Research Article Skew Circulant Type Matrices Involving the Sum of Fibonacci and Lucas Numbers

Transcription

1 Hidawi Publishig Corporatio Abstract ad Applied Aalysis Volume 05, Article ID 95340, 9 pages Research Article Skew Circulat Type Matrices Ivolvig the Sum of Fiboacci ad Lucas Numbers Zhaoli Jiag ad Yula Wei School of Sciece, Liyi Uiversity, Shuaglig Road, Liyi, Shadog 76000, Chia Correspodece should be addressed to Zhaoli Jiag; jzh08@siacom Received 7 Jue 04; Revised 8 August 04; Accepted August 04 Academic Editor: Yogli Sog Copyright 05 Z Jiag ad Y Wei 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 Skew circulat ad circulat matrices have bee a ideal research area ad hot issue for solvig various differetial equatios I this paper, the skew circulat type matrices with the sum of Fiboacci ad Lucas umbers are discussed The ivertibility of the skew circulat type matrices is cosidered The determiat ad the iverse matrices are preseted Furthermore, the maximum colum sum matrix orm, the spectral orm, the Euclidea (or Frobeius) orm, the maximum row sum matrix orm, ad bouds for the spread of these matrices are give, respectively Itroductio As is well-kow, skew circulat ad circulat matrices play a crucial role for solvig various differetial equatios Authors i [] preseted the skew circulat matrices as precoditioers for liear multistep formulae (LMF-)based ordiary differetial equatios (ODEs) codes Claeysse et al [] discussed factor block circulat ad periodic solutios of udamped matrix differetial equatios Usig circulat matrix,karasöze ad Şimşek [3] cosidered periodic boudary coditios such that o additioal boudary terms will appear after semidiscretizatio Meyer ad Rjasaow [4] have preseted a effective direct solutio method for certai boudary elemet equatios i 3D Guo et al cocered o geeric D-Hopf bifurcatio to a delayed Hopfield-Cohe- Grossberg model of eural etworks (57) i [5], where T deoted a itercoectio matrix I particular, they assumed that T is a symmetric circulat matrix I [6], Ji et al proposed the GMRES method with the Strag-type block-circulat precoditioer for solvig sigular perturbatio delay differetial equatios I [7], two ew ormalform realizatios are preseted which utilize circulat ad skew circulat matrices as their state trasitio matrices The well kow secod-order coupled form is a special case of the skew circulat form Compared with cyclic covolutio algorithm, the skew cyclic covolutio algorithm [8] is able to perform filterig procedure i approximately half of computatioal cost for real sigals I [9], a ew fast algorithm for optimal desig of block digital filters (BDFs) was proposed based o skew circulat matrix Spectral decompositios of skew circulat ad skew left circulat matrices were discussed i [0] Li et al [] gave the style spectral decompositio of skew circulat matrix firstly ad the dealt with the optimal backward perturbatio aalysis for the liear system with skew circulat coefficiet matrix Some scholars have give various algorithms for the determiats ad iverses of osigular circulat matrices [, 3] Ufortuately, the computatioal complexity of these algorithms is very amazig with the order of matrix icreasig However, some authors gave the explicit determiats ad iverse of circulat ad skew circulat ivolvig some famous umbers For example, She et al cosidered circulat matrices with Fiboacci ad Lucas umbers ad preseted their explicit determiats ad iverses by costructig the trasformatio matrices i [4] Gao et al [5] gave explicit determiats ad iverses of skew circulat ad skew left circulat matrices with Fiboacci ad Lucas umbers I [6], Jiag et al discussed the osigularity of the skew circulat type matrices ad preseted explicit determiats ad iverse matrices of these special matrices Furthermore, four kids of orms ad bouds for the spread of these matrices are give separately I [7], Jiag ad

