Research Article Norms and Spread of the Fibonacci and Lucas RSFMLR Circulant Matrices

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1 Abstract ad Applied Aalysis Volume 2015, Article ID , 8 pages Research Article Norms ad Spread of the Fiboacci ad Lucas RSFMLR Circulat Matrices Weai Xu ad Zhaoli Jiag Departmet of Mathematics, Liyi Uiversity, Liyi, Shadog , Chia Correspodece should be addressed to Zhaoli Jiag; jzh1208@siacom Received 25 July 2014; Accepted 16 September 2014 Academic Editor: Zidog Wag Copyright 2015 W Xu ad Z Jiag This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited Circulat type matrices have played a importat role i etworks egieerig I this paper, firstly, some bouds for the orms ad spread of Fiboacci row skew first-mius-last right RSFMLR) circulat matrices ad Lucas row skew first-mius-last right RSFMLR) circulat matrices are give Furthermore, the spectral orm of Hadamard product of a Fiboacci RSFMLR circulat matrix ad a Lucas RSFMLR circulat matrix is obtaied Fially, the Frobeius orm of Kroecker product of a Fiboacci RSFMLR circulat matrix ad a Lucas RSFMLR circulat matrix is preseted 1 Itroductio Circulat type matrices have bee put o the firm basis with the work i[1 4] ad so o Circulat type matrices have sigificat applicatios i etworks systems I [5], some prelimiary results o the dyamical behaviours of some specific omootoe Boolea automata etworks which are called xor circulatetworkswere showed I [6], the authors proposed a special class of the feedback delay etwork usig circulat matrices I [7], the impact of iterior symmetries othemultiplicityoftheeigevaluesofthejacobiamatrix at a fully sychroous equilibrium for the coupled cell systems associated with homogeeous etworks was aalyzed by Aguiar ad Rua, which was based o the circulat adjacecy matrices of the etworks iduced by these iterior symmetries Exploitig the circulat structure of the chael matrices, the realistic ear fast fadig scearios with circulat frequecy selective chaels were aalysed by Eghbali et al i [8] The existece of doubly periodic travellig waves i cellular etworks ivolvig the discotiuous Heaviside step fuctio by circulat matrix was studied by Wag ad Cheg i [9] The Fiboacci ad Lucas sequeces F ad L are defied by the recurrece relatios [10, 11]: F 0 =0, F 1 =1, F =F +F 2 for 2, 1) L 0 =2, L 1 =1, L =L +L 2 for 2 2) If we start from =0, the Fiboacci ad Lucas sequeces are give by F ) L I [10], their Biet forms are give by F = 1 5 [ L = 1+ 5 ) 2 ) cos π) 1+ 5 ) ], 2 + cos π) 1+ 5 ) 2 The followig sum formulatios for the Fiboacci ad Lucas umbers are well kow [11]: 4) F 2 s =F F, 5) L 2 s =L L 2, 6) F s F s 1 ={ F2, F 2 1, L L ={ L2 4, s s 1 L 2 +1, eve, odd, eve, odd 7)

