POWERS OF COMPLEX PERSYMMETRIC ANTI-TRIDIAGONAL MATRICES WITH CONSTANT ANTI-DIAGONALS
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1 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 *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 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
2 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
3 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
4 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:
5 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
6 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
7 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
8 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
9 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
10 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
11 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
12 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 C39 Crespo Gut rre-gut rre O the elemetwse covergece of cotuous fuctos of Hermt bded oeplt mtrces IEEE rsctos o Iformto heory C76 3 Guterre owers of comple persymmetrc or sew-persymmetrc t-trdgol mtrces wth costt t-dgols pplthcomput Rms O computg of rbtrry postve teger powers for oe type of symmetrc t-trdgol mtrces of order ppled themtcs d Computto Hogl Wu O computg of rbtrry postve powers for oe type of t-trdgol mtrces of eve order ppl th Comput Qgg O computg of rbtrry postve powers for t-trdgol mtrces of eve order ppl th Comput Rms o computg of rbtrry postve teger power of order t-trdgol mtrces wth eros m sew dgol d elemets ; eghbourg dgols ppl th Comput RHor CRohso tr lyss Cmbrdge versty press ew or LFo pre Chebyshev olyomls umercl lyss Oford versty ress Lodo Rms Iteger powers of rel order sew-persymmetrc t-trdgol mtrces wth costt t-dgols ppl th Comput
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