IRRS 9 y 04 wwwrppresscom/volumes/vol9issue/irrs_9 05pdf OWERS OF COLE ERSERIC I-RIIGOL RICES WIH COS I-IGOLS Wg usu * Q e Wg Hbo & ue College of Scece versty of Shgh for Scece d echology Shgh Ch 00093 *Eml: wgyusu0@6com SRC I ths pper we derve geerl epresso for the pth power p ϵ of y comple persymmetrc t-trdgol Hel costt t-dgols mtrces terms of the Chebyshev polyomls of secod d umercl emples re preseted whch show tht our results geerle the results 457 Keywords: t-trdgol mtrces; Egevlues; Egevectors; Chebyshev polyomls SC00: 548 IROCIO From prctcl pot of vew t-trdgol mtrces rse freuetly my res of mthemtcs d egeerg such s umercl lyss soluto of the boudry vlue problems hgh order hrmoc flterg theory I my of such problems we eed to clculte some mtr fuctos such the powers verse or the epoetl here s lot of wor delg wth the verse of t-trdgol mtr d solvg the resultg ler system c be doe effcet wy However computg the teger powers of t-trdgol mtrces hs bee very populr problem lst few yers here hve bee severl ppers o computg the postve teger powers of vrous ds of sure mtrces by Rms es u s Gut e rre etc 3-7 Rms 4 gve the geerl epresso of the p th power for ths type of symmetrc order t-trdgol mtrces ttrd 0 008 I 5-6 smlr problem ws solved for t-trdgol mtrces hvg eros m sew dgol d uts the eghbourg dgols I 00 the geerl epresso for the etres of the power of order t-trdgol mtrces wth eros m sew dgol d elemets ; eghbourg dgols ws derved by Rms 7 I 0 the geerl epresso for the etres of the power of comple persymmrtrc or sew-persymmetrc t-trdgol mtrces wth costt t-dgols re preseted by es u s Gut e rre 3 I 03 Rms 0 gve the egevlue decomposto for rel order sew-persymmetrc t-trdgol mtrces wth costt t-dgols ttrd g b d derved the geerl epresso for teger powers of such mtrces I the preset pper we derve geerl epresso for the p th power p of y comple persymmetrc t-trdgol mtrces wth costt t-dgols g b b terms of the Chebyshev ttrd polyomls of the secod d hs pper s orged s follows: - I Secto we gve the dervto of geerl epresso - I Secto 3 umercl emples re preseted - I Secto 3 we summre the pper ERIVIO OF GEERL ERESSIO I ths pper we study the etres of postve teger power of wth costt t-dgols comple persymmetrc t-trdgol mtrces g 65
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces d where C b C \ 0 Cosder the followg comple oeplt trdgol mtrces d 3 he et trvl result reltes the mtr detty 3 wth or mtr C wth d wth the 4 bcwrd 66
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces where s the Kroecer delt 5 Lemm Let C bc \ 0 d he 6 C 7 where ttrd g b b trdg b b C ttrdg b0 b trdg b0 b roof d We hve h h C h h h b b 0 h b 0 b 0 f f f f 0 f f other 0 other hs completes the proof We shll fd the th power of the mtrces d heorem reltes ll postve teger powers of the mtr wth d or C wth heorem If C bc \ 0 the d ttrdg b b d C ttrd g b0 b f f s eve s 8 67
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces f s eve C 9 f s where trdg b b trdg b0 b roof We wll proceed by ducto o he cse s obvous Suppose tht the result s true for d cosder tht cse y the ducto hypothess we hve f s eve f s Sce we obt tht Sce s symmetrc d Smlrly we hve hs completes the proof et we hve to solve trdgol mtrces d d we hve C f f f f s eve s s eve s f f s eve s We beg ths wor by revewg theorem regrdg the Hermt oeplt heorem 3 Let C bc \ 0 d he hs egevlues b cos d hs egevlues b cos roof See 6 Wth the trdgol mtr we ssocte the polyoml seuece chrctered by three-term recurrece relto: b b 0 0 we ssocte the polyoml seuece Wth the trdgol mtr recurrece relto: chrctered by three-term b b 0 0 d we c wrte the reltos Wth tl codtos 0 0 d mtr form: 0 0 68
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces Q Q E E Q 0 where E 00 0 R Lemm 4 For 0 the degree of the polyoml s d the polyoml roof See8 s d d Oe c show tht the chrcterstc polyoml of s precsely b If 0 3 hs o commo root 0 d d hs o commo root he degree of s precsely b d the chrcterstc polyoml of Hece the egevlues of re ectly the roots of egevlues of re ectly the roots of re the roots of the polyoml the t follows from 7 tht ech Smlrly we hve the s egevlue of the mtr d Q 0 s correspodg egevector 7-9 For we hve sme results hs observto should be te to ccout elsewhere the muscrpt he polyomls d 0 0 verfy the well-ow Chrstoffel-rbou Idetty: Lemm 5 We hve: y y y for y 4 y 0 y y y for y 5 y 0 roof See8 edg y to formuls 4 d 5 we get: 6 0 Sce the mtr 7 0 hs dstct egevlues 0 E where E dg 0 t where egedecomposto of the mtr where F dg S hus the egedecomposto of the mtr d s the trsformg mtr formed by the egevectors of s 0 s where s mely re defed s bove For we hve sme results: the SFS d S s the trsformg mtr formed by the egevectors of re defed s bove mely 69
