On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers

Transcription

1 Türkmen and Gökbaş Journal of Inequaltes and Applcatons (06) 06:65 DOI 086/s R E S E A R C H Open Access On the spectral norm of r-crculant matrces wth the Pell and Pell-Lucas numbers Ramazan Türkmen * and Hasan Gökbaş * Correspondence: rturkmen@selcukedutr Scence Faculty Selcuk Unversty Konya 403 Turkey Full lst of author nformaton s avalable at the end of the artcle Abstract Let us defne A = C r (a 0 a a n )tobean nr-crculant matrx The entres n the frst row of A = C r (a 0 a a n )area = P a = Q a = P or a = Q ( =0 n )wherep and Q are the th Pell and Pell-Lucas numbers respectvely We fnd some bounds estmaton of the spectral norm for r-crculant matrces wth Pell and Pell-Lucas numbers Keywords: Pell numbers; Pell-Lucas numbers; r-crculant matrx; spectral norm Introducton Specal matrces s a wdely studed subect n matrx analyss Especally specal matrces whose entres are well-known number sequences have become a very nterestng research subect n recent years and many authors have obtaned some good results n ths area For example Bahş and Solak have studed the norms of r-crculant matrces wth the hyper- Fbonacc and Lucas numbers [] Bozkurt and Tam have obtaned some results belong to determnants and nverses of r-crculant matrces assocated wth a number sequence [] Shen and Cen have made a smlar study by usng r-crculant matrces wth the Fbonacc and Lucas numbers [3 4]andHeet al have establshed on the spectral norm nequaltes on r-crculant matrces wth Fbonacc and Lucas numbers [5] Lots of artcle have been wrtten so far whch concern estmates for spectral norms of crculant and r-crculant matrces whch have connectons wth sgnal and mage processng tme seres analyss and many other problems In ths paper we derve expressons of spectral norms for r-crculant matrces We explan some prelmnares and well-known results We thcken the denttes of estmatons for spectral norms of r-crculant matrces wth the Pell and Pell-Lucas numbers The Pell and Pell-Lucas sequences P n and Q n are defned by the recurrence relatons P 0 =0 P = P n =P n + P n for n and Q 0 = Q = Q n =Q n + Q n for n 06 Türkmen and Gökbaş Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 40 Internatonal Lcense ( whch permts unrestrcted use dstrbuton and reproducton n any medum provded you gve approprate credt to the orgnal author(s) and the source provde a lnk to the Creatve Commons lcense and ndcate f changes were made

2 Türkmen and Gökbaş Journal of Inequaltes and Applcatons (06) 06:65 Page of 7 If we start from n = 0 then the Pell and Pell-Lucas sequence are gven by n : P n : Q n : and The followng sum formulas for the Pell and Pell-Lucas numbers are well known [6 7]: n k= n k= P k = P np n+ Q k = Q n+ +( ) n 4 AmatrxC =[c ] M nn (C) s called a r-crculant matrx f t s of the form c = { c rc n+ < Obvously the r-crculant matrx C s determned by the parameter r and ts frst row elements c 0 c c n thuswedenotec = C r (c 0 c c n ) Especally let r =thematrx C s called a crculant matrx [3] The Eucldean norm of the matrx A s defned as ( n ) / A E = a = ThesngularvaluesofthematrxA are ( σ = λ A A ) where λ s an egenvalue of A A and A s conugate transpose of matrx A Thesquare roots of the maxmum egenvalues of A A are called the spectral norm of A and are nduced by A The followng nequalty holds: n A E A A E Defne the maxmum column length norm c andthemaxmumrowlengthnormr of any matrx A by r (A)=max a

3 Türkmen and Gökbaş Journal of Inequaltes and Applcatons (06) 06:65 Page 3 of 7 and c (A)=max a respectvely Let A BandC be m n matrces If A = B C then A r (B)c (C) [8] Result and dscusson Theorem Let A = C r (P 0 P P n ) be a r-crculant matrx where r C Pn P n () r A r Pn P n () r < r A Proof The matrx A s of the form P 0 P P n P n rp n P 0 P n 3 P n A = rp rp 3 P 0 P rp rp rp n P 0 Then we have n A E = n (n )P + r P ; =0 = hence when r weobtan that s =0 = =0 (n ) P np n (n ) P np n n A E n n (n )P + P = n P = n P np n Pn P n A E n Pn P n A On the other hand let the matrces B and C be P 0 r P 0 r r P 0 r r r P 0 P 0 P P n P n P n P 0 P n 3 P n P P 3 P 0 P P P P n P 0

