Devoir de Philosophie

Brandenburg Technical University Cottbus Department 1, Institute of Mathematics Chair for Numerical Mathematics an Scientific Computing Prof.

Publié le 26/02/2014

Extrait du document

Brandenburg Technical University Cottbus Department 1, Institute of Mathematics Chair for Numerical Mathematics an Scientific Computing Prof. Dr. G. Bader, Dr. A. Pawell Problem Session to the Course: Mathematics I Environmental and Resource Management WS 2002/03 Solutions to Sheet No. 13 (Deadline: January, 27/28 2002) Homework H 13.1: Eigenvalues of A: ?1 = -1, ?2/3 = 1 Eigenvectors: ?1 = -1: ? 2 ?0 0 0 1 1 ? 0 1 ? x = 0, 1 x = (0, 1, -1)T , 1 c1 = ? (0, 1, -1)T 2 ?2/3 = 1: 1 c2 = ? (0, 1, 1)T , 2 ? C=? 0 1 ? 2 1 - ?2 c3 = (1, 0, 0)T ? 01 0 ?. 0 1 ? 2 1 ? 2 q (x) = xT Ax = x2 + 2x2 x3 . 1 B = C, C T AC = diag(1, -1, -1) 2 2 2 q (Cy ) = (Cy )T A(Cy ) = y T C T ACy = y T diag(1, -1, -1)y = y1 - y2 - y3 . ? H 13.2: 1-? det(A - ?E ) = det ? 0 0 1 1-? -1 ? 1 ?= 5 -1 - ? (1 - ?)[(1 - ?)(-1 - ?) + 5] = (1 - ?)[4 + ?2 ] = 0 => ?2/3 = ±2i ?1 = 1, Eigenvectors: ? 1-? ?0 0 1 1-? -1 ? 1 ?x = 0 5 -1 - ? ?1 = 1: ? ? 0 ...

« )  1 = 1 ;  2= 3 = 2i Eigenvectors: 0 @ 1  1 1 0 1  5 0 1 1  1 A ~x= ~ 0  1 = 1: 0 @ 0 1 1 0 0 5 0 1 2 1 A ; ~x =t(1 ;0 ;0) T  1 = 2 i: 0 @ 1 2i 1 1 0 1 2i 5 0 1 1 2i 1 A ~x= ~ 0 0 @ 1 2i 1 1 0 1 1 2i 0 0 0 1 A ~x= ~ 0 ; ~x = 2i 1 2i; 1 2i; 1 T  1 = 2i: 0 @ 1 + 2 i 1 1 0 1 + 2 i5 0 1 1 + 2 i1 A ~x= ~ 0 0 @ 1 + 2 i1 1 0 1 1 + 2 i 0 0 0 1 A ~x= ~ 0 ; ~x = 2i 1 + 2 i; 1 + 2 i;1 T H 13.3: det(A E ) = det  1  1 1 3   = (1 )(3 ) 1 = 2 4 + 2 : Eigenvalues: 1= 2 = 2 p 2 Eigenvectors: = 2 + p 2  1 p 2 1 1 1 p 2  ~x= ~ 0 ; ~x = 1 p 4 2p 2 ( 1 + p 2 ;1) T  = 2 p 2  1 + p 2 1 1 1 + p 2  ~x= ~ 0 ; ~x = 1 p 4 + 2 p 2 ( 1 p 2 ;1) T 2. »

↓↓↓ APERÇU DU DOCUMENT ↓↓↓

Liens utiles