2 Abstract ad Applied Aalysis Hog give exact determiats of some special circulat matrices ivolvig four kids of famous umbers Authors [8] discussed the osigularity of the circulat type matrix ad preseted the explicit determiat ad iverse matrices There are several papers o the orms of some special matrices Solak [9] establishedtheloweradupperbouds for the spectral orms of circulat matrices with classical Fiboacci ad Lucas umbers etries İpek [0] ivestigated a improved estimatio for spectral orms of these matrices Begiig with Mirsky []severalauthors[ 4]have obtaied bouds for the spread of a matrix Additioally, skew circulat type matrices iclude skew circulat ad skew left circulat matrices The orm ad spread of skew circulat type matrices have ot bee studied It is hoped that this paper will help i chagig this More work cotiuig the preset paper is forthcomig The sum of Fiboacci ad Lucas sequece is defied by the followig recurrece relatios: L + L + + L where L 0, L, () Lemma Let {L } be the sum of Fiboacci ad lucas umbers; the, (iii) (i) (ii) Proof Accordig to we have L i L + 4, (6) L i L L + 4, (7) il i ( ) L + + L +6 (8) L + L + L where L 0, L, (9) L i L + L + +L for 0Thefirstfewvaluesofthesequecearegivebythe followig table: L () The {L } is give by the formula L α β + α β + α+ β +, (3) where α ad β are the roots of the characteristic equatio x x 0 I this paper, we cosider skew circulat type matrices, icludig the skew circulat ad skew left circulat matrices We defie a sum of Fiboacci ad Lucas skew circulat matrix which is a complex matrix with followig form: SCirc (L, L,,L ) L L L L L L L L ( d ), L 3 L 4 L L L L 3 L L where each row is a cyclic shift of the row above the right Besides, a sum of Fiboacci ad Lucas skew left circulat matrix is give by SLCirc (L, L,,L ) (4) ad hece (L 3 L )+(L 4 L 3 ) + +(L + L + ) L + L, L i L + L + +L L (L L 0 )+L (L 3 L ) + +L (L + L ) L L + L 0 L, il i L +L + +L (L L 0 )+(L L 0 ) + +(L + L ) L i + (+) L +L + L 0 ( ) L + + L +L L 0, L i L + 4, (0) L L L L L L 3 L L ( d ), L L L 3 L L L L L (5) L i L L + 4, il i ( ) L + + L +6 () where each row is a cyclic shift of the row above the left This completes the proof

3 Abstract ad Applied Aalysis 3 Lemma (see [0]) Let A SLCirc(a,a,,a ) be a skew left circulat matrix ad let be odd; the, λ j ± k ω ka (j (/))(k ) λ (+)/ (j,,, ), a k( ) k, k () where λ j (j,,,( )/,(+)/)are the eigevalues of A Determiat ad Iverse of Skew Circulat Matrix with the Sum of Fiboacci ad Lucas Numbers I this sectio, let A SCirc (L,,L ) be a skew circulat matrix First of all, a determiat explicit formula for the matrix A isgiveafterthat,weprovethata is a ivertible matrix for ay positive iterger,adthewefidtheiverse of the matrix A Ithefollowig,let x +L +L +, t, c+l +, d+l, l +tl + k l L k+ x (k+) ( ), k (tl k+ L k+ ) x (k+) ( 3) (3) Proof Obviously, det A satisfies the equatio I the case >3,let ℶ t 0 0, ( c c c ) 0 c c 0 c 0 ( x 0 0 Ω ( 0 x d ) 0 x 0 0 ( ) betwo matrices; the, we have where l c 3 c, c 0 l c 3 c, c ℶA Ω ( 0 0 c 0 0 d d d c ( 0 0 d c So it holds that c j L + j c j tl + j L +3 j (j3,4,,), det ℶ det A det Ω [+tl + ) (5) ), (6) ) (j3,4,,) k (tl k+ L k+ ) x (k+) ] (+L + ) (7) (8) Theorem 3 Let A SCirc (L,,L ) be a skew circulat matrix; the det A [+tl + (tl i+ L i+ )x (i+) ] c ( 3), (4) where L is the th sum of Fiboacci adlucas umbers While takig det ℶdet Ω ( ) ( )( )/,wehave det A [+tl + This completes the proof k (tl k+ L k+ ) x (k+) ] (+L + ) (9) Theorem 4 Let A SCirc (L,,L ) be a skew circulat matrix; the A is a ivertible matrix for ay positive iterger Proof Takig i Theorem 3, we have det A 0 Hece A is ivertible I the case >, sice

