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A7_soln - Math 235 Assignment 7 Solutions 1 For each...

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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 - λI = 1 - λ 3 3 - 7 - λ . The characteristic equation is 0 = det( A - λI ) = λ 2 + 6 λ - 16 = ( λ - 2)( λ + 8) . The roots are 2 and - 8, so these are the eigenvectors of A . For λ = 2 we get A - 2 I = - 1 3 3 - 9 1 - 3 0 0 z 1 = 3 1 . For λ = - 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 ( λ ) = 4 - λ 2 2 2 4 - λ 2 2 2 4 - λ = 4 - λ 0 2 2 2 - λ 2 2 - (2 - λ ) 4 - λ = 4 - λ 0 2 2 2 - λ 2 4 0 6 - λ = - ( λ - 2)( λ 2 - 10 λ + 16) = - ( λ - 2) 2 ( λ - 8) .
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