Harakteristichnij polinom kvadratnoyi matrici A displaystyle A rozmiru n n displaystyle n times n ce mnogochlen stepenya n displaystyle n vid zminnoyi l displaystyle lambda yakij dorivnyuye p A l det A I n l displaystyle p A lambda det A I n lambda de I n displaystyle I n odinichna matricya poryadku n displaystyle n Zmist 1 Motivaciya 2 Vlastivosti 3 Harakteristichne rivnyannya 4 Div takozh 5 DzherelaMotivaciya RedaguvatiSkalyar l displaystyle lambda nbsp ye vlasnim znachennyam matrici A dlya vlasnogo vektora v displaystyle mathbf v nbsp todi i tilki todi koli A v l v displaystyle A mathbf v lambda mathbf v nbsp chi l I n A v 0 displaystyle lambda I n A mathbf v 0 nbsp Oskilki v 0 displaystyle mathbf v neq 0 nbsp to l I n A displaystyle lambda I n A nbsp povinna buti virodzhenoyu a otzhe det l I n A 0 displaystyle det lambda I n A 0 nbsp Vlastivosti RedaguvatiNevazhko perekonatisya shop A l l n tr A l n 1 1 n det A displaystyle p A lambda lambda n operatorname tr A lambda n 1 ldots 1 n det A nbsp Dlya matric elementi yakih komutativnimi ye Q displaystyle mathbb Q nbsp algebrami harakteristichnij mnogochlen mozhna zapisati yak p A l i 0 n 1 i q n i tr A tr A 2 tr A n i l i displaystyle p A lambda sum i 0 n 1 i q n i Bigl operatorname tr A operatorname tr A 2 ldots operatorname tr A n i Bigr lambda i nbsp de q n i displaystyle q n i nbsp mnogochleni iz racionalnimi koeficiyentami sho opisuyut zalezhnist elementarnih simetrichnih mnogochleniv vid stepenevih simetrichnih mnogochleniv u totozhnostyah Nyutona tobto e j q j p 1 p j displaystyle e j q j p 1 p j nbsp Harakteristichni polinomi podibnih matric zbigayutsya p B 1 A B l p A l displaystyle p B 1 AB lambda p A lambda nbsp Harakteristichni polinomi dobutku kvadratnih matric ne zalezhat vid poryadku mnozhnikiv p B A l p A B l displaystyle p BA lambda p AB lambda nbsp Harakteristichnij polinom vid samoyi matrici dorivnyuye nulovij matrici teorema Gamiltona Keli p A A 0 displaystyle p A A 0 nbsp Harakteristichne rivnyannya RedaguvatiHarakteristichnim rivnyannyam abo sekulyarnim rivnyannyam nazivayetsya rivnyannya p A l det l I n A 0 displaystyle p A lambda det lambda I n A 0 nbsp Koreni harakteristichnogo polinoma nazivayutsya harakteristichnimi chislami matrici A displaystyle A nbsp Tilki voni ye vlasnimi znachennyami matrici A displaystyle A nbsp Div takozh RedaguvatiViznachnik matrici Slid matriciDzherela 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 Otrimano z https uk wikipedia org w index php title Harakteristichnij polinom amp oldid 25550370 Harakteristichne rivnyannya
