Eksponenta matrici matrichna funkciya vid kvadratnoyi matrici sho maye bagato vlastivostej analogichnih zvichajnij eksponencijnij funkciyi dijsnih chi kompleksnih chisel Matrichna eksponenta vstanovlyuye zv yazok mizh algebroyu Li matric i vidpovidnoyu grupoyu Li Zmist 1 Viznachennya 1 1 Ekvivalentne viznachennya 2 Vlastivosti 2 1 Osnovni vlastivosti 2 2 Eksponenta sumi 2 2 1 Nerivnist Goldena Tompsona 2 2 2 Teorema Liba 3 Eksponencialne vidobrazhennya 3 1 Diferenciyuvannya 4 Sistemi linijnih diferencialnih rivnyan 4 1 Priklad odnoridnoyi sistemi 4 2 Priklad neodnoridnoyi sistemi 4 3 Uzagalnennya variaciya dovilnoyi staloyi 5 Div takozh 6 Primitki 7 Dzherela 8 PosilannyaViznachennya RedaguvatiDlya dijsnoyi abo kompleksnoyi matrici X displaystyle X nbsp rozmiru n n displaystyle n times n nbsp X M n C displaystyle X in M n mathbb C nbsp eksponenta vid X displaystyle X nbsp sho poznachayetsya yak e X displaystyle e X nbsp abo exp X displaystyle exp X nbsp matricya rozmiru n n displaystyle n times n nbsp viznachena za dopomogoyu ryadu e X k 0 1 k X k displaystyle e X sum k 0 infty 1 over k X k nbsp de X k displaystyle X k nbsp k a stepin matrici X displaystyle X nbsp Danij ryad zavzhdi zbigayetsya absolyutno Yaksho X a displaystyle X alpha nbsp de vzyato matrichnu normu uzgodzhenu z vektornoyu dlya skinchennovimirnih prostoriv usi normi ekvivalentni to 1 k X k 1 k X k 1 k a k displaystyle left frac 1 k X k right leqslant frac 1 k X k leqslant frac 1 k alpha k nbsp Zvidsi k 0 1 k X k k 0 1 k X k k 0 1 k a k e a displaystyle big sum k 0 infty 1 over k X k big leqslant sum k 0 infty 1 over k X k leqslant sum k 0 infty frac 1 k alpha k e alpha nbsp sho dovodit absolyutnu zbizhnist ryadu i korektnist viznachennya Yaksho X displaystyle X nbsp matricya rozmiru 1 1 displaystyle 1 times 1 nbsp to matrichna eksponenta vid X displaystyle X nbsp ye matricya rozmirnosti 1 1 displaystyle 1 times 1 nbsp yedinij element yakoyi dorivnyuye zvichajnij eksponenti vid yedinogo elementa X displaystyle X nbsp Ekvivalentne viznachennya Redaguvati Eksponencijnu funkciyu mozhna takozh viznachiti nastupnoyu rivnistyu exp X lim n I 1 n X n displaystyle exp X lim n to infty left I frac 1 n X right n nbsp de I displaystyle I nbsp odinichna matricya vidpovidnoyi rozmirnosti Cya rivnist ye analogichnoyu do rivnosti e a lim n 1 a n n displaystyle e a lim n to infty left 1 frac a n right n nbsp sho vikonuyetsya dlya dijsnih i kompleksnih chisel Dlya dovedennya rivnosti vikoristovuyetsya formula 1 a m m k 0 m a k k m k displaystyle left 1 frac a m right m sum k 0 m frac a k k m k nbsp de a mozhe buti yak chislom tak i matriceyu Todi yaksho dlya toyi zh normi sho j vishe X a displaystyle X a nbsp to e X I 1 n X n k 0 n a k k a k k n k k n 1 a k k e a 1 a n