Geo 6 - Endomorfismi simmetrici o autoaggiunti
Gen 17th, 2008 by admin
Sia V uno spazio vettoriale euclideo. T: V –> V si dice simmetrico o autoaggiunto se risulta:
<T(v) , w> = <v , T(w)>
T è simmetrico se e solo se in ogni base ortonormale di V si rappresenta con una matrice simmetrica. Se T è simmetrico :
i) I suoi autovalori sono reali
ii) V ammette una base ortonormale di autovettori di T ( Teorema spettrale )
ii - bis) Per ogni matrice A simmetrica esiste una matrice C ortogonale (appartenente a O(n) => Ct = C-1) tale che CtAC = C-1AC = D (matrice diagonale)
iii) Autovettori corrispondenti ad autovalori distinti sono ortogonali
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