U matematici normoyu matrici vvazhayut rozshirennyam terminu vektornoyi normi na matrici Nehaj u prostori vektoriv R m displaystyle mathbb R m viznachena norma vektora displaystyle cdot Todi normoyu matrici A displaystyle A nazivayut chislo A sup x 0 A x x sup x 1 A x displaystyle A sup x neq 0 frac Ax x sup x 1 Ax Zmist 1 Pryami virazi 2 Vektorni normi 2 1 Norma Frobeniusa 3 Vlastivosti normi matrici 4 Uzgodzheni normi 5 Sumisni normi 6 DzherelaPryami virazi RedaguvatiU zalezhnosti vid konkretnoyi normi dlya vektoriv mozhna znajti pryami virazi dlya normi matrici Nizhche navedeni tri poshireni normi x max 1 j m x j displaystyle x infty max 1 leq j leq m left x j right nbsp Todi A max 1 i m j 1 m a i j displaystyle A infty max 1 leq i leq m left sum j 1 m a ij right nbsp x 1 j 1 m x j displaystyle x 1 sum j 1 m left x j right nbsp Todi A 1 max 1 j m i 1 m a i j displaystyle A 1 max 1 leq j leq m left sum i 1 m left a ij right right nbsp x 2 j 1 m x j 2 x x displaystyle x 2 sqrt sum j 1 m left x j right 2 sqrt x x nbsp Todi A 2 max 1 i m l A T A i displaystyle A 2 sqrt max 1 leq i leq m lambda A T cdot A i nbsp de l D i displaystyle lambda D i nbsp vlasni znachennya matrici D displaystyle D nbsp Vektorni normi RedaguvatiMatricyu rozmirnosti m n displaystyle m times n nbsp mozhna traktuvati yak vektor dovzhini m n displaystyle mn nbsp i zastosovuvati do nogo normu vektora Norma Frobeniusa Redaguvati Viglyadaye tak A F i 1 m j 1 n a i j 2 displaystyle A F sqrt sum i 1 m sum j 1 n a ij 2 nbsp Vlastivosti normi matrici RedaguvatiHaj K displaystyle K nbsp poznachaye pole z dijsnih chi kompleksnih chisel Haj K m n displaystyle K m times n nbsp poznachaye vektornij prostir sho mistit vsi matrici z m displaystyle m nbsp ryadkiv ta n displaystyle n nbsp stovpciv z elementami tipu K displaystyle K nbsp Yaksho A displaystyle A nbsp poznachaye normu matrici A displaystyle A nbsp todi dlya neyi vikonuyutsya taki vlastivosti A gt 0 displaystyle A gt 0 nbsp yaksho A 0 displaystyle A neq 0 nbsp ta A 0 displaystyle A 0 nbsp todi i tilki todi koli A 0 displaystyle A 0 nbsp a A a A a K displaystyle alpha A alpha cdot A qquad forall alpha in K nbsp ta A K m n displaystyle forall A in K m times n nbsp A B A B A B K m n displaystyle A B leq A B qquad forall A B in K m times n nbsp Krim togo u vipadku kvadratnih matric deyaki ne vsi normi zadovolnyayut nastupnu vlastivist yaka pov yazana z tim sho matrici ce bilsh nizh vektor A B A B displaystyle AB leq A B nbsp dlya vsih A displaystyle A nbsp ta B displaystyle B nbsp z K n n displaystyle K n times n nbsp Norma matrici sho zadovilnyaye cyu vlastivist nazivayetsya submultiplikativnoyu normoyu deyaki pidruchniki vikoristovuyut termin norma matrici viklyuchno dlya submultiplikativnih norm Mnozhina kvadratnih matric z normoyu sho zadovolnyaye ostannyu vlastivist utvoryuye banahovu algebru Uzgodzheni normi RedaguvatiMatrichna norma displaystyle cdot nbsp na K m n displaystyle K m times n nbsp nazivayetsya uzgodzhenoyu angl consistent z vektornimi normami a displaystyle cdot a nbsp i b displaystyle cdot b nbsp na K n displaystyle K n nbsp i K m displaystyle K m nbsp vidpovidno yaksho A x b A x a displaystyle Ax b leq A x a nbsp dlya vsih A K m n x K n displaystyle A in K m times n x in K n nbsp Usi indukovani normi uzgodzhenni za oznachennyam Sumisni normi RedaguvatiMatrichna norma displaystyle cdot nbsp na K n n displaystyle K n times n nbsp nazivayetsya sumisnoyu angl compatible z vektornoyu normoyu a displaystyle cdot a nbsp na K n displaystyle K n nbsp yaksho A x a A x a displaystyle Ax a leq A x a nbsp dlya vsih A K n n x K n displaystyle A in K n times n x in K n nbsp Indukovana norma sumisna za oznachennyam Dzherela RedaguvatiGantmaher F R Teoriya matric 5 e M Fizmatlit 2010 559 s ISBN 5 9221 0524 8 ros Bahvalov N S Zhidkov N P Kobelkov G G Chislennye metody Otrimano z https uk wikipedia org w index php title Norma matrici amp oldid 36670360 Norma Frobeniusa