n 0 displaystyle left e X left I frac 1 n X right n right leqslant sum k 0 n frac a k k frac a k k n k sum k n 1 infty frac a k k e a left 1 frac a n right n to 0 nbsp pri n displaystyle n to infty nbsp sho j dovodit tverdzhennya Vlastivosti RedaguvatiOsnovni vlastivosti Redaguvati Dlya kompleksnih matric X displaystyle X nbsp i Y displaystyle Y nbsp rozmiru n n displaystyle n times n nbsp dovilnih kompleksnih chisel a displaystyle a nbsp i b displaystyle b nbsp odinichnoyi matrici I displaystyle I nbsp i nulovoyi matrici 0 displaystyle 0 nbsp eksponenta maye nastupni vlastivosti exp 0 I displaystyle exp 0 I nbsp Matrici X displaystyle X nbsp i exp X displaystyle exp X nbsp komutuyut tobto X exp X exp X X displaystyle X exp X exp X X nbsp Ce legko vivoditsya z viznachennya eksponenti yak sumi zbizhnogo ryadu kozhen dodanok yakogo ochevidno komutuye z X displaystyle X nbsp exp a X exp b X exp a b X displaystyle exp aX exp bX exp left a b X right nbsp exp X exp X I displaystyle exp X exp left X right I nbsp Yaksho X Y Y X displaystyle XY YX nbsp to exp X exp Y exp Y exp X exp X Y displaystyle exp X exp Y exp Y exp X exp X Y nbsp Yaksho Y displaystyle Y nbsp nevirodzhena matricya to exp Y X Y 1 Y exp X Y 1 displaystyle exp YXY 1 Y exp X Y 1 nbsp exp X T exp X T displaystyle exp X mathrm T exp X mathrm T nbsp de X T displaystyle X mathrm T nbsp poznachaye transponovanu matricyu do X displaystyle X nbsp ce oznachaye sho yaksho X displaystyle X nbsp ye simetrichnoyu to exp X displaystyle exp X nbsp tezh simetrichna a yaksho X displaystyle X nbsp kososimetrichna matricya to exp X displaystyle exp X nbsp ortogonalna exp X exp X displaystyle exp X exp X nbsp de X displaystyle X nbsp poznachaye ermitovo spryazhenu matricyu dlya X displaystyle X nbsp ce oznachaye sho yaksho X displaystyle X nbsp ermitova matricya to exp X displaystyle exp X nbsp tezh ermitova a yaksho X displaystyle X nbsp antiermitova matricya to exp X displaystyle exp X nbsp unitarna det exp X exp tr X displaystyle det left exp X right exp mbox tr X nbsp de det displaystyle det nbsp viznachnik a tr displaystyle mbox tr nbsp slid matrici Eksponenta sumi Redaguvati Dlya bud yakih dvoh dijsnih chisel skalyariv x displaystyle x nbsp i y displaystyle y nbsp eksponencialna funkciya zadovolnyaye rivnyannyu e x y e x e y displaystyle e x y e x cdot e y nbsp ce zh vlastivist maye misce dlya simetrichnih matric yaksho matrici X displaystyle X nbsp i Y displaystyle Y nbsp komutuyut tobto X Y Y X displaystyle XY YX nbsp to exp X Y exp X exp Y displaystyle exp X Y exp X exp Y nbsp Odnak dlya nekomutativnih matric cya rivnist vikonuyetsya ne zavzhdi v zagalnomu vipadku dlya obchislennya exp X Y displaystyle exp X Y nbsp vikoristovuyetsya formula Bejkera Kempbella Hausdorfa