Brandenburg Technical University Cottbus Department 1, Institute of Mathematics Chair for Numerical Mathematics an Scientific Computing Prof.
Publié le 26/02/2014
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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. »
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