Kvadratna matricya A displaystyle A z kompleksnimi elementami mozhe buti predstavlena yak dobutok unitarnoyi matrici ta nevid yemnoyi ermitovoyi matrici A P U U P 1 displaystyle A PU UP 1 de P P 1 displaystyle P P 1 nevid yemnooznacheni matrici U displaystyle U unitarna matricya Matricya A displaystyle A bude normalnoyu todi i tilki todi koli P U displaystyle P U budut perestavnimi sho rivnoznachno do P P 1 displaystyle P P 1 Dlya dovedennya vikoristayemo singulyarnij rozklad matrici A W S V displaystyle A W Sigma V Zmist 1 Znahodzhennya modulya 2 Znahodzhennya povorotu 3 Polyarnij rozklad normalnoyi matrici 4 DzherelaZnahodzhennya modulya RedaguvatiOskilki A A P U U P P 2 displaystyle AA PUU P P 2 nbsp A A P 1 U U P 1 P 1 2 displaystyle A A P 1 UU P 1 P 1 2 nbsp matrici P P 1 displaystyle P P 1 nbsp odnoznachno viznachayutsya yak P A A W S W displaystyle P sqrt AA W Sigma W nbsp P 1 A A V S V displaystyle P 1 sqrt A A V Sigma V nbsp Yaksho matricya A displaystyle A nbsp normalna to A A A A displaystyle A A AA nbsp za viznachennyam Znahodzhennya povorotu RedaguvatiVikoristavshi U P 1 A displaystyle U P 1 A nbsp otrimayemo U W V displaystyle U WV nbsp Vikoristavshi U A P 1 1 displaystyle U AP 1 1 nbsp znovu zh otrimayemo U W V displaystyle U WV nbsp Polyarnij rozklad normalnoyi matrici RedaguvatiYaksho matricya A displaystyle A nbsp normalna todi matrici P U S displaystyle P U Sigma nbsp ye perestavnimi ta normalnimi otzhe odnochasno diagonalizuyemimi Q S F Q S Q P Q F Q U displaystyle exists Q Sigma Phi quad Q Sigma Q P quad Q Phi Q U nbsp de Q displaystyle Q nbsp unitarna matricya S displaystyle Sigma nbsp nevid yemnooznachena diagonalna matricya F displaystyle Phi nbsp unitarna diagonalna matricya Todi A Q S Q Q F Q Q S F Q displaystyle A Q Sigma Q Q Phi Q Q Sigma Phi Q nbsp vlasnij rozklad matrici Dzherela 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 Polyarnij rozklad matrici amp oldid 28434051
