Exponential Matrix and Their Properties
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1 raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Volu, u, Jauary 6, PP 5-6 SSN 7-7X Pr & SSN 7- Ol Expoal Marx ad hr Propr Mohad bdullah Salh Sala, Collg of Educao & laguag, Dpar of Mahac & Sac, Uvry of ra. ra, Y. Dr. V.C.orar Yhwa Mahavdyalaya, Dpar of Mahac & Sac, Swa Raaad rh Marhwada Uvry, Nadd, da brac: h arx xpoal a vry pora ubcla of arx fuco. h papr, w dcu o of h or coo arx xpoal ad o hod for copug. prcpl, h arx xpoal could b calculad dffr hod o of h hod ar prfrabl o ohr bu o ar rly afacory. Du o ha, w dcud copuao of h arx xpoal ug aylor Sr, Scalg ad Squarg, Egvcor, ad h Schur dcopoo hod horcally. Kyword: Marx Expoal, Coug Marx, No-coug Marx.. NRODUCON h purpo of h o arx fuco, h hory of arx fuco wa ubquly dvlopd by ay ahaca ovr h ug yar. oday, arc of fuco ar wdly ud cc ad grg ad ar of growg r, du o h uccc way hy allow oluo o b xprd ad rc advac urcal algorh for copug h [ ]. gral a rg ara lar algbra, arx aaly ad ar ud ay ara pcally arx Expoal.h arx xpoal a vry pora ubcla of fuco of arc ha ha b udd xvly h la 5 yar [ ]. h copuao of arx fuco ha b o of h o challgg probl urcal lar algbra. og h arx fuco o of h o rg h arx xpoal. larg ubr of hod ha b propod for h arx xpoal, ay of h of pdagogc r oly or of dubou urcal ably. So of h or copuaoally uful hod ar urvyd [ ] prcpl, h arx xpoal could b copud ay way ad ay dffr hod o calcula arx xpoal [,9]. pracc, o of h hod ar prfrabl o ohr, bu o ar coplly afacory.. DEFNONS OF EXP: h fuco of a arx whch w ar rd ca b dfd varou way. ahac, h arx xpoal a fuco o quar arc aalogou o h ordary xpoal fuco [,,,, 7]. L M. h xpoal of, dod by xp, h arx gv by h powr r or... ` Whr = No ha h h gralzao of h aylor r xpao of h adard Expoal fuco. h r covrg aboluly for all C ha radu of covrgc qual o +, o h xpoal of wll-dfd. o prov h Covrgc of h r, w hav h followg hor. RC Pag 5
2 Mohad bdullah Salh Sala & Dr. V.C.orar hor. for or dal [6]: h r covrg aboluly for all ulplcav or o M. h M. Furhror, l b a oralzd ub Proof: h h paral u S So S Sc a ral ubr ad h rgh-had d a par of h covrg r of ral ubr h h quao covrg, f > hr a N uch ha for >, h uffc o prov ha S covrg. Furhror, o ha o ca, a pl ar o xpr h arx xpoal of a coplx arx hall b dod by ad ca b dfd a ubr of quval way [ ]: z z dz Or Or l dx X, X 5 d For dal [7], ad w hav ohr dfo bu w lav o radr o collc h.. COMPUON OF EXPONENL MRX hr ar ay hod ud o copu h xpoal of a arx. pproxao hory, dffral quao, h arx gvalu, ad h arx characrc Polyoal ar o of h varou hod ud. w wll oul varou plc Mhod for fdg h xpoal of a arx. h hod xad ar gv by h yp of arx [,,8,9]. raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Pag 5
3 Expoal Marx ad hr Propr.- Copug Marx Expoal for Dagoal Marx ad for Dagoalzabl Marc f a dagoal arx havg dagoal r h w hav a a a Now, L b Egvcor R v v,..., v, uch ha v l u df h arx = [ yrc ad ha a copl of lar dpd v v,..., of corrpodg o h gvalu of, w hav,,... 6, v v ] who colu ar h gvcor v, v,..., v v, v,... v Sc whr Now ug o copu ad w ca wr a follow Λ Λ Λ d hc Exapl: Codr h arx 5 h by ug h abov forula for dagoal for w g h xpoal arx 5 For dagoalzabl arx w gv h xapl raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Pag 55
4 Mohad bdullah Salh Sala & Dr. V.C.orar Exapl: L 5 afr foud h gvalu ad gvcor ad coruc arx w u h forula o copu a follow -.- Copug Marx Expoal for Gral Squar Marc..- Ug Jorda Noral For Suppo o dagoalzabl arx whch o pobl o fd larly dpd gvcor of h arx, h ca ca u h Jorda for of. Suppo h Jorda for of, wh P h rao arx. h Whr - - dag,,..., dag... h J... hu, h probl o fd h arx xpoal of a Jorda bloc whr h Jorda bloc ha h for J N M ad gral N a o o h h uppr dagoal ad h ull arx f h do of h arx. by ug h abov xpro w hav J J N h ca b wr Exapl: N N N J N h w calcula h gvalu of whch ar [ ] W hav whch P - 5 PJP h w calcula P d raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Pag 56
