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Unformatted text preview: Math 235 Assignment 7 Solutions 1. For each quadratic form Q ( ~x ), determine the corresponding symmetric matrix A . By diagonalizing A , Write Q so that it has no cross terms and give the change of variables which brings it into this form. Classify each quadratic form as positive definite, negative definite or indefinite. a) Q ( x, y ) = x 2 + 6 xy 7 y 2 . Solution: We have A = 1 3 3 7 so A I = 1 3 3 7 . The characteristic equation is 0 = det( A I ) = 2 + 6  16 = (  2)( + 8) . The roots are 2 and 8, so these are the eigenvectors of A . For = 2 we get A 2 I = 1 3 3 9 1 3 ~ z 1 = 3 1 . For = 8 we get A + 8 I = 9 3 3 1 3 1 0 0 ~ z 2 = 1 3 . Normalizing the vectors, we get that the orthogonal matrix which diagonalizes A is P = 3 / 10 1 / 10 1 / 10 3 / 10 , and Q = 2 x 2 1 8 y 2 1 , where x 1 y 1 = P T x y . Since one of the eigenvalues of A are positive and the other is negative, it follows that Q ( x, y ) is indefinite. b) Q ( x, y, z ) = 4 x 2 + 4 xy + 4 y 2 + 4 xz + 4 yz + 4 z 2 Solution: We have A = 4 2 2 2 4 2 2 2 4 so the characteristic polynomial is C ( ) = 4 2 2 2 4 2 2 2 4 =...
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Math

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