2 2 Abstract ad Applied Aalysis Lately, some authors studied the problems of the orms of some special matrices [11 21] The author [11] foud upper ad lower bouds for the spectral orms of Toeplitz matrices such that a ij F i j ad b i j L i j I[13], the authors obtai upper ad lower bouds for the spectral orms of matrices A = C r F k,0,f k,1,,f k, ) ad B = C r L k,0,l k,1,,l k, ),{F k, } N ad {L k, } N are k- Fiboacci ad k-lucas sequeces, respectively, ad they also give the bouds for the spectral orms of Kroecker ad Hadamard products of these special matrices, respectively [14] Solak ad Bozkurt [16] have foud out upper ad lower bouds for the spectral orms of Cauchy-Toeplitz ad Cauchy-Hakel matrices Solak [18 20] has defied A=[a ij ] ad B = [b ij ] as circulat matrices, a ij F modj i,)) ad b ij L modj i,)) ;thehehasgivesome bouds for the A ad B matrices cocered with the spectral ad Euclidea orms I this paper, we defie two kids of special matrices as follows A Fiboacci row skew first-mius-last right RSFMLR) circulat matrix is defied as a square matrix of the form F 0 F 1 F F F 0 F F 1 F 2 F F 2 d ) 8) F 2 d d F 1 F 1 F 2 F 1 F 0 F A Lucas row skew first-mius-last right RSFMLR) circulat matrix is defied as a square matrix of the form L 0 L 1 L L L 0 L L 1 L 2 L L 2 d ) 9) L 2 d d L 1 L 1 L 2 L 1 L 0 L Obviously, the RSFMLR circulat matrix is determied by its first row, ad RSFMLR circulat matrix is a x +x+1circulat matrix [22] We defie Θ 1, 1) as the basic RSFMLR circulat matrix; that is, d d d Θ 1, 1) = ) d d ) = RSFMLRcircfr 0, 1, 0,, 0) 10) It is easily verified that gx) = x +x+1has o repeated roots i its splittig field ad gx) = x +x+1isboth the miimal polyomial ad the characteristic polyomial of the matrix Θ 1, 1) I additio, Θ 1, 1) is oderogatory ad satisfies = RFMLRcircfr0,,0,1,0,,0 ) ad Θ 1, 1) = Θ j 1, 1) I +Θ 1, 1) j j 1 As we all kow, lettig A = RSFMLRcircfra 0,a 1,, a ) be a RSFMLR circulat matrix with the first row a 0, a 1,,a ),itisclearthat A=RSFMLRcircfr a 0,a 1,,a ) = a i Θ i 1, 1) 11) Thus, A is a RSFMLR circulat matrix if ad oly if A=fΘ 1, 1) ) for some polyomial fx) Thepolyomial fx) = a ix i will be called the represeter of the RSFMLR circulat matrix A By11), itisclearthata is a RSFMLR circulatmatrixifadolyifa commutes with Θ 1, 1) ;that is, AΘ 1, 1) =Θ 1, 1) A I additio to the algebraic properties that ca be easily derived from the represetatio 11), wemetiothat RSFMLR circulat matrices have very ice structure The product of two RSFMLR circulat matrices is a RSFMLR circulat matrix ad A 1 is a RSFMLR circulat matrix too Let A = a ij ) be a matrix The Euclidea or Frobeius) orm, the spectral orm, the maximum colum sum matrix orm, ad the maximum row sum matrix orm of the matrix A are, respectively [11], A F = i,j=1 1/2 a 2 ij ), 12) A 2 =max 1 i λ ia A)) 1/2, 13) A 1 = max a ij, 14) 1 j A = max a ij, 15) 1 i j=1 A deotes the cojugate traspose of AThefollowig iequality holds: 1 A F A 2 A F 16) Let A=[a ij ] ad B=[b ij ] be matrices The Hadamard product of A ad B is defied by A B=[a ij b ij ]If is ay orm o mmatrices, the [18, 23] A B A B 17) Kroecker product of A ad B isgivetobe[18] The [18] A B= [ a 11 B a 1m B ] 18) [ a 1 B a m B] A B F = A F B F 19)

