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Vektor p (\displaystyle p)-norma
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| bet | 7/12 | | Sana | 06.06.2023 | | Hajmi | 1 Mb. | | #70424 |
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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
Frobenius normasi, yoki evklid normasi uchun p-normasining 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)), kabi ‖ 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
matritsasidir.
- ‖ 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))- matritsaning yagona qiymatlari A (\displaystyle A).
- ‖ A ‖ F (\displaystyle \|A\|_(F)) matritsani ko'paytirishda o'zgarmaydi A (\displaystyle A) ortogonal (unitar) matritsalarga chapga yoki o'ngga.
Maksimal modul
Maksimal modul normasi p-normasining yana bir maxsus holatidir p = ∞ .
‖ A ‖ max = max ( | a i j |). (\ displaystyle \ | A \ | _ (\ matn (maks)) = \ max \ (| a_ (ij) | \).)
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