4 Türkmen and Gökbaş Journal of Inequaltes and Applcatons (06) 06:65 Page 4 of 7 such that A = B CThen r (B)=max b n = r (n )= r (n ) and c (C)=max c n = A r (n ) P np n When r < we also obtan that s =0 n P Pn P n = n A E n (n ) r P + r P = n r P np n = =0 Pn P n A E r n Pn P n A r On the other hand let the matrces B and C be P 0 r P 0 r r P 0 r r r P 0 such that A = B CThen r (B)=max b = n b n = n and c (C)=max c = A (n ) P np n =0 P 0 P P n P n P n P 0 P n 3 P n P P 3 P 0 P P P P n P 0 n c n = n P Pn P n = =0 =0 Thus the proof s completed Corollary Let A = C r (P 0 P P n ) be a r-crculant matrx where r C r ; we have

5 Türkmen and Gökbaş Journal of Inequaltes and Applcatons (06) 06:65 Page 5 of 7 A (n ) r P np n where s the spectral norm and P n denotes the nth Pell number Proof Snce A = C r (P0 P P n )sar-crculant matrx f the matrces C r(p 0 P P n )andc = C(P0 P P n )wegeta = B C;thusweobtan A (n ) r P np n Theorem 3 Let A = C r (Q 0 Q Q n ) be a r-crculant matrx where r C () r () r < r r A r n Q n +6 nodd A r n Q n + neven Q n +6 A n Q n +6 nodd Q n + A n Q n + neven Q n +6 Q n + Proof The matrx A s of the form Q 0 Q Q n Q n rq n Q 0 Q n 3 Q n A = rq rq 3 Q 0 Q rq rq rq n Q 0 Then we have n A E = n (n )Q + r Q ; =0 = hence when r weobtan n A E n (n )Q + n Q = n Q = =0 = =0 that s Q n +6 n odd A E A n Q n + n even On the other hand let the matrces B and C be r r r r r r n Q n +6 n odd n Q n + n even Q 0 Q Q n Q n Q n Q 0 Q n 3 Q n Q Q 3 Q 0 Q Q Q Q n Q 0

6 Türkmen and Gökbaş Journal of Inequaltes and Applcatons (06) 06:65 Page 6 of 7 such that A = B CThen r (B)=max b = n b n = r (n )+ and c (C)=max c = A When r < we also obtan =0 n c n = n Q = =0 =0 ( r (n ) + )( Q n +6 ) n odd ( r (n ) + )( Q n + ) n even Q n +6 n odd Q n + n even n A E n (n ) r Q + r n( Q n +6 ) n odd r Q = =0 = r n( Q n + ) n even that s Q r n +6 n odd A E A n Q r n + n even On the other hand let the matrces B and C be r r r r r r such that A = B CThen r (B)=max Q 0 Q Q n Q n Q n Q 0 Q n 3 Q n Q Q 3 Q 0 Q Q Q Q n Q 0 b = n b n = n and c (C)=max c = A =0 n c n = n Q = =0 n Q n +6 n odd n Q n + n even =0 Q n +6 n odd Q n + n even Thus the proof s completed

7 Türkmen and Gökbaş Journal of Inequaltes and Applcatons (06) 06:65 Page 7 of 7 Corollary 4 Let A = C r (Q 0 Q Q n ) be a r-crculant matrx where r C r A { n r Q n +6 nodd n r Q n + neven where s the spectral norm and Q n denotes the nth Pell-Lucas number Proof Snce A = C r (Q 0 Q Q n )sar-crculant matrx f the matrces C r(q 0 Q Q n )andc = C(Q 0 Q Q n )wegeta = B C;thusweobtan A { n r Q n +6 n odd n r Q n + n even Competng nterests The authors declare that they have no competng nterests Authors contrbutons All authors contrbuted equally to the wrtng of ths paper All authors read and approved the fnal manuscrpt Author detals Scence Faculty Selcuk Unversty Konya 403 Turkey Sems Tebrz Anatolan Relgous Vocatonal Hgh School Konya Turkey Acknowledgements The authors wsh to express ther heartfelt thanks to the referees for ther detaled and helpful suggestons for revsng the manuscrpt Receved: 9 September 05 Accepted: February 06 References Bahş M Solak S: On the norms of r-crculant matrces wth the hyper-fbonacc and Lucas numbers J Math Inequal 8(4) (04) Bozkurt D Tam TY: Determnants and nverses of r-crculant matrces assocated wth a number sequence Lnear Multlnear Algebra (04) do:0080/ Shen S Cen J: On the bounds for the norms of r-crculant matrces wth the Fbonacc and Lucas numbers Appl Math Comput (00) 4 Shen S Cen J: On the spectral norms of r-crculant matrces wth the k-fbonacc and k-lucas numbers Int J Contemp Math Sc 5() (00) 5 He C Ma J Zhang K Wang Z: The upper bound estmaton on the spectral norm r-crculant matrces wth the Fbonacc and Lucas numbers J Inequal Appl (05) do:086/s Halc S: On some nequaltes and Hankel matrces nvolvng Pell Pell-Lucas numbers Math Rep 65(5) -0 (03) 7 Koshy T: Pell and Pell-Lucas Numbers wth Applcatons Sprnger Berln (04) 8 Horn RA Johnson CR: Topcs n Matrx Analyss pp Cambrdge Unversty Press Cambrdge (99)