4 4 Abstract ad Applied Aalysis L (α β )/(α β) + (α β )/(α β) + (α + β + )/(α β),whereα+β,αβ,wehave f(ω k η) j j j L j (ω k η) j ( αj β j + αj β j + αj+ β j+ )(ω k η) j ( αj +α j +α j+ )(ω k η) j j +α+α ( βj +β j +β j+ )(ω k η) j +β+β +α αω k η +β βω k η +L + +(+L )ω k η ω k η ω k η (k,,, ), (0) where ωexp(πi/), ηexp(πi/) If there exists ω l η(l,,, )such that f(ω l η) 0,wehave+L + +(+ L )ω l η0for ω l η ω l η 0, ad hece it follows that ω l η (+L + )/( + L ) is a real umber Sice ω l ηexp ( cos (l + ) πi ) (l + ) π +isi (l + ) π, () it yields that si((l + )π)/ 0,sowehaveω l η for 0< (l + )π/ < πsicex is ot the root of the equatio +L + +(+L )x0( )Weobtaif(ω k η) 0for ay ω k η (k,,, ), while f(η) j L j η j +L + +(+L )η η η 0 () It follows from Lemma i [5] that the coclusio holds Lemma 5 Let the matrix H [h ij ] i,j be of the form +L { +, i j, h ij +L {, i j+, { 0, otherwise (3) The the iverse H [h i,j ] i,j of the matrix H is equal to h ij { ( d) i j { c i j+, i j, { 0, i < j (4) Proof Let e ij k h ikh kj Obviously,e ij 0for i<jithe case ij,weobtaie ii h ii h ii (+L +) /(+L + ) For i j+,weobtai e ij k h ik h kj h i,i h i,j +h iih ij d ( d)i j c i j +c ( d)i j c i j+ 0 (5) Hece, we get HH I ;herei is ( ) ( ) idetity matrix Similarly, we ca verify H H I Thus, the proof is completed Theorem 6 Let A SCirc(L,,L ) be a skew circulat matrix; the where (A ) SCirc (y l,y,,y ) ( 4), (6) y [(6 4t) ( d) 3 c y t 3 + y 3 (6 4t) c, y 4 y k Proof Let (L + i tl + i ) ( d)i c i ] (L + i tl i ) ( d)i (L +i tl i ) ( d)i (L +i tl r+i ) ( d)k 5+i c k 4+i (k5,6,,) ( 4), (7) l ω 3 ω 4 ω 0 ω 3 ω 4 ω Ω , ( ) (8) d ( )

5 Abstract ad Applied Aalysis 5 where ω j [l l (tl + j L +3 j ) L + j ] ω j l (L +3 j tl + j ) (j3,4,,), (j3,4,,) (9) y k T, k+3 T, k+4 T, k+5 l y T 3 T 4 T 5 (L +i tl i ) ( d)k 5+i c k 4+i, The we have l ℶA Ω Ω ( 0 0 c d c 0), (30) d ( c) so ℶA Ω Ω D H, adhered diag(, l ) is a diagoal matrix, ad D H is the direct sum of D ad H If we deote Ω Ω Ω,theweobtaiA Ω(D H )Σ Sice the last row elemets of the matrix Ω are (0,, ω 3,ω 4,,ω,,ω ), the last row elemets of the matrix Ω(D H ) are (0, /l,t 3,T 4,,T ), where ω,+i ( d)i c i 4 l y l T 3 T 4 ω,4+i ( d)i c i 3 (L +i tl i ) ( d) 5+i c 4+i, l l [(6 4t) ( d) 3 c ω,3+i ( d)i c i T 3 T k + k ω,+i ( d)i c i ( 3), ω,k +i ( d)i c i (k3,4,,) (3) 3 + (L + i tl + i ) ( d)i c i ] ( 4) (3) Hece it follows from Lemma 5 that by lettig A SCirc (y,y,,y ) the its last row elemets are ( y, y 3,, y,y ) which are give by the followig equatios: y t l +T 3 t + l l y 3 T, l (6 4t) c, y 4 T, T l (L + i tl i ) ( d)i (L +i tl i ) ( d)i y 5 T, T T l (L +i tl i ) ( d)i c i+, Hece, we obtai y l l [(6 4t) ( d) 3 c y t l l + 3 y 3 l (6 4t) c, y 4 l y 5 l (L + i tl + i ) ( d)i c i ], (L + i tl i ) ( d)i (L +i tl i ) ( d)i (L +i tl i ) ( d)i c i+,

