U linijnij algebri vlasnij rozklad abo spektralnij rozklad ce rozklad matrici v kanonichnu formu takim chinom mi predstavlyayemo matricyu v terminah yiyi vlasnih znachen i vlasnih vektoriv Tilki diagonalizovni matrici mozhna tak rozklasti Zmist 1 Fundamentalna teoriya vlasnih vektoriv i znachen matrici 2 Vlasnij rozklad matrici 2 1 Priklad 2 2 Obernena matricya cherez vlasnij rozklad 3 DzherelaFundamentalna teoriya vlasnih vektoriv i znachen matrici RedaguvatiDokladnishe Vlasnij vektorVektor nenulovij v rozmirnosti N ye vlasnim vektorom kvadratnoyi N N matrici A todi i tilki todi koli vin zadovolnyaye linijnomu rivnyannyu A v l v displaystyle mathbf A mathbf v lambda mathbf v nbsp de l ce skalyar termin vlasne znachennya stosuyetsya v Tobto vlasni vektori ce taki vektori yaki linijne peretvorennya A lishe roztyaguye abo skorochuye i koeficiyent roztyaguvannya skorochennya i ye vlasnim znachennyam Zvidsi pohodit rivnyannya dlya vlasnih znachen p l det A l I 0 displaystyle p left lambda right det left mathbf A lambda mathbf I right 0 nbsp Mi zvemo p l harakteristichnim mnogochlenom a rivnyannya nazivayut harakteristichnim rivnyannyam vono yavlyaye soboyu mnogochlenom poryadku N z nevidomoyu l Ce rivnyannya matime Nl vidminnih rozv yazkiv de 1 Nl N Mnozhinu rozv yazkiv tobto vlasnih znachen inodi zvut spektrom A Mi mozhemo rozklasti p na mnozhniki p l l l 1 n 1 l l 2 n 2 l l k n k 0 displaystyle p left lambda right lambda lambda 1 n 1 lambda lambda 2 n 2 cdots lambda lambda k n k 0 nbsp Cile ni nazivayetsya algebrichnoyu kratnistyu vlasnogo znachennya li Suma vsih algebrayichnim kratnostej dorivnyuye N i 1 N l n i N displaystyle sum limits i 1 N lambda n i N nbsp Dlya kozhnogo vlasnogo znachennya li mi mayemo osoblive rivnyannya A l i I v 0 displaystyle left mathbf A lambda i mathbf I right mathbf v 0 nbsp Vsogo bude 1 mi ni linijno nezalezhnih rozv zyakiv dlya kozhnogo vlasnogo znachennya mi rozv yazkiv budut vlasnimi vektorami pov yazanimi z vlasnim znachennyam li Cile mi nazivayut geometrichnoyu kratnistyu li Vazhlivo pam yatati sho algebrayichne ni i geometrichne mi kratni mozhut buti odnakovimi i riznimi ale zavzhdi mi ni Najprostishij vipadok ce koli mi ni 1 Zagalna kilkist linijno nezalezhnih vlasnih vektoriv Nv mozhna diznatis dodavshi geometrichni kratnosti i 1 N l m i N v displaystyle sum limits i 1 N lambda m i N mathbf v nbsp Vlasni vektori mozhna proindeksuvati po yih vlasnim znachennyam tobto iz vikoristannyam podvijnogo indeksuvannya z vi j de jj vlasnij vektor igo vlasnogo znachennya Takozh ce mozhna zrobiti z odnim indeksom vk z k 1 2 Nv Vlasnij rozklad matrici RedaguvatiNehaj A bude kvadratnoyu N N matriceyu z N linijno nezalezhnimi vlasnimi vektorami q i i 1 N displaystyle q i i 1 dots N nbsp Todi A mozhna rozklasti yak A Q L Q 1 displaystyle mathbf A mathbf Q mathbf Lambda mathbf Q 1 nbsp de Q ce kvadratna N N