3 Abstract ad Applied Aalysis 3 Let A = a ij ) be a matrix with eigevalues λ i, i = 1,2,,ThespreadofA is defied as [24, 25] s A) = max i,j λ i λ j 20) AupperboudforthespreadduetoMirsky[24]statesthat s A) 2 A 2 F 2 tr A 2, 21) A F deotes the Frobeius orm of A ad tr A is the trace of A 2 Norms ad Spread of Fiboacci RSFMLR Circulat Matrices Theorem 1 Let A = RSFMLRcircfrF 0,F 1,,F ) be a Fiboacci RSFMLR circulat matrix, {F i } 0 i deote Fiboacci umbers give by 1); thetwokidsoformsofa are give by A 1 = A =2F +1 1) 22) Proof The matrix A is of the form 8), by14), 15); thewe have A 1 = max a ij = 1 j F i +F +1 F 2, A = max a ij = 1 i j=1 F i +F +1 F 2 23) Sice the Fiboacci sequeces F are defied by the recurrece relatios 1),the we obtai To sum up, we ca get The F =F F ) which completes the proof F s =F +1 F 2 25) A 1 = A =2F +1 1), 26) Theorem 2 Let A = RSFMLRcircfrF 0,F 1,,F ) be a Fiboacci RSFMLR circulat matrix, {F i } 0 i deote Fiboacci umbers give by 1);the Γ A 2, A 2 2F +1 1), 27) Γ=F ) F2 + ) F 3F ) F F ) F ) Proof Sice F +2 =F +1 +F ad F0) = 0 give by 1), the matrix A is of the form F 0 F 1 F 2 F F F F 2 F F 2 d ) 29) F 2 d F1 F 1 F 3 F F 2 F ) We kow that 1/ ) A F A 2 A F from equivalet orms By 5),wecaget The We have A 2 F = = = F 2 i + F 2 i + F 2 i k=1 if 2 i +2 k=1 i= k k=1 2 F i F i+1 2 if i F i+1 F 2 2 i +2 F 2 i 2 k=1 i= k 1 k 1 k 2 F 2 i ) F i F i+1 ) F i F i+1 =F ) F2 + ) F 3F ) F F ) F ) 1 A F = Γ, 31) Γ=F ) F2 + ) F 3F ) F F ) F ) Γ A 2 33)

4 4 Abstract ad Applied Aalysis O the other had, suppose that The We ca get d 0 0 M 1 = d 0 0 d d ), ) M 2 = d d d ), d ) M 3 = d ) ) A= 34) F i M i 2 1 F i 1 M i 2 +F M 3 35) A 2 = F i M i i 1 M F i 2 +F M 3 2 Furthermore, F i M 1 i 2 2 F i 1 M 2 i 2 +F M M H 1 M = d ), ) The other result is obtaied as follows: A 2 =2 F i M 1 i F i 1 M 2 i 2 +F M 3 2 F i =2F +1 1), which completes the proof 39) Theorem 3 Let A = RSFMLRcircfrF 0,F 1,,F ) be a Fiboacci RSFMLR circulat matrix, {F i } 0 i deote Fiboacci umbers give by 1); the the boud for the spread of A is s A) τ 1 ) 2 τ 2 ), 40) τ 1 ) =2F ) F2 + ) F 3F 2 τ 2 ) =[) F ] ) F F ) F 2 2 ), 41) Proof The trace of A is tr A=F 0 +)F ByTheorem 2 ad iequatio 21),we have s A) 2 A 2 F 2 tr A2, 42) A 2 F =F ) F2 + ) F 3F 2 We ca get + 2 4) F F ) F 2 2, tr A=F 0 ) F 43) s A) τ 1 ) 2 τ 2 ), 44) τ 1 ) =2F ) F2 + ) F 3F 2 We obtai M H 2 M 2 = d ), M H 3 M = d ) ) M 1 2 = M 2 2 = M 3 2 =1 38) τ 2 ) =[) F ] 2, which completes the proof + 2 4) F F ) F 2 2 ), 3 Norms ad Spread of Lucas RSFMLR Circulat Matrices 45) Theorem 4 Let B = RSFMLRcircfrL 0,L 1,,L ) be a Lucas RSFMLR circulat matrix, {L i } 0 i deote Lucas umbers give by 2); thetwokidsoformsofb are give by B 1 = B =2L +1 3)+2 46)

5 Abstract ad Applied Aalysis 5 Proof The matrix B is of the form 9), by14), 15); thewe get B 1 = max bij = L i +L 0 +L +1 L 2, 1 j B = max bij = L i +L 0 +L +1 L 2 1 i j=1 47) Sice the Lucas sequeces L are defied by the recurrece relatios 2),the we obtai To sum up, we ca get L =L L ) = 2 +2 L 2 i + k=1 k=1 2 4) L L i L i+1 L 2 k 1 i L 2 i ) k 2 L i L i+1 ) =L ) L2 + ) L 3L ) L ) L + 2 4) L L ) The which completes the proof L s =L +1 L 2 49) B 1 = B =2L +1 3) +2, 50) Theorem 5 Let B = RSFMLRcircfrL 0,L 1,,L ) be a Lucas RSFMLR circulat matrix, {L i } 0 i deote Lucas umbers give by 2);the The We have 1 B F = Π 55) Π B 2, 56) Π=L ) L2 + ) L 3L 2 Π B 2, B 2 2L +1 2), Π=L ) L2 + ) L 3L ) L ) L + 2 4) L L ) 52) Proof Sice L +2 =L +1 +L ad L0) = 2, thematrixb is of the form L 0 L 1 L 2 L L L 0 L L 2 L L 2 d ) L 2 d L1 L 1 L 3 L L 2 L 0 L ) 53) We kow that 1/ ) B F B 2 B F from equivalet orms By 6),wecaget B 2 F = L 2 i + il 2 2 i +2 il i L i+1 4) L = L 2 i + k=1 i= k 4) L L 2 2 i +2 2 k=1 i= k 1 L i L i ) L ) L + 2 4) L L O the other had, supposig that the d 0 0 M 1 = d 0 0 d d ), ) M 2 = d d d ), d ) M 3 = d ), ) B= 57) 58) L i M i 2 1 L i 1 M i 2 +L M 3 59)

