Teore ma Ga miltona Ke li na chest Vilyama Gamiltona ta Artura Keli stverdzhuye sho rezultat pidstanovki kvadratnoyi matrici A displaystyle A do yiyi harakteristichnogo polinoma totozhno dorivnyuye nulyu p A A 0 displaystyle p A A 0 Teorema Gamiltona Keli dozvolyaye viraziti polinomi visokogo stepenya vid n n displaystyle n times n matrici A displaystyle A yak linijni kombinaciyi A n 1 A I displaystyle A n 1 ldots A I Tverdzhennya teoremi ye spravedlivim dlya matric iz elementami iz bud yakogo komutativnogo kilcya z odiniceyu zokrema bud yakogo polya Zmist 1 Poyasnennya ta prikladi 1 1 Priklad 2 Dovedennya 2 1 Chastkovi vipadki 2 2 Zagalnij vipadok 3 DzherelaPoyasnennya ta prikladi RedaguvatiOskilki rezultatom dodavannya mnozhennya ta mnozhennya na skalyar kvadratnih matric ye kvadratna matricya to mozhna konstruyuvati polinomi z matric Tomu dlya dovilnogo polinoma f x a 0 x k a 1 x k 1 a k 1 x a k displaystyle f x a 0 x k a 1 x k 1 ldots a k 1 x a k nbsp mozhlivo rozglyanuti viraz f A a 0 A k a 1 A k 1 a k 1 A a k I displaystyle f A a 0 A k a 1 A k 1 ldots a k 1 A a k I nbsp yakij ye kvadratnoyu matriceyu togo samogo poryadka sho j A displaystyle A nbsp Priklad Redaguvati A 0 1 2 3 p A l l 2 3 l 2 displaystyle A begin bmatrix 0 amp 1 2 amp 3 end bmatrix quad p A lambda lambda 2 3 lambda 2 nbsp Todi A 2 0 1 2 3 0 1 2 3 2 3 6 11 p A A A 2 3 A 2 I 2 3 6 11 0 3 6 9 2 0 0 2 0 0 0 0 displaystyle A 2 begin bmatrix 0 amp 1 2 amp 3 end bmatrix begin bmatrix 0 amp 1 2 amp 3 end bmatrix begin bmatrix 2 amp 3 6 amp 11 end bmatrix qquad p A A A 2 3A 2I begin bmatrix 2 amp 3 6 amp 11 end bmatrix begin bmatrix 0 amp 3 6 amp 9 end bmatrix begin bmatrix 2 amp 0 0 amp 2 end bmatrix begin bmatrix 0 amp 0 0 amp 0 end bmatrix nbsp Dovedennya RedaguvatiChastkovi vipadki Redaguvati Dovedemo teoremu dlya matric 2x2 Mayemo A a b c d p A l l 2 tr A l det A displaystyle A begin bmatrix a amp b c amp d end bmatrix quad p A lambda lambda 2 operatorname tr A lambda det A nbsp tomup A A A 2 a d A a d b c I a 2 b c a b b d c a d c c b d 2 a d a a d b a d c a d d a d b c 0 0 a d b c 0 0 0 0 displaystyle p A A A 2 a d A ad bc I begin bmatrix a 2 bc amp ab bd ca dc amp cb d 2 end bmatrix begin bmatrix a d a amp a d b a d c amp a d d end bmatrix begin bmatrix ad bc amp 0 0 amp ad bc end bmatrix begin bmatrix 0 amp 0 0 amp 0 end bmatrix nbsp Rozglyanemo vipadok diagonalnih matric Yaksho A diag l 1 l n displaystyle A operatorname diag lambda 1 ldots lambda n nbsp diagonalna matricya i f x displaystyle f x nbsp polinom to f A diag f l 1 f l n displaystyle f A operatorname diag f lambda 1 ldots f lambda n nbsp Dlya harakteristichnogo polinoma p A l 1 p A l n 0 displaystyle p A lambda 1 ldots p A lambda n 0 nbsp tomu oderzhuyemo p A A diag 0 0 displaystyle p A A operatorname diag 0 ldots 0 nbsp Zagalnij vipadok Redaguvati Poznachimo cherez B displaystyle B nbsp soyuznu matricyu dlya harakteristichnoyi matrici l I n A displaystyle lambda I n A nbsp Elementi matrici V ye algebrayichnimi dopovnennyami elementiv viznachnika l I n A displaystyle lambda I n A nbsp i tomu ye mnogochlenami vid l stepeni ne vishe n 1 Otzhe matricyu V mozhna predstaviti u viglyadi polinoma z matrichnimi koeficiyentami B i 0 n 1 l i B i displaystyle B sum i 0 n 1 lambda i B i nbsp Za vlastivostyami soyuznih matric B l I n A det l I n A I n p l I n displaystyle B cdot lambda I n A det lambda I n A I n p lambda I n nbsp Nehaj p l l n l n 1 c n 1 l c 1 c 0 displaystyle p lambda lambda n lambda n 1 c n 1 cdots lambda c 1 c 0 nbsp Pidstavimo i otrimayemo i 0 n 1 l i B i l I n A l n I n l n 1 c n 1 I n l c 1 I n c 0 I n displaystyle sum i 0 n 1 lambda i B i lambda I n A lambda n I n lambda n 1 c n 1 I n cdots lambda c 1 I n c 0 I n nbsp Rozkrivayuchi duzhki i pririvnyavshi koeficiyenti pri odnakovih stepenyah l oderzhimo I n B n 1 displaystyle I n B n 1 nbsp c i I n B i 1 B i A 0 lt i lt n displaystyle c i I n B i 1 B i cdot A qquad 0 lt i lt n nbsp c 0 I n B 0 A displaystyle c 0 I n B 0 cdot A nbsp Pomnozhimo ci rivnosti vidpovidno na A n A n 1 I n displaystyle A n A n 1 ldots I n nbsp sprava i dodamo Vsi chleni pravoyi chastini skorotyatsya i mi oderzhimo p A A n c n 1 A n 1 c 1 A c 0 I n 0 displaystyle p A A n c n 1 A n 1 cdots c 1 A c 0 I n 0 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 Malcev A I Osnovy linejnoj algebry 3 e izd Novosibirsk Nauka 1970 400 s ros Otrimano z https uk wikipedia org w index php title Teorema Gamiltona Keli amp oldid 34244580