matricya chiyi i ti stovpchiki ye vlasnimi vektorami q i displaystyle q i nbsp A i L ce diagonalna matricya chiyi diagonalni elementi ye vidpovidnimi vlasnimi znachennyami tobto L i i l i displaystyle Lambda ii lambda i nbsp Zauvazhte sho tilki diagonoalizovni matrici mozhna rozklasti takim chinom Napriklad matricyu sho na maye N 2 nezalezhnih vlasnih vektoriv 1 1 0 1 displaystyle begin pmatrix 1 amp 1 0 amp 1 end pmatrix nbsp ne mozhna diagonalizuvati Zazvichaj vlasni vektori q i i 1 N displaystyle q i i 1 dots N nbsp normalizuyut ale v comu nemaye potrebi Nenormalizovanij nabir vlasnih vektoriv v i i 1 N displaystyle v i i 1 dots N nbsp takozh mozhna vikoristovuvati yak stovpchiki dlya Q Ce mozhna zrozumiti zauvazhivshi sho velichina vlasnih vektoriv u Q znikaye v rozkladi zavdyaki prisutnosti Q 1 Priklad Redaguvati Yaksho za priklad dlya dekompoziciyi cherez mnozhennya na nesingulyarnu matricyu B a b c d a b c d R displaystyle mathbf B begin bmatrix a amp b c amp d end bmatrix a b c d in mathbb R nbsp v diagonalnu matricyu vzyati dijsnu matricyu A 1 0 1 3 displaystyle mathbf A begin bmatrix 1 amp 0 1 amp 3 end bmatrix nbsp Todi a b c d 1 1 0 1 3 a b c d x 0 0 y displaystyle begin bmatrix a amp b c amp d end bmatrix 1 begin bmatrix 1 amp 0 1 amp 3 end bmatrix begin bmatrix a amp b c amp d end bmatrix begin bmatrix x amp 0 0 amp y end bmatrix nbsp dlya deyakoyi dijsnoyi diagonalnoyi matrici x 0 0 y displaystyle begin bmatrix x amp 0 0 amp y end bmatrix nbsp Perenesemo B displaystyle mathbf B nbsp na pravij bik 1 0 1 3 a b c d a b c d x 0 0 y displaystyle begin bmatrix 1 amp 0 1 amp 3 end bmatrix begin bmatrix a amp b c amp d end bmatrix begin bmatrix a amp b c amp d end bmatrix begin bmatrix x amp 0 0 amp y end bmatrix nbsp Poperednye rivnyannya mozhna roznesti v sistemu z dvoh rivnyan 1 0 1 3 a c a x c x 1 0 1 3 b d b y d y displaystyle begin cases begin bmatrix 1 amp 0 1 amp 3 end bmatrix begin bmatrix a c end bmatrix begin bmatrix ax cx end bmatrix begin bmatrix 1 amp 0 1 amp 3 end bmatrix begin bmatrix b d end bmatrix begin bmatrix by dy end bmatrix end cases nbsp Vinesemo vlasni znachennya x displaystyle x nbsp i y displaystyle y nbsp 1 0 1 3 a c x a c 1 0 1 3 b d y b d displaystyle begin cases begin bmatrix 1 amp 0 1 amp 3 end bmatrix begin bmatrix a c end bmatrix x begin bmatrix a c end bmatrix begin bmatrix 1 amp 0 1 amp 3 end bmatrix begin bmatrix b d end bmatrix y begin bmatrix b d end bmatrix end cases nbsp Poklavshi a a c b b d displaystyle overrightarrow a begin bmatrix a c end bmatrix overrightarrow b begin bmatrix b d end bmatrix nbsp otrimayemo dva vektornih rivnyannya A a x a A b y b displaystyle begin cases A overrightarrow a x overrightarrow a A overrightarrow b y overrightarrow b end cases nbsp I ce mozhna predstaviti yak odne vektorne rivnyannya yake maye dva rozv yazki yak vlasni znachennya A u l u displaystyle