Minorom k displaystyle k go poryadku matrici A displaystyle A nazivayetsya viznachnik matrici utvorenoyi elementami na peretini k displaystyle k stovpciv ta k displaystyle k ryadkiv Zmist 1 Formalne oznachennya 1 1 Pov yazani oznachennya 2 Prikladi 3 Vlastivosti 4 Div takozh 5 DzherelaFormalne oznachennya RedaguvatiNehaj A a i j displaystyle A a ij nbsp matricya rozmiru m n displaystyle m times n nbsp v yakij vibrano dovilni k displaystyle k nbsp k n k m displaystyle k leqslant n k leqslant m nbsp ryadkiv z nomerami i 1 lt i 2 lt lt i k displaystyle i 1 lt i 2 lt ldots lt i k nbsp ta stovpciv z nomerami j 1 lt j 2 lt lt j k displaystyle j 1 lt j 2 lt ldots lt j k nbsp Elementi sho znahodyatsya na peretini obranih ryadkiv ta stovpciv utvoryuyut kvadratnu matricyu poryadku k displaystyle k nbsp Viznachnik matrici yaka oderzhuyetsya z A displaystyle A nbsp vikreslyuvannyam vsih ryadkiv ta stovpciv okrim vibranih nazivayetsya minorom k displaystyle k nbsp go poryadku roztashovanim v ryadkah z nomerami i 1 i 2 i k displaystyle i 1 i 2 ldots i k nbsp ta stovpcyah z nomerami j 1 j 2 j k displaystyle j 1 j 2 ldots j k nbsp M j 1 j k i 1 i k det a i 1 j 1 a i 1 j 2 a i 1 j k a i k j 1 a i k j 2 a i k j k displaystyle M j 1 ldots j k i 1 ldots i k det begin pmatrix a i 1 j 1 amp a i 1 j 2 amp ldots amp a i 1 j k vdots amp vdots amp ddots amp vdots a i k j 1 amp a i k j 2 amp ldots amp a i k j k end pmatrix nbsp Yaksho A displaystyle A nbsp ye kvadratnoyu matriceyu viznachnik matrici yaka oderzhuyetsya vikreslyuvannyam tilki vibranih ryadkiv ta stovpciv z matrici A displaystyle A nbsp nazivayetsya dopovnyuvalnim minorom do minoru A j 1 j k i 1 i k displaystyle A j 1 ldots j k i 1 ldots i k nbsp M j 1 j k i 1 i k det a i k 1 j k 1 a i k 1 j k 2 a i k 1 j n a i n j k 1 a i n j k 2 a i n j n displaystyle overline M j 1 ldots j k i 1 ldots i k det begin pmatrix a i k 1 j k 1 amp a i k 1 j k 2 amp ldots amp a i k 1 j n vdots amp vdots amp ddots amp vdots a i n j k 1 amp a i n j k 2 amp ldots amp a i n j n end pmatrix nbsp de i k 1 lt lt i n displaystyle i k 1 lt ldots lt i n nbsp ta j k 1 lt lt j n displaystyle j k 1 lt ldots lt j n nbsp nomeri ne vibranih ryadkiv i stovpciv Pov yazani oznachennya Redaguvati Minorom M i j displaystyle M ij nbsp elementa a i j displaystyle a ij nbsp kvadratnoyi matrici A displaystyle A nbsp poryadku n displaystyle n nbsp nazivayetsya viznachnik n 1 poryadku yakij oderzhuyemo z viznachnika A displaystyle A nbsp n go poryadku shlyahom vikreslyuvannya i go ryadka ta j go stovpcya na peretini yakih znahoditsya element a i j displaystyle a ij nbsp Nehaj D k displaystyle Delta k nbsp deyakij minor poryadku k displaystyle k nbsp matrici A displaystyle A nbsp Minor poryadku k 1 displaystyle k 1 nbsp matrici nazivayetsya otochuyuchim dlya minora D k displaystyle Delta k nbsp yaksho jogo matricya mistit v sobi matricyu minoru D k displaystyle Delta k nbsp Takim chinom otochuyuchij minor dlya minora D k displaystyle Delta k nbsp mozhna