On The Circulant K Fibonacci Matrices

Transcription

1 IOSR Jou of Mthetcs (IOSR-JM) e-issn: p-issn: X. Voue 3 Issue Ve. II (M. - Ap. 07) PP O he Ccut K bocc Mtces Sego co (Deptet of Mthetcs Uvesty of Ls Ps de G C Sp) Abstct: We stted oog fo fou to spfy the ccuto of the dffeece of two bocc ubes depedg o the d of subscpts. he we study the vue of the detet of ccut tces whose etes e bocc ubes. We cotue ccutg the egevues d fsh wth the ccuto of the egevues of the tx obted utpyg the bocc Keywods: bocc d Lucs ubes Egevues Ccut tx. I. Itoducto he cssc bocc sequece { } hd bee exteded y wys [ ]. Oe o whch they e wog oe tesey ecet yes s due to co d Pz [3 4] whch we eebe. by the ecuece eto o gve tege ube we defe the bocc sequece N fo wth t codtos 0 0. Accodg to ths defto the gee expesso of the fst tes of the bocc sequece e I ptcu fo = the cssc bocc sequece s obted whe fo = we get the Pe sequece Chctestc equto of ths sequece s 4. It s esy to pove these soutos vefy whose soutos e I ptcu the Bet Idetty fo the bocc ubes s d Moeove we defe the bocc ubes wth egtve subscpt s ( ). Sy we defe the Lucs ubes s L L L wth t codtos L0 L. [5]. he Bet Idetty fo the Lucs ubes tes the fo L d cosequety L. Moeove L ( ) L. Wth these stuctos t s etvey esy to pove L L ( ) () ( 4) 0 Now s we w te eed ths fou we w spfy p odd. o the Bet Idetty d tg to ccout p p p p p p p p p p p p p eve: ccodg to whethe t s eve o L p p p p p L p p odd: p p p DOI: / Pge

2 O he Ccut K bocc Mtces I shot: p p L p f s eve pl f s odd (). Mtx os he foowg tx os e defed [6 7]. Let A be tx. he obeus o Eucde o of A s defed s A E he cou o of A s defed s A x whch s spy the xu bsoute cou su of the tx. he ow o of A s A x whch s spy the xu bsoute ow su of the tx. he spect o of tx A s the gest sgu vue of A.e. the sque oot of the gest egevue of the postve-sedefte tx AA whee A deotes the cougte tspose of A; tht s A A A A x ( ) x ( ). Ccut tx 0 0 Gve the ubes 0... the tx C 0 3 s ced ccut tx 3 0 [8 9 0] becuse the ety s equ to the ety fo... If C s ccut tx ts tspose tx C s so ccut. It s ow the detet of the ccut tx whee w e e the th oots of uty. C s [8] det( C ) C w (3) 0 0 We w use the otto C CIRC ( 0... ) fo the ccut tx whose top ow s c Ad te we w eed the foowg popetes: ) he p : CIRC ( ) s the egevue p o e ccut tces to copex -vectos. hus f C CIRC( ) the ( C) s the set of egevues of the tx C. b) ( CIRC( 0... ) w 0 ([] heoe.6()). c) λ s geb soophs ([] Cooy.8.). o the os of ccut tces see [ 3 4 8]..3 Poposto If b b 0 d + b s egevue of e ccut tx A the poduct tx AA wth utpcty whee A s the tspose tx of A. Poof. Suppose A CIRC ( 0... ). he A CIRC( 0... ). / b s egevue of the DOI: / Pge

3 O he Ccut K bocc Mtces We e gve tht b ( A) fo soe 0 wth b 0. heefoe b ( A) s so egevue fo the bove Popety (c). (f the subscpt s the b = 0 coty to wht s gve). A b A b. Ag fo the Popety (c) d AA AA b d ts utpcty s. Hece he poof st wos cse b = 0 povded s odd d 0 ( A) othewse f fo expe b = 0 d s eve the egevue c be o-degeete AA. But ths cse the utpcty s becuse the egevue s 0 wth utpcty. 0 II. A Ccut K bocc Mtx Accodg to pevous defto fo 0 3 s ced 3 ccut bocc tx. Next we ty to spfy the expesso of the detet of ths tx. It s obvous tht > o >0 t s 0. heoe (Detet of the bocc ccut tx) he vue of the ccut bocc detet s L ( ) (4) Poof. Accodg to ou (.3) w. he 0 0 w w w w w w 0 w w w w w w ( w )( w ) 0 w w w w w w 0 w w w w w w w w 0 w w 0 ( ) ( ) ( ) ( ) w w w w 0 becuse w d. DOI: / Pge