6 6 Abstract ad Applied Aalysis y k l y l (L +i tl i ) ( d)k 5+i c k 4+i, (L +i tl i ) ( d) 5+i c 4+i, A SCirc (y l,y,,y ), (33) Thus ( A F ) a ij j L i (L L + 4) (37) A F (L L + 4) (38) where y [(6 4t) ( d) 3 c + y t 3 y 3 (6 4t) c, y 4 y k (L + i tl + i ) ( d)i c i ] (L + i tl i ) ( d)i (L +i tl i ) ( d)i ( 4), (L +i tl i ) ( d)k 5+i c k 4+i, (k5,6,,) This completes the proof 3 Norm ad Spread of Skew Circulat Matrix with the Sum of Fiboacci ad Lucas Numbers (34) Theorem 7 Let A SCirc(L,,L ) be a skew circulat matrix The three kids orms of A are give by A A L + 4, (35) A F (L L + 4) (36) Proof By Defiitio 4 i [6], (6), ad (7), we have A A L i L + 4, Theorem 8 Let A SCirc(L, L,, L, L ) be a odd-order alterative skew circulat matrix ad let be odd The A L i L + 4 (39) Proof By Lemma i [5], we have λ j (A ) ( ) i L i (ω j η) i (40) Therefore λ j (A ) ( )i L i (ωj η) i L i (j0,,, ) Sice is odd, L i is a eigevalue of A,whichis L L L L L L ( L L L ) ( ) L L 3 L ( ) To sum up, we have L i ( ) max 0 j ( ) (4) (4) λ j (A ) L i (43) Sice all skew circulat matrices are ormal, by Lemma 7 i [6], (6), ad (43), we have which completes the proof A L i L + 4, (44)

7 Abstract ad Applied Aalysis 7 Theorem 9 Let A SCirc(L,,L ) be a skew circulat matrix; the the bouds for the spread of A are s(a ) (L L + 8), s(a ) (45) L +4 L Proof The trace of A is deoted by tr A L By Defiitio 5 i [6]ad(36), we have Sice a ij ij by (6)ad(8), s (A ) (L L + 8) (46) k (+) [ (k )] L k k L k k k kl k, (k ) L k a ij (+)(L + 6) (L + L +3 6+L 4 ) ij L +4 L By Lemma 6 i [6], we have (47) (48) S(A ) L +4 L (49) 4 Determiat ad Iverse of Skew Left Circulat Matrix with the Sum of Fiboacci ad Lucas Numbers I this sectio, let A SLCirc(L,,L ) be a skew left circulat matrix By usig the obtaied coclusios i Sectio, we give a determiat explicit formula for the matrix A Adthe,weprovethatA is a ivertible matrix for ay positive iterger TheiverseofthematrixA is also preseted AccordigtoLemmas4ad5i[5] ad Theorems 3, 4, ad 6, we ca obtai the followig theorems Theorem 0 Let A circulat matrix; the det A ( )( )/ [+tl + SLCirc(L,,L ) be a skew left k (tl +i L +i )x i ] c where L is the th sum of Fiboacci ad Lucas umber ( 3), (50) Theorem Let A SLCirc(L,,L ) be a skew left circulat matrix for ay positive iterger ; thea is a ivertible matrix Theorem Let A SLCirc(L,,L ) be a skew left circulat matrix; the where (A ) SLCirc (y l,y,,y ) ( 4), (5) y [(6 4t) ( d) 3 c y k y k+ 3 + (L + i tl + i ) ( d)i c i ], (L +i tl i ) ( d) k 3+i c k +i, (k,3,, ) y y 3 (6 4t) c, y y t+ (L + i tl i ) ( d)i c i (5) 5 Norm ad Spread of Skew Left Circulat Matrix with the Sum of Fiboacci ad Lucas Numbers Theorem 3 Let A SLCirc(L,,L ) be a skew left circulat matrix The three kids orms of A are give by A A L + 4, A F (L L + 4) (53) Proof Usig the similar method i Theorem 7, the coclusio is obtaied Theorem 4 Let A SLCirc(L, L,, L, L ) be a odd-order alterative skew left circulat matrix; the A L i L + 4 (54) Proof Accordig to Lemma, λ j (A )± ( ) i L i ω (j (/))(k ), (55)