en U zagalnomu vipadku z rivnosti exp X Y exp X exp Y displaystyle exp X Y exp X exp Y nbsp ne viplivaye sho X displaystyle X nbsp i Y displaystyle Y nbsp komutuyut Dlya ermitovih matric isnuye dvi prosti teoremi pov yazani z slidom eksponent matric Nerivnist Goldena Tompsona Redaguvati Yaksho A displaystyle A nbsp i H displaystyle H nbsp ermitovi matrici to 1 tr exp A H tr exp A exp H displaystyle operatorname tr exp A H leqslant operatorname tr exp A exp H nbsp de tr X displaystyle operatorname tr X nbsp slid matrici X displaystyle X nbsp Komutativnist dlya vikonannya cogo tverdzhennya ne potribna Isnuyut kontrprikladi yaki pokazuyut sho nerivnist Goldena Tompsona ne mozhe buti uzagalnena na tri matrici a tr exp A exp B exp C displaystyle operatorname tr exp A exp B exp C nbsp ne zavzhdi ye dijsnim chislom dlya ermitovih matric A displaystyle A nbsp B displaystyle B nbsp i C displaystyle C nbsp Teorema Liba Redaguvati Teorema Liba nazvana im yam Eliota Liba stverdzhuye sho dlya fiksovanoyi ermitovoyi matrici H displaystyle H nbsp funkciya f A tr exp H log A displaystyle f A operatorname tr exp left H log A right nbsp ye uvignutoyu na konusi dodatnooznachenih matric 2 Eksponencialne vidobrazhennya RedaguvatiEksponenta matrici zavzhdi ye nevirodzhenoyu matriceyu Obernena do exp X displaystyle exp X nbsp matricya rivna exp X displaystyle exp X nbsp ce analog togo faktu sho eksponenta vid kompleksnogo chisla nikoli ne dorivnyuye nulyu Takim chinom matrichna eksponenta viznachaye vidobrazhennya exp M n C G L n C displaystyle exp colon M n mathbb C to mathrm GL n mathbb C nbsp z prostoru vsih matric rozmirnosti n n displaystyle n times n nbsp na zagalnu linijnu grupu poryadku n displaystyle n nbsp tobto grupu vsih nevirodzhenih matric rozmirnosti n n displaystyle n times n nbsp Ce vidobrazhennya ye syur yekciyeyu tobto kozhna nevirodzhena matricya mozhe buti zapisana yak eksponenta vid deyakoyi inshoyi matrici shob ce tverdzhennya bulo spravedlivim neobhidno rozglyadati pole kompleksnih chisel C displaystyle mathbb C nbsp a ne dijsnih chisel R displaystyle mathbb R nbsp Dlya bud yakih dvoh matric X displaystyle X nbsp i Y displaystyle Y nbsp maye misce nerivnist e X Y e X Y e X e Y displaystyle e X Y e X leqslant Y e X e Y nbsp de displaystyle cdot nbsp poznachaye dovilnu matrichnu normu Zvidsi viplivaye sho eksponencialne vidobrazhennya ye neperervnim i lipshicevim na kompaktnih pidmnozhinah M n C displaystyle M n mathbb C nbsp Zagalom eksponencijne vidobrazhennya ne ye in yektivnim Ale vono bude in yektivnim napriklad na pidmnozhini X B 0 ln 2 M n C displaystyle X in B 0 ln 2 subset M n mathbb C nbsp de B 0 ln 2 displaystyle B 0 ln 2 nbsp mnozhina matric norma yakih uzgodzhena z vektornoyu normoyu mensha nizh ln 2 Na cij