FM5002-HW1-1.25.12

X y 7 1 7 x2 72 the lower half would b 2 y2 n egative

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Unformatted text preview: oint for f−− that is, show that ( f) (0, 0) = (0, 0). b. Compute M. c. Find the characterisitc polynomial of M. d. Find the eigenvalues of M. e. For each eigenvalue of M, find a basis of the corresponding eigenspace. f. Find a 2 x2 rotation matrix R such that R 1 MR is diagonal. 6x f a. 4y x 2 3 x2 1 4x 6y 3 x2 4xy f , , so y 3 y2 4xy 2 1 f 0, 0 0, 0 . 3 y2 b. We compute : 2 6x f 2 x 2 3x 4x xy 2 2 41 f 4y 4xy 6y 6x 4x 2 y 41 6y 2 3x 3 2 d . Solving 5 Λ2 3 Λ 6 Λ Λ2 3 x2 1 4xy 4y 3 y2 3. x2 3 y2 x 0, y 0 2 , 32 1 2 3x 4xy 3y 2 , 2 32 3y 1 3 x2 4xy 3 y2 g xy 2 3 32 f 2 2 4xy Therefore, H f 0, 0 , 3y 4xy 2 3 2 32 3 x2 41 f c. det 2 2. x 0, y 0 f y2 3. x 0, y 0 . 2 3 5 6 Λ Λ2 0 gives Λ 5 or Λ 1, which are the eigenvalues. e. Abasis for the eigenspace corresponding to the eigenvalue 5 is given by the vector 1, 1 . The eigenspace corresponding to 1 is spanned by 1, 1 . f. Normalizing the eigenvectors and arranging them as the columns of the matrix R, we obtain 1 11 R . 11 2 . 5...
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