Kocer T Mansour and N Tuglu ldquoNorms of circulantand semicirculant matrices with Horadamrsquos numbersrdquo ArsCombinatoria vol 85 pp 353ndash359 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Research Article On the Spectrum and Spectral …downloads.hindawi.com/archive/2014/795175.pdf-Circulant Matrices In this section, we consider the spectral norms of normal - circulant

6 International Journal of Computational Mathematics

[3] W Bani-Domi and F Kittaneh ldquoNorm equalities and inequali-ties for operator matricesrdquo Linear Algebra and Its Applicationsvol 429 no 1 pp 57ndash67 2008

[4] S Shen and J Cen ldquoOn the bounds for the norms of 119903-circulant matrices with the Fibonacci and Lucas numbersrdquoApplied Mathematics and Computation vol 216 no 10 pp2891ndash2897 2010

[5] S Shen and J Cen ldquoOn the spectral norms of r-circulant matri-ces with the k-Fibonacci and k-Lucas numbersrdquo InternationalJournal of Contemporary Mathematical Sciences vol 5 no 9ndash12 pp 569ndash578 2010

[6] Y Yazlik and N Taskara ldquoA note on generalized 119896-Horadamsequencerdquo Computers ampMathematics with Applications vol 63no 1 pp 36ndash41 2012

[7] Y Yazlik and N Taskara ldquoOn the norms of an r-circulantmatrix with the generalized k-Horadam numbersrdquo Journal ofInequalities and Applications vol 2013 article 394 2013

[8] Y Yazlik and N Taskara ldquoSpectral norm eigenvalues anddeterminant of circulant matrix involving the generalized k-Horadam numbersrdquo Ars Combinatoria vol 104 pp 505ndash5122012

[9] Z Jiang and Z Zhou ldquoNonsingularity of 119903-circulant matricesrdquoApplied Mathematics A Journal of Chinese Universities vol 10no 2 pp 222ndash226 1995

[10] R A Horn and C R Johnson Matrix Analysis CambridgeUniversity Press 1985

[11] W Rudin Principles of Mathematical Analysis McGraw-Hill3rd edition 1976

[12] E G Kocer T Mansour and N Tuglu ldquoNorms of circulantand semicirculant matrices with Horadamrsquos numbersrdquo ArsCombinatoria vol 85 pp 353ndash359 2007

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Research Article On the Spectrum and Spectral …downloads.hindawi.com/archive/2014/795175.pdf-Circulant Matrices In this section, we consider the spectral norms of normal - circulant

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


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