Operator normalariga misollar - O’zbekiston raqamli tehnalogiyalar vazirligi muhammad al xorazmiy nomidagi

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O’zbekiston raqamli tehnalogiyalar vazirligi muhammad al xorazmiy nomidagi
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bet | 5/11 | Sana | 13.05.2024 | Hajmi | 108,14 Kb. | | #229637 |
Bog'liq O’zbekiston raqamli tehnalogiyalar vazirligi muhammad al xorazmi Operator normalariga misollar
Spektral normaning xususiyatlari:
Operatorning spektral normasi maksimalga teng yagona raqam bu operator.
Spektral norma oddiy operator ga teng mutlaq qiymat maksimal modul o'z qiymati bu operator.
Matritsani ko'paytirishda spektral norma o'zgarmaydi ortogonal (unitar) matritsa.
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.
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)).
‖ 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.
‖ 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).
‖ A ‖ F (\displaystyle \|A\|_(F)) matritsani ko'paytirishda o'zgarmaydi A (\displaystyle A) chapga yoki o'ngga ortogonal (unitar) matritsalar.
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