U Vikipediyi ye statti pro inshi znachennya cogo termina slid Slid matrici operaciya sho vidobrazhaye prostir kvadratnih matric u pole nad yakim viznachena matricya div funkcional Slid matrici ce suma usih yiyi diagonalnih elementiv tobto yaksho a i j displaystyle a ij elementi matrici A displaystyle A yiyi slid dorivnyuye t r A S p A i a i i displaystyle mathop rm tr A mathop rm Sp A sum i a ii V matematichnih tekstah zustrichayetsya dva poznachennya operaciyi vzyattya slidu t r A displaystyle mathop rm tr A trejs vid angl Trace slid i S p A displaystyle mathop rm Sp A shpur vid nim Spur slid Zmist 1 Vlastivosti 2 Vnutrishnij dobutok 3 Div takozh 4 DzherelaVlastivosti RedaguvatiLinijnistt r a A b B a t r A b t r B displaystyle mathop rm tr alpha A beta B alpha mathop rm tr A beta mathop rm tr B nbsp Ciklichnistt r A B t r B A displaystyle mathop rm tr AB mathop rm tr BA nbsp t r A B C t r B C A t r C A B displaystyle mathop rm tr ABC mathop rm tr BCA mathop rm tr CAB nbsp t r A t r A T displaystyle mathop rm tr A mathop rm tr A T nbsp de T oznachaye operaciyu transponuvannya Slid podibnih matric odnakovijtr P 1 A P tr A P P 1 tr A displaystyle operatorname tr P 1 AP operatorname tr APP 1 operatorname tr A nbsp Yaksho A B displaystyle A otimes B nbsp dobutok Kronekera matric A ta B tot r A B t r A t r B displaystyle mathop rm tr A otimes B mathop rm tr A mathop rm tr B nbsp Slid matrici dorivnyuye sumi yiyi vlasnih znachen f t r A t r f A displaystyle f rm trA mathop rm tr f A nbsp Vnutrishnij dobutok RedaguvatiDlya matrici A rozmirom m na n z kompleksnimi chi dijsnimi elementami de A poznachaye ermitovo spryazhenu matricyu mayemo nerivnist tr A A 0 displaystyle operatorname tr A A geq 0 nbsp yaka peretvoryuyetsya v rivnist todi i tilki todi koli A 0 displaystyle A 0 nbsp Prisvoyennya A B tr A B displaystyle langle A B rangle operatorname tr A B nbsp daye vnutrishnij dobutok na prostori vsih kompleksnih chi dijsnih matric rozmiru m na n Normu yaku otrimuyut z vishenavedenogo vnutrishnogo dobutku nazivayut normoyu Frobeniusa yaka zadovilnyaye vlastivist submultiplikativnosti dlya norm matric Spravdi ce prosto Evklidova norma yaksho vvazhati matricyu vektorom dovzhini mn Yaksho A i B dijsni dodatnonapivoznacheni matrici odnakovogo rozmiru to vikonuyetsya rivnist 0 tr A B 2 tr A 2 tr B 2 tr A 2 tr B 2 displaystyle 0 leq operatorname tr AB 2 leq operatorname tr A 2 operatorname tr B 2 leq operatorname tr A 2 operatorname tr B 2 nbsp Yiyi mozhna dovesti vikoristavshi nerivnist Koshi Bunyakovskogo Div takozh RedaguvatiTeoriya matric Viznachnik matrici Rang matriciDzherela RedaguvatiGantmaher F R Teoriya matric 5 e M Fizmatlit 2010 559 s ISBN 5 9221 0524 8 ros Otrimano z https uk wikipedia org w index php title Slid matrici amp oldid 24203381