6 6 Abstract ad Applied Aalysis We obtai B 2 = L i M i i 1 M L i 2 +L M 3 2 We have L i M 1 i 2 2 L i 1 M 2 i 2 +L M M H 1 M = d ), ) B 2 F =L ) L2 + ) L 3L ) L ) L 67) + 2 4) L L , tr B=L 0 ) L We obtai s B) κ 1 ) 2 κ 2 ), 68) We get M H 2 M 2 = d ), M H 3 M = d ) ) M 1 2 = M 2 2 = M 3 2 =1 62) The other result is obtaied as follows: B 2 =2 L i M 1 i L i 1 M 2 i 2 +L M 3 2 L i =2L +1 2), which completes the proof 63) Theorem 6 Let B = RSFMLRcircfrL 0,L 1,,L ) be a Lucas RSFMLR circulat matrix, {L i } 0 i deote Lucas umbers give by 2);the s B) κ 1 ) 2 κ 2 ), 64) κ 1 ) =2L ) L2 + ) L 3L ) L ) L + 2 4) L L ), 65) κ 1 ) =2L ) L2 + ) L 3L ) L ) L + 2 4) L L ), κ 2 ) =[L 0 ) L ] 2, which completes the proof 69) Corollary 7 Let A = RSFMLRcircfrF 0,F 1,,F ) be a Fiboacci RSFMLR circulat matrix ad let B = RSFMLRcircfrL 0,L 1,,L ) be a Lucas RSFMLR circulat matrix, {F i } 0 i ad {L i } 0 i deote Fiboacci umbers ad Lucas umbers, respectively; the the spectral orm of Hadamard product of A ad B satisfies the followig iequality: A B 2 4F +1 1) L +1 2) 70) Proof The proof is trivial by Theorems 2 ad 5;weobtai A 2 2F +1 1), B 2 2L +1 2) 71) By iequatio 17),wehave A B 2 4F +1 1) L +1 2), 72) which completes the proof Corollary 8 Let A = RSFMLRcircfrF 0,F 1,,F ) be a Fiboacci RSFMLR circulat matrix ad let B = RSFMLRcircfrL 0,L 1,,L ) be a Lucas RSFMLR circulat matrix, {F i } 0 i ad {L i } 0 i deote Fiboacci umbers ad Lucas umbers, respectively; the the Frobeius orm of Kroecker product of A ad B is A B F = Γ Π, 73) Γ=F ) F2 + ) F 3F 2 κ 2 ) =[L 0 ) L ] 2 Proof The trace of B is tr B=L 0 +)L ByTheorem 5 ad by iequatio 21),wehave s B) 2 B 2 F 2 tr B2, 66) + 2 4) F F ) F 2 2, Π=L ) L2 + ) L 3L ) L ) L + 2 4) L L )

7 Abstract ad Applied Aalysis 7 Proof Sice the proof is trivial by Theorems 2 ad 5, we obtai A 2 F =F ) F2 + ) F 3F ) F F ) F 2 2, B 2 F =L ) L2 + ) L 3L 2 By 19),the + 2 3) L ) L + 2 4) L L ) A B F = Γ Π, 76) Γ=F ) F2 + ) F 3F ) F F ) F 2 2, Π=L ) L2 + ) L 3L ) L ) L + 2 4) L L , which completes the proof 4 Coclusio 77) I this study, we defie matrices of the followig forms: let A=RSFMLRcircfrF 0,F 1,,F ) be a Fiboacci RSFMLR circulat matrix ad let B=RSFMLRcircfrL 0,L 1,,L ) be a Lucas RSFMLR circulat matrix Firstly, we get lower adupperboudsforthespectralormsofthesematrices UpperboudsforthespreadofthematrixA ad the matrix B are give Afterwards, we obtai some corollaries related to orms of Hadamard ad Kroecker products of these matrices Based o the existig problems i [26 28], we will explore solvig these problems by circulat matrices techology Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper Ackowledgmets The