Targadda n Kuci–Cwarz

Amnni ad icqqa ad t tfhmt ! Amnni ad gis kra n iwalwn nna ur iẓḍaṛ yan ad tn ifhm iɣ ur issin ɣ umawal iẓlin n tutlayt taclḥiyt tatrart. Mad igan afssay ? Fad ad tfhmt mad gis illan tzḍart ad tawst i yixf nnk s umawal lli illan ɣ izddar akkʷ n tasna. Iɣ t ur tufit, tzḍart ad nn taggʷt ɣ imawaln d isgzawaln.

Ɣ tusnakt, Targadda nɣ tamyagart n Kuci–Scwarz (s tanglizt : Cauchy–Schwarz), tettawsan ula s yism n targadda n Cauchy–Bunyakovsky–Schwarz, tga yat targadda bahra istawhman ɣ tusnakt kullut, ar sis nswurri ɣ aljibr imzirg, taslṭ, tiẓri n tsqqart d kigan n igran yaḍni. Targadda ad issufɣt-id yan umusnak afransis iga s yism Augustin-Louis Cauchy ɣ usggwas n 1821, sliɣ yufa Viktor Bunyakovsky yat targadda trwast akk ɣ mad izdin d aɣrd ɣ usggwas n 1859, mn bɛd yufatt daɣ yan umusnak almani Hermann Amandus Schwarz ɣ usggwas n 1888.

Mad tettini

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

$|\langle \mathbf {u} ,\mathbf {v} \rangle |^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \cdot \langle \mathbf {v} ,\mathbf {v} \rangle$

maɣ $\langle \cdot ,\cdot \rangle$ iga yan ufaris agnsan. S umdya afaris agnsan gis afaris afsnan n ilawn d ismlaln. S ɣmkad, iɣ nusi aẓur uzmir-sin ɣ tsgiwin s snat, d iɣ asn nga alugn. Ar nttafa targadda ad :

$|\langle \mathbf {u} ,\mathbf {v} \rangle |\leq \|\mathbf {u} \|\|\mathbf {v} \|.$

D yat tɣawsa yaḍn, tasgiwin s snat gaddan iɣ ${\textstyle \mathbf {u} }$ d ${\textstyle \mathbf {v} }$ gan ilelliyn imzirgn (yaɛni gan imsadaɣn).

Iɣ ${\textstyle u_{1},\ldots ,u_{n}\in \mathbb {C} }$ d ${\textstyle v_{1},\ldots ,v_{n}\in \mathbb {C} }$ d ufaris agnsan iga afaris agnsan asmlal anaway, hann targadda tzḍar ad ttyura zund ɣikad (maɣ tirra n taɛṛṛaḍt ar sis nmmal unaftay n ismlaln): i $\mathbf {u} ,\mathbf {v} \in \mathbb {C} ^{n}$, darnɣ

$\left|\langle \mathbf {u} ,\mathbf {v} \rangle \right|^{2}=\left|\sum _{k=1}^{n}u_{k}{\bar {v}}_{k}\right|^{2}\leq \langle \mathbf {u} ,\mathbf {u} \rangle \langle \mathbf {v} ,\mathbf {v} \rangle =\left(\sum _{k=1}^{n}u_{k}{\bar {u}}_{k}\right)\left(\sum _{k=1}^{n}v_{k}{\bar {v}}_{k}\right)=\sum _{j=1}^{n}|u_{j}|^{2}\sum _{k=1}^{n}|v_{k}|^{2}$

Ad t igan,

$|u_{1}{\bar {v}}_{1}+\cdots +u_{n}{\bar {v}}_{n}|^{2}\leq (|u_{1}|^{2}+\cdots +|u_{n}|^{2})(|v_{1}|^{2}+\cdots +|v_{n}|^{2})$

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