8 8 Abstract ad Applied Aalysis for j,,,( )/,ad So λ (+)/ (A ) L i (56) λ j (A ) ( )i L i ( ) i By (56)ad(57), we have L i, (j,,, ) (57) max 0 i + λ i (A ) L i (58) Sice all skew left circulat matrices are symmetrical, by Lemma 7 i [6], (6), ad (58), we obtai A L + 4 (59) Theorem 5 Let A SLCirc(L,,L ) be a skew left circulat matrix; the the bouds for the spread of A are M L s(a ) { N, if is odd, { M (60) { N, if is eve, where M(L L + 4), N L L,N 0 Proof Sice A is a symmetric matrix, by Lemma 6 i [6], we get s(a ) max ij a ij L The trace of A is, if is odd, the tr(a )L L + L 3 +L L + L + L 3 + +L 4+ 3 L i,by (6), we have tr (A )L L N (6) Let M(L L + 4);thebyDefiitio5i[6], (53), ad (6), we obtai If is eve, the s(a ) M N (6) tr (A )N L L + L 3 L 3 + L 0, (63) hece So the result follows tr (A )N 0 (64) 6 Coclusio We discuss the ivertibility of the skew circulat type matrices with the sum of Fiboacci ad Lucas umbers ad preset the determiat ad the iverse matrices by costructig the trasformatio matrices The four kids of orms ad bouds for the spread of these matrices are give, respectively 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 0GGX05) ad the AMEP of Liyi Uiversity, Chia Refereces [] D Bertaccii ad M K Ng, Skew circulat precoditioers for systems of LMF-based, i Numerical Aalysis ad Its Applicatios, Lecture Notes i Computer Sciece, pp 93 0, 00 [] J R Claeysse, M Davila, ad T Tsukaza, Factor circulat block matrices ad eve order udamped matrix differetial equatios, Matemática Aplicada e Computacioal,vol3,o, pp8 96,983 [3] B Karasöze ad G Şimşek, Eergy preservig itegratio of bi-hamiltoia partial differetial equatios, Applied ad Egieerig Mathematics,vol6,o,pp5 33,03 [4] A Meyer ad S Rjasaow, A effective direct solutio method for certai boudary elemet equatios i 3D, Mathematical Methods i the Applied Scieces,vol3,o,pp43 53,990 [5] SJGuo,YMChe,adJHWu, Equivariatormalforms for parameterized delay differetial equatios with applicatios to bifurcatio theory, Acta Mathematica Siica, vol8,o4, pp , 0 [6] X Q Ji, S L Lei, ad Y Wei, Circulat precoditioers for solvig sigular perturbatio delay differetial equatios, Numerical Liear Algebra with Applicatios,vol,o-3,pp , 005 [7] V C Liu ad P P Vaidyaatha, Circulat ad skew-circulat matrices as ew ormal-form realizatio of IIR digital filters, IEEE Trasactios o Circuits ad Systems, vol35,o6,pp , 988 [8] MJNarasimha, Liearcovolutiousigskew-cycliccovolutios, IEEE Sigal Processig Letters, vol 4, o 3, pp 73 76, 007 [9]DQFu,ZLJiag,YFCui,adSTJhag, Newfast algorithm for optimal desig of block digital filters by skewcyclic covolutio, IET Sigal Processig,vol8,o6,pp , 04 [0] H Karer, J Scheid, ad C W Ueberhuber, Spectral decompositio of real circulat matrices, Liear Algebra ad Its Applicatios, vol 367, pp 30 3, 003 [] J Li, Z Jiag, N She, ad J Zhou, O optimal backward perturbatio aalysis for the liear system with skew circulat