research is supported by the Developmet Project of Sciece & Techology of Shadog Provice Grat o 2012GGX10115) ad the AMEP of Liyi Uiversity, Chia Refereces [1] P J Davis, Circulat Matrices, Joh Wiley & Sos, New York, NY, USA, 1979 [2] Z L Jiag ad Z X Zhou, Circulat Matrices, Chegdu Techology Uiversity Publishig Compay, Chegdu, Chia, 1999 [3] Z Jiag, O the miimal polyomials ad the iverses of multilevel scaled factor circulat matrices, Abstract ad Applied Aalysis,vol2014,ArticleID521643,10pages,2014 [4] X Jiag ad K Hog, Exact determiats of some special circulat matrices ivolvig four kids of famous umbers, Abstract ad Applied Aalysis, vol 2014, Article ID , 12 pages, 2014 [5] M Noual, D Regault, ad S Seé, About o-mootoy i Boolea automata etworks, Theoretical Computer Sciece,vol 504,pp12 25,2013 [6] DRocchessoadJOSmith, Circulatadellipticfeedback delay etworks for artificial reverberatio, IEEE Trasactios o Speech ad Audio Processig,vol5,o1,pp51 63,1997 [7] MAAguiaradHRua, Iteriorsymmetriesadmultiple eigevalues for homogeeous etworks, SIAM Joural o Applied Dyamical Systems,vol11,o4,pp ,2012 [8]HEghbali,SMuhaidat,SAHejazi,adYDig, Relay selectio strategies for sigle-carrier frequecy-domai equalizatio multi-relay cooperative etworks, IEEE Trasactios o Wireless Commuicatios,vol12,o5,pp ,2013 [9] G-Q Wag ad S S Cheg, 6-periodic travellig waves i a artificial eural etwork with bag-bag cotrol, Differece Equatios ad Applicatios,vol18,o2,pp , 2012 [10] E G Kocer, N Tuglu, ad A Stakhov, O the m-extesio of the Fiboacci ad Lucas p-umbers, Chaos, Solitos ad Fractals,vol40,o4,pp ,2009 [11] M Akbulak ad D Bozkurt, O the orms of Toeplitz matrices ivolvig Fiboacci ad Lucas umbers, Hacettepe Mathematics ad Statistics,vol37,o2,pp89 95,2008 [12] R Mathias, The spectral orm of a oegative matrix, Liear Algebra ad Its Applicatios,vol139,pp ,1990 [13] S Q She ad J M Ce, O the spectral orms of r- circulat matrices with the k-fiboacci ad k-lucas umbers, Iteratioal Cotemporary Mathematical Scieces, vol 5, o 12, pp , 2010 [14] S Q She ad J M Ce, O the orms of circulat matrices with the k, h)-fiboacci ad k, h)-lucas umbers, Iteratioal Cotemporary Mathematical Scieces, vol6, o18,pp ,2011 [15] SSolakadDBozkurt, Someboudsol p matrix ad l p operator orms of almost circulat, CAUchy-Toeplitz ad CAUchy- Hakel matrices, Mathematical & Computatioal Applicatios, vol 7, o 3, pp , 2002 [16] S Solak ad D Bozkurt, O the spectral orms of Cauchy- Toeplitz ad Cauchy-Hakel matrices, Applied Mathematics ad Computatio,vol140,o2-3,pp ,2003 [17] S Solak ad D Bozkurt, A ote o boud for orms of Cauchy- Hakel matrices, Numerical Liear Algebra with Applicatios, vol 10, o 4, pp , 2003 [18] S Solak, O the orms of circulat matrices with the Fiboacci ad Lucas umbers, Applied Mathematics ad Computatio, vol160,o1,pp ,2005 [19] S Solak, Erratum to o the orms of circulat matrices with the Fiboacci ad Lucas umbers, Applied Mathematics ad Computatio,vol160,o1,pp ,2005 [20] S Solak, Erratum to O the orms of circulat matrices with the Fiboacci ad Lucas umbers [Appl Math Comput 160

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