mathbf A mathbf u lambda mathbf u nbsp de l displaystyle lambda nbsp predstavlyaye dva vlasnih znachennya x displaystyle x nbsp i y displaystyle y nbsp u displaystyle mathbf u nbsp predstavlyaye vektori a displaystyle overrightarrow a nbsp i b displaystyle overrightarrow b nbsp Perenesemo l u displaystyle lambda mathbf u nbsp livoruch i vinesemo za duzhki u displaystyle mathbf u nbsp A l I u 0 displaystyle mathbf A lambda mathbf I mathbf u 0 nbsp Cherez te sho B displaystyle mathbf B nbsp nesingulyarna tut vazhlivo sho u displaystyle mathbf u nbsp ne nul A l I 0 displaystyle mathbf A lambda mathbf I mathbf 0 nbsp Rozglyadayuchi viznachnik A l I displaystyle mathbf A lambda mathbf I nbsp 1 l 0 1 3 l 0 displaystyle begin bmatrix 1 lambda amp 0 1 amp 3 lambda end bmatrix 0 nbsp Otzhe 1 l 3 l 0 displaystyle 1 lambda 3 lambda 0 nbsp Otrimavshi l 1 displaystyle lambda 1 nbsp i l 3 displaystyle lambda 3 nbsp yak rozv yazki vlasnih znachen dlya matrici A displaystyle mathbf A nbsp mayemo v rezultati diagonalnu matricyu 1 0 0 3 displaystyle begin bmatrix 1 amp 0 0 amp 3 end bmatrix nbsp vlasnogo rozkladu A displaystyle mathbf A nbsp Vpishemo rozv yazki v sistemu rivnyan 1 0 1 3 a c 1 a c 1 0 1 3 b d 3 b d displaystyle begin cases begin bmatrix 1 amp 0 1 amp 3 end bmatrix begin bmatrix a c end bmatrix 1 begin bmatrix a c end bmatrix begin bmatrix 1 amp 0 1 amp 3 end bmatrix begin bmatrix b d end bmatrix 3 begin bmatrix b d end bmatrix end cases nbsp Rozv yazavshi rivnyannya mi mayemo a 2 c a R displaystyle a 2c a in mathbb R nbsp and b 0 b R displaystyle b 0 b in mathbb R nbsp Otzhe matricya B displaystyle mathbf B nbsp potribna dlya vlasnogo rozkladu matrici A displaystyle mathbf A nbsp ye 2 c 0 c d c d R displaystyle begin bmatrix 2c amp 0 c amp d end bmatrix c d in mathbb R nbsp tobto 2 c 0 c d 1 1 0 1 3 2 c 0 c d 1 0 0 3 c d R displaystyle begin bmatrix 2c amp 0 c amp d end bmatrix 1 begin bmatrix 1 amp 0 1 amp 3 end bmatrix begin bmatrix 2c amp 0 c amp d end bmatrix begin bmatrix 1 amp 0 0 amp 3 end bmatrix c d in mathbb R nbsp Obernena matricya cherez vlasnij rozklad Redaguvati Dokladnishe Obernena matricyaYaksho matricya A maye vlasnij rozklad i yaksho zhodne z yiyi vlasnih znachen ne dorivnyuye nulyu todi A nesingulyarna tobto moye obernenu i obernena zadayetsya tak A 1 Q L 1 Q 1 displaystyle mathbf A 1 mathbf Q mathbf Lambda 1 mathbf Q 1 nbsp Dali bilshe cherez te sho L diagonalna yiyi obernenu duzhe legko obchisliti L 1 i i 1 l i displaystyle left Lambda 1 right ii frac 1 lambda i nbsp Dzherela RedaguvatiGantmaher F R Teoriya matric 5 e M Fizmatlit 2010 559 s ISBN 5 9221 0524 8 ros Gelfand I M Lekcii po linejnoj algebre 5 e Moskva Nauka 1998 320 s ISBN 5791300158 ros Weisstein Eric W Vlasnij rozklad matrici angl na sajti Wolfram MathWorld Otrimano z https uk wikipedia org w index php title Vlasnij rozklad matrici amp oldid 36886380