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces 70 Lemm 6 If d y S the 8 y 9 roof y usg the reltos 4 d 8 or reltos 5 d 9 we obt: t s y where f d 0 f hs completes the proof For we hve E d F We get mmedtely: heorem 7 ssume tht d he: y usg the Cuchy Itegrl Formul we c gve other epressos of the coeffcets ; d 0 d where s closed curve cotg the roots of d o roots of s closed curve cotg the roots of d o roots of roof: Obvously ths theorem holds Corollry 8 If the mtr s osgulr wth the mtr s osgulr wth E the E S SF 3
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces 7 d 4 5 y usg the Cuchy Itegrl Formul we c gve other epressos of the coeffcets d d 6 d 7 where s closed curve cotg the roots of d o roots of s closed curve cotg the roots of d o roots of heorem 9 ssume tht d C he: s f eve s f 8 s f eve s f 9 y usg the Cuchy Itegrl Formul we c gve other epressos of the coeffcets d : s f d eve s f d 30 s f d eve s f d 3 where s closed curve cotg the roots of d o roots of s closed curve cotg the roots of d o roots of
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces 7 roof From heorem we get: s f eve s f s f eve s f h h mely s f eve s f From heorem 7 t follows: s f eve s f y usg the Cuchy Itegrl Formul we c gve other epressos of the coeffcets : s f d eve s f d Smlrly we hve s f d eve s f d hs completes the proof Corollry 0 ssume tht d C he: y usg the Cuchy Itegrl Formul we c gve other epressos of the coeffcets d : d 3 d 33 where s closed curve cotg the roots of d o roots of s closed curve cotg the
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces roots of d o roots of 3 ERICL ELES he ersymmetrc Cse Cosder the order t-trdgol mtr of the followg type: ssume tht d where d re mtr he polyoml seuece verfes 0 Wth tl codtos 0 d y smple clculto we c show tht: 0 0 where re the Chebyshev polyomls 9 of the secod d whch stsfes the three-term recurrece reltos: wth tl codtos Ech d 0 stsfes 73
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces 74 srccos rccos s d thus the roots of re cos he the egevlues of re cos y heorem 7 We get : ssume tht d the: cos cos cos cos cos If the mtr s osgulr d he: cos cos cos cos cos l We c obt: cos cos cos cos cos cos cos cos cos cos s f eve s f d cos cos cos cos cos If 0 we hve the followg theorems heorem 3 Cosder turl umber m m Let 0 g ttrd d cos for every he 4
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces for ll d where f s eve f s re the egevlues of the mtr s the th degree Chebyshev polyoml of the secod d heorem 3 Cosder eve turl umber m cos for every he m Let ttrd g 0 4 d f s eve for ll d where 0 f s For eve order mtr the followg codto s fulflled: 0 4 hs mes tht eve order mtr ttrd g 0 s osgulr ts determt s ot eul to ero d derved epresso of c be ppled for computg egtve teger powers s well g we get the followg epresso for elemets of the verse mtr 4 0 If we hve the followg theorems heorem 33 Cosder eve turl umber m cos he s s b he Sew-persymmetrc Cse For C ttrd g b0 b heorem 34 Cosder ~ we hve I other cse we get f m Let ttrd g d s s b we hve the followg theorem by heorem 9 Let C ttrd g 0 d cos ~ h h C 4 h ~ h C 0 If 75
IRRS 9 y 04 usu et l owers of Comple ersymmetrc t-rdgol trces where ~ f s eve f s From heorem 34 we c fd y of these postve teger power of C ttrd 0 s emple we cosder the cses 3 g 4 where c 3 4 c c c c 3 wth 4 COCLSIO I ths pper we derve geerl epresso for the p th power p of y comple persymmetrc t-trdgol Hel costt t-dgols mtrces wth costt t-dgols ttrd g b b d ttrd g b0 b umercl emples re preseted hs ovel epresso s both eteso of the oe ttrd wth obted by Rms for the powers of the mtr 0 g 4 see4 for the cse d 7 for the eve cse d eteso of the oe obted by Hogl Wu for the powers of the mtr ttrd g wth see 5 for the eve cse We my sfely drw the cocluso: our results geerle the results 457 REFERECES R Gry oeplt d crcult mtrces: revew Foudtos d reds Commuctos d Iformto heory 3 006 55 C39 Crespo Gut rre-gut rre O the elemetwse covergece of cotuous fuctos of Hermt bded oeplt mtrces IEEE rsctos o Iformto heory 53 3 007 68 C76 3 Guterre owers of comple persymmetrc or sew-persymmetrc t-trdgol mtrces wth costt t-dgols pplthcomput 7 0 65-63 4 Rms O computg of rbtrry postve teger powers for oe type of symmetrc t-trdgol mtrces of order ppled themtcs d Computto 03 008 573-58 5 Hogl Wu O computg of rbtrry postve powers for oe type of t-trdgol mtrces of eve order ppl th Comput 7 00 750-756 6 Qgg O computg of rbtrry postve powers for t-trdgol mtrces of eve order ppl th Comput 03 008 5-57 7 Rms o computg of rbtrry postve teger power of order t-trdgol mtrces wth eros m sew dgol d elemets ; eghbourg dgols ppl th Comput000964-7 8 RHor CRohso tr lyss Cmbrdge versty press ew or 990 9 LFo pre Chebyshev olyomls umercl lyss Oford versty ress Lodo 968 0 Rms Iteger powers of rel order sew-persymmetrc t-trdgol mtrces wth costt t-dgols ppl th Comput9037075-7088 76