Matricya ye diagonalno panivnoyu yaksho dlya kozhnogo ryadku velichina diagonalnogo elementu kozhnogo ryadka bilsha abo dorivnyuye sumi velichin usih inshih nediagonalnih elementiv cogo ryadka Tochnishe u matrici A displaystyle A panivna diagonal yaksho a i i j i a i j i displaystyle a ii geq sum j neq i a ij forall i Zauvazhte sho ce viznachennya poslugovuyetsya slabkoyu nerivnistyu i cherez ce inodi jogo nazivayut slabke diagonalne panuvannya Yaksho vikoristati strogu nerivnist gt jogo nazivayut stroge diagonalne panuvannya Termin diagonalne panuvannya mozhe oznachati yak stroge tak i slabke diagonalne panuvannya zalezhno vid kontekstu 1 Zmist 1 Prikladi 2 Zastosuvannya i vlastivosti 3 Dzherela 4 PrimitkiPrikladi RedaguvatiMatricya A 3 2 1 1 3 2 1 2 4 displaystyle mathbf A begin bmatrix 3 amp 2 amp 1 1 amp 3 amp 2 1 amp 2 amp 4 end bmatrix nbsp daye a 11 a 12 a 13 displaystyle a 11 geq a 12 a 13 nbsp oskilki 3 2 1 displaystyle 3 geq 2 1 nbsp a 22 a 21 a 23 displaystyle a 22 geq a 21 a 23 nbsp oskilki 3 1 2 displaystyle 3 geq 1 2 nbsp a 33 a 31 a 32 displaystyle a 33 geq a 31 a 32 nbsp oskilki 4 1 2 displaystyle 4 geq 1 2 nbsp Cherez te sho velichina kozhnogo diagonalnogo elementu bilsha abo dorivnyuye sumi velichin elementiv u ryadku kazhut sho matricya A displaystyle mathbf A nbsp diagonalno panivna abo maye panivnu diagonal Matricya B 2 2 1 1 3 2 1 2 0 displaystyle mathbf B begin bmatrix 2 amp 2 amp 1 1 amp 3 amp 2 1 amp 2 amp 0 end bmatrix nbsp Ale tut b 11 lt b 12 b 13 displaystyle b 11 lt b 12 b 13 nbsp oskilki 2 lt 2 1 displaystyle 2 lt 2 1 nbsp b 22 b 21 b 23 displaystyle b 22 geq b 21 b 23 nbsp oskilki 3 1 2 displaystyle 3 geq 1 2 nbsp b 33 lt b 31 b 32 displaystyle b 33 lt b 31 b 32 nbsp oskilki 0 lt 1 2 displaystyle 0 lt 1 2 nbsp Z togo sho velichini b 11 displaystyle b 11 nbsp i b 33 displaystyle b 33 nbsp menshi nizh velichini sum elementiv u vidpovidnih ryadkah B displaystyle mathbf B nbsp ne ye diagonalno panivnoyu Matricya C 4 2 1 1 6 2 1 2 5 displaystyle mathbf C begin bmatrix 4 amp 2 amp 1 1 amp 6 amp 2 1 amp 2 amp 5 end bmatrix nbsp daye c 11 c 12 c 13 displaystyle c 11 geq c 12 c 13 nbsp oskilki 4 gt 2 1 displaystyle 4 gt 2 1 nbsp c 22 c 21 c 23 displaystyle c 22 geq c 21 c 23 nbsp oskilki 6 gt 1 2 displaystyle 6 gt 1 2 nbsp c 33 c 31 c 32 displaystyle c 33 geq c 31 c 32 nbsp oskilki 5 gt 1 2 displaystyle 5 gt 1 2 nbsp Tut u kozhnomu ryadku velichina diagonalnogo elementu bilsha nizh vidpovidna suma elementiv ryadka C displaystyle mathbf C nbsp ye strogo diagonalno panivnoyu matriceyu Zastosuvannya i vlastivosti RedaguvatiStrogo diagonalno panivna matricya ye oborotnoyu Cej rezultat mozhna dovesti vikoristovuyuchi teoremu kil Gershgorina Ermitova matricya z panivnoyu diagonallyu A displaystyle A nbsp z dijsnimi nevid yemnimi diagonalnimi elementami ye nevid yemno oznachenoyu Dzherela RedaguvatiGantmaher F R Teoriya matric 5 e M Fizmatlit 2010 559 s ISBN 5 9221 0524 8 ros R Horn Ch Dzhonson Matrichnyj analiz M Mir 1989 653 s ros Primitki Redaguvati Napriklad Horn and Johnson 1985 p 349 vikoristovuyut ce dlya poznachennya slabkogo diagonalnogo panuvannya Otrimano z https uk wikipedia org w index php title Diagonalno panivna matricya amp oldid 36887072