9 Abstract ad Applied Aalysis 9 coefficiet matrix, Computatioal ad Mathematical Methods i Medicie,vol03,ArticleID70738,7pages,03 [] P J Davis, Circulat Matrices, Joh Wiley & Sos, New York, NY, USA, 979 [3] Z L Jiag ad Z X Zhou, Circulat Matrices, Chegdu Techology Uiversity, Chegdu, Chia, 999 [4] S Q She, J M Ce, ad Y Hao, O the determiats ad iverses of circulat matrices with Fiboacci ad Lucas umbers, Applied Mathematics ad Computatio, vol 7, o 3, pp , 0 [5]YGao,ZLJiag,adYPGog, Othedetermiats ad iverses of skew circulat ad skew left circulat matrices with Fiboacci ad Lucas umbers, WSEAS Trasactios o Mathematics,vol,o4,pp47 48,03 [6] Z L Jiag, J J Yao, ad F L Lu, O skew circulat type matrices ivolvig ay cotiuous Fiboacci umbers, Abstract ad Applied Aalysis,vol04,ArticleID4830,0pages,04 [7] X Y Jiag ad K Hog, Exact determiats of some special circulat matrices ivolvig four kids of famous umbers, Abstract ad Applied Aalysis, vol04,articleid73680, pages, 04 [8] Z Jiag ad D Li, The ivertibility, explicit determiats, ad iverses of circulat ad left circulat ad g-circulat matrices ivolvig ay cotiuous Fiboacci ad Lucas umbers, Abstract ad Applied Aalysis, vol04,articleid9345,4 pages, 04 [9] S Solak, O the orms of circulat matrices with the Fiboacci ad Lucas umbers, Applied Mathematics ad Computatio, vol 60, o, pp 5 3, 005 [0] A İpek, O the spectral orms of circulat matrices with classical Fiboacci ad Lucas umbers etries, Applied Mathematics ad Computatio,vol7,o,pp60 60,0 [] L Mirsky, The spread of a matrix, Mathematika, vol 3, pp 7 30, 956 [] R Sharma ad R Kumar, Remark o upper bouds for the spread of a matrix, Liear Algebra ad its Applicatios, vol 438, o, pp , 03 [3] JWu,PZhag,adWLiao, Upperboudsforthespreadofa matrix, Liear Algebra ad Its Applicatios,vol437,o,pp 83 8, 0 [4] C R Johso, R Kumar, ad H Wolkowicz, Lower bouds for the spread of a matrix, Liear Algebra ad its Applicatios,vol 7, pp 6 73, 985 [5] Z Jiag, Y Gog, ad Y Gao, Ivertibility ad explicit iverses of circulat-type matrices with $k$-fiboacci ad $k$-lucas umbers, Abstract ad Applied Aalysis, vol 04, ArticleID 38953, 9 pages, 04 [6] JLi,ZLJiag,adFLLu, Determiats,orms,adthe spread of circulat matrices with Triboacci ad geeralized Lucas umbers, Abstract ad Applied Aalysis,vol04,Article ID 3889, 9 pages, 04

10 Advaces i Operatios Research Hidawi Publishig Corporatio Advaces i Decisio Scieces Hidawi Publishig Corporatio Joural of Applied Mathematics Algebra Hidawi Publishig Corporatio Hidawi Publishig Corporatio Joural of Probability ad Statistics The Scietific World Joural Hidawi Publishig Corporatio Hidawi Publishig Corporatio Iteratioal Joural of Differetial Equatios Hidawi Publishig Corporatio Submit your mauscripts at Iteratioal Joural of Advaces i Combiatorics Hidawi Publishig Corporatio Mathematical Physics Hidawi Publishig Corporatio Joural of Complex Aalysis Hidawi Publishig Corporatio Iteratioal Joural of Mathematics ad Mathematical Scieces Mathematical Problems i Egieerig Joural of Mathematics Hidawi Publishig Corporatio Hidawi Publishig Corporatio Hidawi Publishig Corporatio Discrete Mathematics Joural of Hidawi Publishig Corporatio Discrete Dyamics i Nature ad Society Joural of Fuctio Spaces Hidawi Publishig Corporatio Abstract ad Applied Aalysis Hidawi Publishig Corporatio Hidawi Publishig Corporatio Iteratioal Joural of Joural of Stochastic Aalysis Optimizatio Hidawi Publishig Corporatio Hidawi Publishig Corporatio