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Mustaqil ishi-4
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| bet | 6/12 | | Sana | 13.05.2024 | | Hajmi | 61,74 Kb. | | #229485 |
Bog'liq Mustaqil ishi-4-kompy.infoOperator normalari
Matritsa me'yorlarining muhim sinfi operator normalari, deb ham ataladi bo'ysunuvchilar yoki qo'zg'atilgan . Operator normasi har qanday
matritsaning mavjudligiga asoslanib, va da belgilangan ikkita normaga muvofiq
yagona tarzda tuzilgan. m × n dan chiziqli operator bilan ifodalanadi K n
(\displaystyle K^(n)) ichida K m (\displaystyle K^(m)). Xususan,
‖ A ‖ = sup (‖ A x ‖: x ∈ K n, ‖ x ‖ = 1) = sup (‖ A x ‖ ‖ x ‖: x ∈ K n, x ≠ 0). (\displaystyle (\begin(hizalangan)\|A\|&=\sup\(\|Ax\|:x\in K ^(n),\ \|x\|=1\)\\&=\ sup
\left\((\frac (\|Ax\|)(\|x\|)):x\in K^(n),\ x\neq 0\right\).\end(hizalangan)))
Vektor bo'shliqlari bo'yicha me'yorlar izchil ko'rsatilgan holda, bunday norma submultiplikativ hisoblanadi (qarang).
Operator normalariga misollar
Spektral normaning xususiyatlari:
Operatorning spektral normasi ushbu operatorning maksimal singulyar qiymatiga
teng.
Oddiy operatorning spektral normasi ushbu operatorning maksimal modul o'z qiymatining mutlaq qiymatiga teng.
Matritsa ortogonal (unitar) matritsaga ko'paytirilganda spektral norma o'zgarmaydi.
Matritsalarning operator bo'lmagan normalari
Operator normalari bo'lmagan matritsa normalari mavjud. Matritsalarning operator bo'lmagan normalari tushunchasini Yu.I.Lyubich kiritgan va G.R.Belitskiy tomonidan o'rganilgan.
Operator bo'lmagan normaga misol
Misol uchun, ikki xil operator normalarini ko'rib chiqing ‖ A ‖ 1 (\displaystyle
\|A\|_(1)) va ‖ A ‖ 2 (\displaystyle \|A\|_(2)) qator va ustun normalari kabi. Yangi
normani shakllantirish ‖ A ‖ = m a x (‖ A ‖ 1 , ‖ A ‖ 2) (\displaystyle
\|A\|=max(\|A\|_(1),\|A\|_(2)). Yangi norma halqali xususiyatga ega ‖ A B ‖ ≤ ‖ A ‖ ‖ B ‖ (\displaystyle \|AB\|\leq \|A\|\|B\|), birlikni saqlaydi ‖ I ‖ = 1 (\displaystyle
\|I\|=1) va operator emas.
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