Talɣa:Aggur:Tusnakt/Tamyagart n ɣassa

Ɣ tusnakt, Targadda nɣ tamyagart n 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 tɣawsiwin 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 aɣ 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,

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

(Ɣr uggar...)