Curve evolution and image processing by Cao F. PDF

By Cao F.

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N xn yn für alle v ∈ V und w ∈ W dargestellt durch Koordinaten x = κB (v) = (x1 , . . , xn )t und y = κB (w) = (y1 , . . , yn )t . Die Bilinearform ist perfekt ⇐⇒ λi = 0 für alle i = 1, . . , n. Beweis. 12 gesichtert ist. Die Gram’sche Matrix D = MB,B (f ) ist dann eine Diagonalmatrix D = diag(λ1 , . . 1). 5. 14 (Normalformensatz). Sei 2 ∈ K × . Zu einer symmetrischen Matrix A = At ∈ Mn (K) gibt es S ∈ GLn (K) und eine Diagonalmatrix D ∈ Mn (K), so daß A = S t DS. Beweis. Wir betrachten die Bilinearform x, Ay = xt Ay auf K n .

Dann gilt: (1) f ist symmetrisch ⇐⇒ At = A ist symmetrisch. (2) f ist antisymmetrisch ⇐⇒ At = −A ist antisymmetrisch. Geometrie 31 Beweis. Wir benutzen das Vorzeichen ε = 1 für den Fall „symmetrisch“, und das Vorzeichen ε = −1 für den Fall „antisymmetrisch“. 1) und das ist äquivalent dazu, daß MB,B (f ) (anti-)symmetrisch ist. 1) folgern, daß f (anti-)symmetrisch ist. Seien v = i xi bi und w = i yi bi beliebig. Dann gilt f (v, w) = f ( xi bi , i yj bj ) = j xi yj f (bi , bj ) i,j =ε· xi yj f (bj , bi ) = ε · f ( i,j xi bi ) = ε · f (w, v).

Sei (V, , ) ein endlichdimensionaler R-Vektorraum mit symmetrischer Bilinearform, und sei A die Gram’sche Matrix bezüglich einer Basis von V . Sei Ar die Matrix aus den ersten r Zeilen und Spalten von A. Dann gilt: (1) (2) , , ist positiv definit ⇐⇒ für alle 1 ≤ r ≤ dim(V ) gilt det(Ar ) > 0. ist negativ definit ⇐⇒ für alle 1 ≤ r ≤ dim(V ) gilt (−1)r det(Ar ) > 0. Beweis. Sei A die Gram’sche Matrix zur Basis B = (b1 , . . , bn ). Dann ist Ar die Gram’sche Matrix für die Einschränkung von , auf die lineare Hülle Ur = b1 , .

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Curve evolution and image processing by Cao F.


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