oderzhati dopisuyuchi do nogo odin ryadok i odin stovpchik Bazisnim minorom nenulovoyi matrici A displaystyle A nbsp isnuye nenulovij element nazivayetsya minor yakij ne dorivnyuye nulyu a vsi jogo otochuyuchi minori dorivnyuyut nulyu abo yih ne isnuye Dovedennya isnuvannya bazisnogo minora utvorimo minor z yedinogo nenulovogo elementa i budemo rekursivno shukati nenulovi otochuyuchi minori azh do najbilshogo V zagalnomu vipadku v matrici mozhe isnuvati bagato bazisnih minoriv Rozmir bazisnogo minora matrici nazivayetsya rangom matrici Dlya m n displaystyle m times n nbsp matrici A displaystyle A nbsp minori vidu M i 1 i k i 1 i k displaystyle M i 1 ldots i k i 1 ldots i k nbsp de i 1 lt i 2 lt lt i k displaystyle i 1 lt i 2 lt ldots lt i k nbsp i k n k m displaystyle k leqslant n k leqslant m nbsp nazivayutsya golovnimi minorami Tobto dlya cih minoriv obirayutsya odnakovi nomeri dlya ryadkiv i stovpciv Golovni minori perevazhno rozglyadayut dlya kvadratnih matric Prikladi RedaguvatiRozglyanemo matricyu A displaystyle A nbsp rozmiru m n displaystyle m times n nbsp A a 11 a 12 a 13 a 1 n a 21 a 22 a 23 a 2 n a 31 a 32 a 33 a 3 n a m 1 a m 2 a m 2 a m n M 2 3 1 2 a 12 a 13 a 22 a 23 displaystyle A begin pmatrix a 11 amp a 12 amp a 13 amp cdots amp a 1n a 21 amp a 22 amp a 23 amp cdots amp a 2n a 31 amp a 32 amp a 33 amp cdots amp a 3n vdots amp vdots amp vdots amp ddots amp vdots a m1 amp a m2 amp a m2 amp cdots amp a mn end pmatrix qquad M 2 3 1 2 begin array cc a 12 amp a 13 a 22 amp a 23 end array nbsp minor 2 go poryadku Zagalom dlya ciyeyi matrici ye C m 2 C n 2 displaystyle C m 2 C n 2 nbsp minoriv drugogo poryadku Minor M 23 displaystyle M 23 nbsp kvadratnoyi matrici A displaystyle A nbsp viznachnik matrici otrimanij shlyahom vikreslyuvannya ryadka 2 ta stovpchika 3 A 1 4 7 3 0 5 1 9 11 M 23 1 4 1 9 displaystyle A begin pmatrix 1 amp 4 amp 7 3 amp 0 amp 5 1 amp 9 amp 11 end pmatrix qquad M 23 begin vmatrix 1 amp 4 amp Box Box amp Box amp Box 1 amp 9 amp Box end vmatrix nbsp displaystyle longrightarrow nbsp 1 4 1 9 9 4 13 displaystyle begin vmatrix 1 amp 4 1 amp 9 end vmatrix left 9 left 4 right right 13 nbsp Vlastivosti RedaguvatiDlya matrici A displaystyle A nbsp rozmiru m n displaystyle m times n nbsp isnuye C m k C n k displaystyle C m k cdot C n k nbsp riznih minoriv poryadku k displaystyle k nbsp de k n k m displaystyle k leqslant n k leqslant m nbsp Teorema Laplasa Nehaj A a i j displaystyle A a ij nbsp kvadratna matricya rozmiru n n displaystyle n times n nbsp v yakij vibrano dovilni k displaystyle k nbsp ryadkiv Todi viznachnik matrici A displaystyle A nbsp rivnij sumi vsilyakih dobutkiv minoriv k displaystyle k nbsp go poryadku roztashovanih v cih ryadkah na yih algebrayichni dopovnennya det A j 1 lt lt j k M j 1 j k i 1 i k A j 1 j k i 1 i k displaystyle det A sum j 1 lt ldots lt j k M j 1 ldots j k i 1 ldots i k A j 1 ldots j k i 1 ldots i k nbsp de pidsumovuvannya vedetsya po vsih nomerah stovpciv j 1 j k displaystyle j 1 