mnozhini eksponencijna funkciya ye difeomorfizmom i obernena funkciya mozhe buti podana yak suma zbizhnogo ryadu log Y k 1 1 k 1 k Y I k displaystyle log Y sum k 1 infty 1 k 1 over k Y I k nbsp Diferenciyuvannya Redaguvati Vidobrazhennya t e t X t R displaystyle t mapsto e tX qquad t in mathbb R nbsp viznachaye gladku krivu v zagalnij linijnij grupi yaka prohodit cherez odinichnij element pri t 0 displaystyle t 0 nbsp Pohidna cogo vidobrazhennya viznachayetsya formuloyu d d t e t X X e t X displaystyle frac rm d rm d t e tX Xe tX nbsp Spravdi z viznachen pohidnoyi i vlastivostej eksponenti oderzhuyetsya poslidovnist rivnostej d d t e t X lim t 0 0 e t t 0 X e t X t 0 lim t 0 0 e t 0 X I t 0 e t X lim t 0 0 X O t 0 e t X X e t X displaystyle frac rm d rm d t e tX lim t 0 to 0 frac e t t 0 X e tX t 0 lim t 0 to 0 frac e t 0 X I t 0 e tX lim t 0 to 0 X O t 0 e tX Xe tX nbsp Bilsh zagalno dlya matrici X t zalezhnoyi vid parametra t spravedlivoyu ye rivnist 3 d d t e X t e X 1 e a d X a d X d X d t displaystyle frac rm d rm d t e X t e X frac 1 e rm ad X rm ad X frac rm d X rm d t nbsp de a d X M n C M n C displaystyle rm ad X M n mathbb C to M n mathbb C nbsp linijne vidobrazhennya viznachene a d X Y X Y X Y Y X displaystyle rm ad X Y X Y XY YX nbsp dlya dovilnoyi matrici Y M n C displaystyle Y in M n mathbb C nbsp U poperednij formuli dlya virazu v pravij chastini spravedliva formula 1 e a d X a d X k 0 1 k k 1 a d X k displaystyle frac 1 e rm ad X rm ad X sum k 0 infty 1 k over k 1 rm ad X k nbsp Vzyavshi v formuli dlya diferenciyuvannya X t X t Y displaystyle X t X tY nbsp otrimuyemo formulu dlya diferenciala eksponencijnogo vidobrazhennya v tochci X M n C displaystyle X in M n mathbb C nbsp d X exp Y exp d d t e X t Y e X 1 e a d X a d X Y Y M n C T X M n C displaystyle rm d X exp Y exp frac rm d rm d t e X tY e X frac 1 e rm ad X rm ad X Y forall Y in M n mathbb C simeq T X M n mathbb C nbsp Pri X Y Y X displaystyle XY YX nbsp cya rivnist sproshuyetsya do d X exp Y e X Y displaystyle rm d X exp Y e X Y nbsp Sistemi linijnih diferencialnih rivnyan RedaguvatiOdna z prichin yaki zumovlyuyut vazhlivist matrichnoyi eksponenti polyagaye v tomu sho vona mozhe buti vikoristana dlya rozv yazku sistem zvichajnih diferencialnih rivnyan 4 Rozv yazok sistemi d d t y t A y t y 0 y 0 displaystyle frac d dt y t Ay t quad y 0 y 0 nbsp de A displaystyle A nbsp stala matricya dayetsya virazom y t e A t y 0 displaystyle y t e At y 0 nbsp Matrichna eksponenta mozhe buti takozh vikoristana dlya rozv yazuvannya neodnoridnih rivnyan vidu d d t y t A y t z t y 0 y 0 displaystyle frac d dt y t Ay t z t quad y 0 y 0 nbsp Ne isnuye zamknutogo analitichnogo virazu dlya rishen neodnoridnih diferencialnih rivnyan vidu d d t y t A t y t y 0 y 0 