O’zbekiston raqamli tehnalogiyalar vazirligi muhammad al xorazmiy nomidagi

Operator normalariga misollar

Spektral normaning xususiyatlari:

1. 
   Operatorning spektral normasi maksimalga teng yagona raqam bu operator.
   
2. 
   Spektral norma oddiy operator ga teng mutlaq qiymat maksimal modul o'z qiymati bu operator.
   
3. 
   Matritsani ko'paytirishda spektral norma o'zgarmaydi ortogonal (unitar) matritsa.
   

3. 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 halqasimon xususiyatga ega ‖ A B ‖ ≤ ‖ A ‖ ‖ B ‖ (\displaystyle \|AB\|\leq \|A\|\|B\|), birlikni saqlaydi ‖ I ‖ = 1 (\displaystyle \|I\|=1) va operator emas.

4. Normativlarga 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'ngda)^(1/p))

Frobenius normasi

Frobenius normasi, yoki evklid normasi uchun p-normaning alohida holatidir p = 2 : ‖ A ‖ F = ∑ i = 1 m ∑ j = 1 n a i j 2 (\displaystyle \|A\|_(F)=(\sqrt (\sum _(i=1)^(m)\sum _(j) =1)^(n)a_(ij)^(2)))).
Frobenius normasini hisoblash oson (masalan, spektral norma bilan solishtirganda). U quyidagi xususiyatlarga ega:
‖ A x ‖ 2 2 = ∑ i = 1 m | ∑ j = 1 n a i j x j | 2 ≤ ∑ i = 1 m (∑ j = 1 n | a i j | 2 ∑ j = 1 n | x j | 2) = ∑ j = 1 n | x j | 2 ‖ A ‖ F 2 = ‖ A ‖ F 2 ‖ x ‖ 2 2 . (\displaystyle \|Ax\|_(2)^(2)=\sum _(i=1)^(m)\chap|\sum _(j=1)^(n)a_(ij)x_( j)\right|^(2)\leq \sum _(i=1)^(m)\left(\sum _(j=1)^(n)|a_(ij)|^(2)\sum _(j=1)^(n)|x_(j)|^(2)\o'ng)=\sum _(j=1)^(n)|x_(j)|^(2)\|A\ |_(F)^(2)=\|A\|_(F)^(2)\|x\|_(2)^(2).)

• 
  Submultiplikativlik: ‖ A B ‖ F ≤ ‖ A ‖ F ‖ B ‖ F (\displaystyle \|AB\|_(F)\leq \|A\|_(F)\|B\|_(F)), chunki ‖ A B ‖ F 2 = ∑ i, j | ∑ k a i k b k j | 2 ≤ ∑ i , j (∑ k | a i k | | b k j |) 2 ≤ ∑ i , j (∑ k | a i k | 2 ∑ k | b k j | 2) = ∑ i , k | a i k | 2 ∑ k , j | b k j | 2 = ‖ A ‖ F 2 ‖ B ‖ F 2 (\displaystyle \|AB\|_(F)^(2)=\sum _(i,j)\left|\sum _(k)a_(ik) b_(kj)\right|^(2)\leq \sum _(i,j)\left(\sum _(k)|a_(ik)||b_(kj)|\right)^(2)\ leq \sum _(i,j)\left(\sum _(k)|a_(ik)|^(2)\sum _(k)|b_(kj)|^(2)\o'ng)=\sum _(i,k)|a_(ik)|^(2)\sum _(k,j)|b_(kj)|^(2)=\|A\|_(F)^(2)\| B\|_(F)^(2)).
  
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  ‖ A ‖ F 2 = t r ⁡ A ∗ A = t r ⁡ A A ∗ (\displaystyle \|A\|_(F)^(2)=\mathop (\rm (tr)) A^(*)A=\ mathop (\rm (tr)) AA^(*)), Qayerda t r ⁡ A (\displaystyle \mathop (\rm (tr)) A) - matritsa izi A (\displaystyle A), A ∗ (\displaystyle A^(*)) - Hermit konjugati matritsasi.
  
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  ‖ A ‖ F 2 = r 1 2 + r 2 2 + ⋯ + r n 2 (\displaystyle \|A\|_(F)^(2)=\rho _(1)^(2)+\rho _ (2)^(2)+\nuqtalar +\rho _(n)^(2)), Qayerda r 1 , r 2 , … , r n (\displaystyle \rho _(1),\rho _(2),\nuqtalar,\rho _(n)) - birlik raqamlar matritsalar A (\displaystyle A).
  
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  ‖ A ‖ F (\displaystyle \|A\|_(F)) matritsani ko'paytirishda o'zgarmaydi A (\displaystyle A) chapga yoki o'ngga ortogonal (unitar) matritsalar.