ldots j k nbsp Chislo minoriv po yakih beretsya suma v teoremi Laplasa rivne chislu sposobiv vibrati k displaystyle k nbsp stovpciv z n displaystyle n nbsp tobto binomialnomu koeficiyentu n k displaystyle textstyle n choose k nbsp Oskilki ryadki i stovpci matrici rivnosilni shodo vlastivostej viznachnika teoremu Laplasa mozhna sformulyuvati i dlya stovpciv matrici Teorema pro bazisnij minorRyadki nenulovoyi matrici A displaystyle A nbsp na yakih buduyetsya yiyi bazisnij minor D r displaystyle Delta r nbsp ye linijno nezalezhnimi Vsi inshi ryadki matrici linijno virazhayutsya cherez nih Nehaj A displaystyle A nbsp ye matriceyu rozmiru m n displaystyle m times n nbsp B displaystyle B nbsp ye matriceyu rozmiru n p displaystyle n times p nbsp i C A B displaystyle C AB nbsp ye yih dobutkom Poznachatimemo A M B M C M displaystyle A M B M C M nbsp minori vidpovidnih matric Todi dlya minora C M j 1 j k i 1 i k displaystyle C M j 1 ldots j k i 1 ldots i k nbsp de k p k m displaystyle k leqslant p k leqslant m nbsp i i 1 lt i 2 lt lt i k displaystyle i 1 lt i 2 lt ldots lt i k nbsp ye nomerami ryadkiv a j 1 lt j 2 lt lt j k displaystyle j 1 lt j 2 lt ldots lt j k nbsp nomerami stovpciv yaksho k n displaystyle k geqslant n nbsp to C M j 1 j k i 1 i k 0 displaystyle C M j 1 ldots j k i 1 ldots i k 0 nbsp V inshomu vipadku cej minor oderzhuyetsya cherez minori matric A displaystyle A nbsp i B displaystyle B nbsp za dopomogoyu formuli C M j 1 j k i 1 i k 1 l 1 lt lt l k n A M l 1 l k i 1 i k B M j 1 j k l 1 l k displaystyle C M j 1 ldots j k i 1 ldots i k sum 1 leqslant l 1 lt ldots lt l k leqslant n A M l 1 ldots l k i 1 ldots i k cdot B M j 1 ldots j k l 1 ldots l k nbsp Dana formula ye uzagalnennyam formuli Bine Koshi Iz poperednogo uzagalnennya formuli Bine Koshi viplivaye sho suma golovnih minoriv odnakovogo poryadku matric A B displaystyle AB nbsp i B A displaystyle BA nbsp ye odnakovo Harakteristichnij mnogochlen p A l det A I n l displaystyle p A lambda det A I n lambda nbsp kvadratnoyi matrici A displaystyle A nbsp mozhna zapisati yak p A l l n i 1 n 1 i m i A l n i displaystyle p A lambda lambda n sum i 1 n 1 i m i A lambda n i nbsp de m i A displaystyle m i A nbsp poznachaye sumu golovnih minoriv poryadku i displaystyle i nbsp matrici A displaystyle A nbsp Yak naslidok sumi golovnih minoriv odnakovogo poryadku dvoh podibnih matric ye rivnimi Zokrema yedinim golovnim minorom maksimalnogo poryadku ye viznachnik a suma golovnih minoriv poryadku 1 nazivayetsya slidom matrici Div takozh Redaguvati nbsp Portal Matematika Teoriya matric Matricya matematika Viznachnik matrici Rang matrici Formula Bine KoshiDzherela RedaguvatiGantmaher F R Teoriya matric 5 e M Fizmatlit 2010 559 s ISBN 5 9221 0524 8 ros S S Shestakov S I Docenko Viznachniki matrici ta sistemi linijnih rivnyan nedostupne posilannya z lipnya 2019 Kurs lekcij z algebri dlya studentiv fakultetu kibernetiki Otrimano z https uk wikipedia org w index php title Minor matrici amp oldid 33200425