displaystyle frac d dt y t A t y t quad y 0 y 0 nbsp de A displaystyle A nbsp matricya elementi yakoyi ne ye konstantami ale Rozklad Magnusa en dozvolyaye otrimati podannya rozv yazku u viglyadi neskinchennoyi sumi Priklad odnoridnoyi sistemi Redaguvati Dlya sistemi x 2 x y z y 3 y 1 z z 2 x y 3 z displaystyle begin matrix x amp amp 2x amp y amp z y amp amp amp 3y amp 1z z amp amp 2x amp y amp 3z end matrix nbsp matricya rivna A 2 1 1 0 3 1 2 1 3 displaystyle A begin bmatrix 2 amp 1 amp 1 0 amp 3 amp 1 2 amp 1 amp 3 end bmatrix nbsp Mozhna pokazati sho eksponenta vid matrici t A displaystyle tA nbsp ye e t A 1 2 e 2 t 1 e 2 t 2 t 2 t e 2 t e 2 t 1 e 2 t e 2 t 1 e 2 t 2 t 2 t 1 e 2 t e 2 t 1 e 2 t e 2 t 1 e 2 t 2 t 2 t e 2 t e 2 t 1 e 2 t displaystyle e tA frac 1 2 begin bmatrix e 2t 1 e 2t 2t amp 2te 2t amp e 2t 1 e 2t e 2t 1 e 2t 2t amp 2 t 1 e 2t amp e 2t 1 e 2t e 2t 1 e 2t 2t amp 2te 2t amp e 2t 1 e 2t end bmatrix nbsp takim chinom zagalnim rozv yazkom ciyeyi sistemi rivnyan ye x y z x 0 2 e 2 t 1 e 2 t 2 t e 2 t 1 e 2 t 2 t e 2 t 1 e 2 t 2 t y 0 2 2 t e 2 t 2 t 1 e 2 t 2 t e 2 t z 0 2 e 2 t 1 e 2 t e 2 t 1 e 2 t e 2 t 1 e 2 t displaystyle begin bmatrix x y z end bmatrix frac x 0 2 begin bmatrix e 2t 1 e 2t 2t e 2t 1 e 2t 2t e 2t 1 e 2t 2t end bmatrix frac y 0 2 begin bmatrix 2te 2t 2 t 1 e 2t 2te 2t end bmatrix frac z 0 2 begin bmatrix e 2t 1 e 2t e 2t 1 e 2t e 2t 1 e 2t end bmatrix nbsp Priklad neodnoridnoyi sistemi Redaguvati Dlya rozv yazku neodnoridnoyi sistemi x 2 x y z e 2 t y 3 y z z 2 x y 3 z e 2 t displaystyle begin matrix x amp amp 2x amp amp y amp amp z amp amp e 2t y amp amp amp amp 3y amp amp z amp z amp amp 2x amp amp y amp amp 3z amp amp e 2t end matrix nbsp vvodyatsya poznachennya A 2 1 1 0 3 1 2 1 3 displaystyle A left begin array rrr 2 amp 1 amp 1 0 amp 3 amp 1 2 amp 1 amp 3 end array right nbsp i b e 2 t 1 0 1 displaystyle mathbf b e 2t begin bmatrix 1 0 1 end bmatrix nbsp Tak yak suma zagalnogo rozv yazku odnoridnogo rivnyannya i chastkovogo rozv yazku dayut zagalnij rozv yazok neodnoridnogo rivnyannya zalishayetsya lishe znajti chastkovij rozv yazok Tak yak y p e t A 0 t e u A e 2 u 0 e 2 u d u e t A c displaystyle mathbf y p e tA int 0 t e u A begin bmatrix e 2u 0 e 2u end bmatrix du e tA mathbf c nbsp y p e t A 0 t 2 e u 2 u e 2 u 2 u e 2 u 0 2 e U 2 u 1 e 2 u 2 u 1 e 2 u 0 2 u e 2 u 2 u e 2 u 2 e u e 2 u 0 e 2 u d u e t A c displaystyle mathbf y p e tA int 0 t begin bmatrix 2e u 2ue 2u amp 2ue 2u amp 0 2e U 2 u 1 e 2u amp 2 u 1 e 2u amp 0 2ue 2u amp 2ue 2u amp 2e u end bmatrix begin bmatrix e 2u 0 e 2u end bmatrix du e tA mathbf c nbsp y p e t A 0 t e 2 u 2 e u 2 u e 2 u e 2 u 2 e u 2 1 u e 2 u 2 e 3 u 2 u e 4 u d u e t A c displaystyle mathbf y p e tA int 0 t begin bmatrix e 2u 2e u 2ue 2u e 