Matritsa normasi

Operator 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:

1. 
   Operatorning spektral normasi ushbu operatorning maksimal singulyar qiymatiga teng.
   
2. 
   Oddiy operatorning spektral normasi ushbu operatorning maksimal modul o'z qiymatining mutlaq qiymatiga teng.
   
3. 
   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.

Normlarga misollar

Vektor p (\displaystyle p)-norma

Ko'rib chiqish mumkin m × n (\displaystyle m\times n) matritsa o'lcham vektori sifatida m n (\displaystyle mn) va standart vektor normalaridan foydalaning:
‖ A ‖ p = ‖ v e c (A) ‖ p = (∑ i = 1 m ∑ j = 1 n | a i j | p) 1 / p (\displaystyle \|A\|_(p)=\|\mathrm ( vec) (A)\|_(p)=\left(\sum _(i=1)^(m)\sum _(j=1)^(n)|a_(ij)|^(p)\ o'ng)^(1/p))

Frobenius normasi