5 Expoal Marx ad hr Propr J 6 6 hrfor, by ug h Jorda caocal for o copu h xpoal of arx Ug Halo hor Cayly hor. Cayly Halo L a quar arx ad Proof:. characrc polyoal h Codr a quar arx ad a polyoal px ad x b h characrc polyoal of. h wr px h for p x x q x r x by Cayly-Halo x, h p = r uch ha w ca wr polyoal X r X Whr r x h radr of log dvo of ca b wr a x by x, h h arx xpoal r hu a polyoal of of dgr l ha Codr ow a gvcor v wh h corrpodg gvalu alogouly v a v a v a v v v, h ad hu f w hav dc gvalu o ha h abov quao a rpolao probl whch ca b ud o copu h coffc a. h ca of ulpl gvalu w u h corrpodg gralzd gvcor...- Ug Nurcal grao Codr h ODE x x, x,,...,,..., h wh collc h oluo fro o w g raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Pag 57
6 Mohad bdullah Salh Sala & Dr. V.C.orar,..., x,..., x,..., h h gral oluo for abov ODE X ha pl Now, by ug urcal graor wh p X X, X, wh X w g,..., -..-h Marx Expoal Va rpolao Hr w hav wo d a follow:... -Lagrag rpolao Forula L,,..., b h dc gvalu of a arx wll dfd a h gvalu of, h h Lagrag forula for X M ad f ay fuco ha, Nwo' Dvdd Dffrc rpolao L M b a arx wh gvalu,,..., Now w df f a follow f f [,,..., ] 8 Whr [,,..., ] h dvdd dffrc a,,..., whch dfd f [,..., ] f [,..., ] f [,..., ] whr h valu of dvdd dffrc dpd of h ordr of h argu.,.5- Ug a L of Powr Fro calculu w ow ha for ay ubr a ad h xpoal a l fro quao o ca df h arx xpoal a a l of powr a l 9 h forula h l of h fr ordr aylor xpao of rad o h powr Z.. SCLNG ND SQURNG W drv a calg propry fro a fudaal olar dffral quao who oluo h o-calld q-xpoal fuco. calg propry ha b blvd o b gv by a powr fuco oly, bu acually or gral xpro for h calg propry foud o b a oluo of h abov fudaal o-lar dffral quao. h raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Pag 58
7 Expoal Marx ad hr Propr hod wll hlp o corol o of h roud off rror ad or ubr of r would a o fd a aylor approxao h calg ad quarg hod h o wdly ud hod for copug h arx xpoal. h hod cal h arx by a powr of o rduc h or o ordr.h advaag of h calg hod ha h cald rao arx ca b ad o hav a or l ha uy. 5. YLOR SERES L M h xpoal of, dod by h aylor powr r or xp, h arx gv by... Whr. No ha h h gralzao of h aylor r xpao of h adard xpoal fuco. h abov r alway covrg ad wll-dfd. Now o calcula h arx xpoal w u copur ad w cao calcula h xpoal arx xac oly w wll b abl o approxa wh a rucad aylor r of r. h rucad aylor r dod by R h ordr of h approxao o b h hgh powr of h rucad aylor r whch rprd by bu hr ar ay ohr facor ha ca affc h accuracy of a oluo chqu uch ha For accuracy ad ffccy, h rul for h arx xpoal dpd o h arx or wh h arx or vry larg h h ur ay cau accuracy du o urcal roud off h probl occur wh h r of ar larg,ohrw f h or all h accuracy ad ffccy a w drd. 6. EGENVECORS ND SCHUR DECOMPOSON MEHODS h hod bad o h lary raforao of a arx a follow P P whr a ral yrc arx ad P a ral uary arx ad h gvalu of whch ar ral, h ay o copu wh o dfcv dagoalzabl bu, wh ay o b dagoalzabl ad hu dfcv uch ha hr o vrbl arx of gvcor P. Wh P o vrbl h ha ll codod o h rror wll b larg. Du o h obrvao h hod rl o dagoalzag h arx. 7. PROPERES h co of h papr w collc for rfrc addoal pora propr of h arx xpoal ha ar o dd h dvlop [,8,9]. L, M ad l ad b arbrary coplx ubr. W do h Zro arx by. h arx xpoal af h followg propr Propry f do h zro arx, h Propry f vrbl, h -. h dy arx. Propry f,...,, h dag,...,. Propry d d. rac., wh coplx quar arx ad rac= h Propry 5. follow ha f yrc, h yrc, ad f w yrc h orhogoal. alo raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Pag 59