2u 2e u 2 1 u e 2u 2e 3u 2ue 4u end bmatrix du e tA mathbf c nbsp y p e t A 1 24 e 3 t 3 e t 4 t 1 16 1 24 e 3 t 3 e t 4 t 4 16 1 24 e 3 t 3 e t 4 t 1 16 2 e t 2 t e 2 t 2 t e 2 t 0 2 e T 2 t 1 e 2 t 2 t 1 e 2 t 0 2 t e 2 t 2 t e 2 t 2 e t c 1 c 2 c 3 displaystyle mathbf y p e tA begin bmatrix 1 over 24 e 3t 3e t 4t 1 16 1 over 24 e 3t 3e t 4t 4 16 1 over 24 e 3t 3e t 4t 1 16 end bmatrix begin bmatrix 2e t 2te 2t amp 2te 2t amp 0 2e T 2 t 1 e 2t amp 2 t 1 e 2t amp 0 2te 2t amp 2te 2t amp 2e t end bmatrix begin bmatrix c 1 c 2 c 3 end bmatrix nbsp de c y p 0 displaystyle mathbf c mathbf y p 0 nbsp pochatkova umova Uzagalnennya variaciya dovilnoyi staloyi Redaguvati U razi neodnoridnoyi sistemi mozhna vikoristovuvati metod variaciyi dovilnoyi staloyi Shukayetsya chastkovij rozv yazok u viglyadi y p t exp t A z t displaystyle mathbf y p t exp tA mathbf z t nbsp y p t e t A z t e t A z t A e t A z t e t A z t A y p t e t A z t displaystyle begin aligned mathbf y p t amp e tA mathbf z t e tA mathbf z t 6pt amp Ae tA mathbf z t e tA mathbf z t 6pt amp A mathbf y p t e tA mathbf z t end aligned nbsp Shob y p displaystyle mathbf y p nbsp bula rozv yazkom maye vikonuvatisya nastupne e t A z t b t z t e t A 1 b t z t 0 t e u A b u d u c displaystyle begin aligned e tA mathbf z t amp mathbf b t 6pt mathbf z t amp e tA 1 mathbf b t 6pt mathbf z t amp int 0 t e uA mathbf b u du mathbf c end aligned nbsp Takim chinom y p t e t A 0 t e u A b u d u e t A c 0 t e t u A b u d u e t A c displaystyle begin aligned mathbf y p t amp e tA int 0 t e uA mathbf b u du e tA mathbf c amp int 0 t e t u A mathbf b u du e tA mathbf c end aligned nbsp de c displaystyle mathbf c nbsp viznachayetsya z pochatkovih umov zadachi Div takozh RedaguvatiEksponenta teoriya grup Li Primitki Redaguvati Bhatia R 1997 Matrix Analysis Graduate Texts in Mathematics 169 Springer ISBN 978 0 387 94846 1 EH Lieb 1973 Convex trace functions and the Wigner Yanase Dyson conjecture Adv Math 11 3 267 288 doi 10 1016 0001 8708 73 90011 X Rossman Wulf 2002 Lie Groups An Introduction Through Linear Groups angl Oxford Science Publications s 15 16 Yurij Golovatij Linijni sistemi zi stalimi koeficiyentami Arhivovano 13 zhovtnya 2016 u Wayback Machine Dzherela RedaguvatiBaker Andrew J 2003 Matrix Groups An Introduction to Lie Group Theory Berlin DE New York NY Springer Verlag ISBN 978 1 85233 470 3 Rossmann Wulf 2002 Lie Groups An Introduction Through Linear Groups Oxford Graduate Texts in Mathematics Oxford Science Publications ISBN 0 19 859683 9 Posilannya RedaguvatiWeisstein Eric W Matrix Exponential Arhivovano 19 listopada 2016 u Wayback Machine MathWorld Module for the Matrix Exponential Otrimano z https uk wikipedia org w index php title Eksponenta matrici amp oldid 38906241