4 O he Ccut K bocc Mtces O the othe hd ( bw) b w b b 0 0 b b. heefoe w 0 w w w w L Cosequety ths estbshes the equto (.4). _ o ths equto s postve o egtve ccodg s odd o eve espectvey. hs fou c be spfed f s eve. Copg the fous () d (4) t s. he s eve f 0 (od 4) d the s odd f (od 4) d the L L L L L L L L L L L. Mtx os of the bocc ccut tx g to ccout the defto of the Eucde tx o d s the ow vectos hve the se E 0 etes the Eucde o of the bocc ccut tx s. Ad ppyg the fou () t s L ( ) ( ) L. E ( 4) Logcy the Eucde o of the bocc ccut tx s tes ts ow o ts cou o..3 Egevues d egevectos he egevues of of the uty d s the gy ut. e gve by w [ 0] whee w 0 he coespodg ozed egevectos e gve by g to ccout f p q p () A the egevues e spe. q the egevues of () If s odd oy oe egevue s e:. 0 0 exp e the th oots DOI: / Pge e w w... w vefy the foowg popetes: p 0 (3) If s eve = p the tx get oy two e egevues: λ 0 d (4) Hf the othe egevues of o stce f = 3 the egevues of 3 ) w0 0 gets copex d the othe hf e the cougtes. e: w )

5 O he Ccut K bocc Mtces ) w Evdety III. O he Mtx Poduct Let us cosde the tx M whee DOI: / Pge (()) s the tspose tx of. Evdety M s doube syetc tht s d. Cosequety ts egevues e e. y M s so ccut. c c... s the fst ow vecto of ths tx the If 0 c : c c ; c ( c) ( c) 0 c g to ccout Poposto we c deduce the foowg theoe. 3.. heoe If λ s egevue of the ccut tx M. 3. Cooy If b b 0 s copex egevue of M. If λ = s e egevue of the the sque of ts o the s egevue of b s doube egevue of s spe egevue of M. Acowedgeets I th AuWy-oes the suggestos de fo the pubcto of ths tce s we s the poof of Poposto. hs esech dd ot eceve y specfc gt fo fudg geces the pubc coec o ot-fo-poft sectos. Refeeces [] V.E. Hoggt bocc d Lucs ubes Po Ato CA: Houghto-Mff; 969. [] A.. Hod A geezed bocc sequece Mth. Mg. Vo [3] Sego co A. Pz O the bocc -ubes Chos Sot. & ct. Vo. 3(5) [4] Sego co A. Pz he -bocc sequece d the Psc -tge Chos Sot. & ct. Vo. 33() [5] Sego co O the Lucs ubes It. J. Cotep. Mth. Sceces Vo. 6() [6] So ouct [7] o [8] Iw K d Stgo R. Sc O ccut tces [9] D.A. Ld A bocc ccut bocc Qutey Vo. 8(5) [0] [] Au Wy-oes Ccuts [] D. Bozut. O the spect os of the tces coected to tege ube sequeces. Apped Mthetcs d Coputto 9 (03) [3] Php Dvs. Ccut Mtces. Joh Wey & Sos Ic New Yo 979. [4] E. Goce Apte. Msou d N. ugu Nos of Ccut d Seccut tces d Hods sequece. As cobto 85 (007) [5] S. She J. Ce. O the Spect Nos of -Ccut Mtces wth the -bocc d -Lucs Nubes. It. J. Cotep. Mth. Sceces 5() [6] Y. Yz N. s. O the vese of ccut tx v geezed -Hod ubes. Apped Mthetcs d Coputto 3 (03) [7] J. Zhou. he Idetc Esttes of Spect Nos fo Ccut Mtces wth Bo Coeffcets Cobed wth bocc Nubes d Lucs Nubes Etes. Jou of ucto Spces Voue 04 Atce ID pges. [8] J. Zhou. he spect os of g-ccut tces wth cssc bocc d Lucs ubes etes. Apped Mthetcs d Coputto 33 (04)