8 Mohad bdullah Salh Sala & Dr. V.C.orar Propry 6 f h ad falar propr of h calar xpoal fuco y uforualy o all carry ovr o h arx xpoal. For xapl, w ow fro calculu wh ad ar ubr. Howvr h of o ru for xpoal of arc. ohr word, pobl o hav arc ad uch ha. Exacly wh w hav qualy dpd o pcfc propr of h arx ad ha dcud h co. Propry 7 l b a coplx quar Marx h for gr. Propry 8 L b a coplx quar arx, ad l, C h. Sg = ad =- propry w g. ohr word, rgardl of h arx, h arx xpoal alway vrbl, ad ha vr. Propry 9. follow ha f Hra arx, h Hra, ad f w-hra, h Propry.. Propry l, M h f ad oly f for all Propry l Proof ad Rar., M b gv. f, h.. alo uch ha h co w dcu o proof ad o rar o h prvou co ad w wll gv o xapl rlad o h for propry ay o proof. For propry w proof Rcall ha, for all gr, w hav ug h dfo o g bu f a arx dagoalzabl arx, h hr x a vrbl o ha D, whr D a dagoal arx of gvalu of, ad a arx havg gvcor of D a colu. h ca, P P.For propry ay o proof. For propry f,,..., ar h gvalu of,h,,..., ar h gvalu of by h pcral appg propry for dagoalzabl arc uch ha f Pf P whr p vrbl arx.h h rac h u of h gvalu, ad h dra h produc of h gvalu of, o d... yrc uch ha... rac h by ug h dfo of xp ha. For Propry 5 ay f raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Pag 6
9 Expoal Marx ad hr Propr raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Pag 6 h. For propry 6w u h dfo of xp h l l by l h w ca u duco o proof propry7fro propry 6 o g for propry8 w u h dfo of xp h w hav Pu, h h fro h boal hor ha propry 9 larly propry5 ad propry ay for radr propry f = ad by ug h powr r xpao of ad w g Cou ad dcal. Covrly, f for all ad by dffrag wc wh rpc o ad pu = w g =. For propry whch h o pora arx xpoal o w wll proof a follow: W u h powr r for o proof h propry a follow Pu, h h fro h boal hor ha h follow ha. Cocluo, alhough coua- vy a uffc codo for h d o hold, bu o cary a h xapl
10 Mohad bdullah Salh Sala & Dr. V.C.orar, how. u f ad hav algbrac r h hr couavy cary for o hold. algbrac ubr dfd by h propry ha a roo of a polyoal wh raoal or quvally, gr coffc. 8. CONCLUSON uary, w hav ha h couavy a uffc codo for h d o hold wh, hav algbrac r, bu o cary gral a xapl abov ha a h covr gral o ru. d w ocd ha wh calculad h arx xpoal by h Jorda for h way vry borg for bg arx z. Epcally wh h arx dfcv whch dffcul o dr urcally bcau ay Sall chag a dfcv arx wll chag Jorda for oally. u wh w u aylor Sr urcally whch alway covrg horcal w hav larg caclao rror du o rucad aylor r, o ha h covrgc ca b low f larg. u w ca b avodd h probl by carful Scalg ad Squarg hod. Egvalu-Egvcor hod do o wor wh o dagoalzabl ad w hav probl wh ao-dagoalzabl arx. Fally W hav ha hr o uforly b hod for h copuao arx xpoal h choc of hod dpd o h applcao ad h parcular arx REFERENCES [] Molr. C ad Va Loa. C, N Dubou Way o Copu h Expo- al of a Marx, wy-fv Yar Lar, SM Rvw,,-9 pp [] Nchola J. Hgha ad wad H. l-mohy, Copug Marx Fuc- o, Machr u for Mahacal Scc,h Uvry of Machr, SSN pp-7 [] Clv. Molr ad Charl F. Va Loa, N dubou way o copu h xpoal of a arx, wy-fv yar lar. SM Rv., 5:9,. [] N. J. Hgha, Fuco of Marc hory ad Copuao, SM, Socy for dural ad ppld Mahac, Phladlpha,. US, 8. SN xx+5 pp. [5] R.. Hor ad C. R. Joho, opc Marx a l y, Cabrdg Uvry Pr, 99,SN , v+67 pp. [6] R.. Hor ad C. R. Joho, M a r x a l y, Cabrdg Uvr y Pr,985, SN , x+56 pp. [7] R. lla, roduco o Marx aly, d d. Nw Yor: McGraw- Hll, 96, rprd by SM, Phladlpha,995. [8] Syd Muhaad Ghufra, h Copuao of Marx Fuco Parcular, h Marx Expoal, Mar h, Uvry of rgha,eglad,9,69 pp. [9] Nahl Sall, h xpoal of arc, Mar h, Uvry of Gorga, Gorga, 7,9 pp raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Pag 6
11 Expoal Marx ad hr Propr UHORS OGRPHY Mr Mohad.S. Sala. worg a a aa achr a h dpar of Mahac ad ac, ra Uvry, H a rarch ud worg h Swa Raaad rh Marahwada Uvry, Nadd. H ara of rarch lar algbra ad applcao varou fld. Dr. V. C. orar, worg a oca Profor ad Had Dpar of Mahac ad Sac Yhwa Mahavdyalaya Nadd, Udr Swa Raaad rh Marahwada Uvry, Nadd M.S da. H ara of pcalzao fucoal aly. H ha ar abou 6 yar rarch xprc. H copld o rarch proc o Dyacal y ad hr applcao. h proc wa poord by UGC, Nw Dlh. raoal Joural of Scfc ad ovav Mahacal Rarch JSMR Pag 6
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