Targadda n Kuci–Cwarz

Ɣ tusnakt, Targadda nɣ tamyagart n Kuci–Scwarz (s tanglizt : Cauchy–Schwarz), ⵜⵜⵡⴰⵢⵙⴰⵏ ula s ⵉⵙⵎ n targadda n Cauchy–Bunyakovsky–Schwarz, tga yat targadda ⵜⴰⴼⵔⴰⵔⵜ ɣ tusnakt ⴰⴽⴽⵯ , ar sis nswurri ɣ ⴰⵍⴳⴰⴱⵔ ⵉⵎⵣⵔⵉⴳ , tiẓri n tsqqart d kigan n igran yaḍni. Targadda ad issufɣt-id yan umusnak afransis iga s yism Augustin-Louis Cauchy ɣ ⵓⵙⴳⴳⵯⴰⵙ ⵏ 1821, sliɣ yufa Viktor Bunyakovsky yat targadda trwast akk ɣ mad izdin d aɣrd ɣ ⵓⵙⴳⴳⵯⴰⵙ n 1859, v tgira yufatt daɣ yan umusnak [ⴳⵓ ⴰⵍⵉⵎⴰⵏ ]] Hermann Amandus Schwarz ɣ ⵓⵙⴳⴳⵯⴰⵙ n 1888.

ⵎⴰⴷ ⵜⵉⵏⵉ

Ar ttini targadda n Cauchy–Schwarz mas i akk sin imawayn $u$ d $v$ n yat tallunt gis afaris agnsan hann rad darnɣ tili

$|⟨𝐮,𝐯⟩{|}^2\le ⟨𝐮,𝐮⟩\cdot ⟨𝐯,𝐯⟩$

maɣ $⟨\cdot ,\cdot ⟩$ iga yan ufaris agnsan. S umdya afaris agnsan gis afaris afsnan n utul d arafrar S ɣmka-d, iɣ nusi aẓur uzmir-sin ɣ ⵜⵉⵙⴳⴳⵯⵉⵏ s snat, d iɣ asn nga alugn. Ar nttafa targadda ad :

$|⟨𝐮,𝐯⟩|\le \Vert 𝐮\Vert \Vert 𝐯\Vert .$

D yat tɣawsa yaḍn, tasgiwin s snat gaddan iɣ $𝐮$ d $𝐯$ gan ilulliyn imzirgn (ra nini gan imsadaɣn).

Iɣ $u_1,\dots ,u_n\in ℂ$ d $v_1,\dots ,v_n\in ℂ$ d ufaris agnsan iga afaris agnsan usniy anaway, hann targadda tZḍaR ad ttyara zund ɣika-d (maɣ tirra n arafrar ar sis nmmal unaftay n ismlaln): i $𝐮,𝐯\in {ℂ}^n$, darnɣ

${|⟨𝐮,𝐯⟩|}^2={\left|\sum _{k=1}^n{u}_k{\overline{v}}_k\right|}^2\le ⟨𝐮,𝐮⟩⟨𝐯,𝐯⟩=\left(\sum _{k=1}^n{u}_k{\overline{u}}_k\right)\left(\sum _{k=1}^n{v}_k{\overline{v}}_k\right)=\sum _{j=1}^n|u_j{|}^2\sum _{k=1}^n|v_k{|}^2$

Ad t igan,

$|u_1{\overline{v}}_1+\cdots +u_n{\overline{v}}_n{|}^2\le (|u_1{|}^2+\cdots +|u_n{|}^2)(|v_1{|}^2+\cdots +|v_n{|}^2)$

Amawal

• Targadda = inegalite
• Aljibr imzirg = Algebre lineaire
• Taslṭ = analyse
• Tiẓri = theorie
• Tasqqart = probabilite
• Amusnak = mathematicien
• Aɣrd = intagral
• Amaway = vecteur
• Tallunt = espace
• Afaris = produit
• Agnsan = interne
• Afsnan = scalaire
• Ilawn = reel
• Ismlaln = complexe
• Aẓur = racine
• Uzmir-sin = carree
• Alugn = norme
• Tilelli timzirgt = independance lineaire
• Imsadaɣn = paralleles
• Anaway = standard
• Anaftay = conjuge