A7_soln

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

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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 ) = 5 x 2 + 6 xy- 3 y 2 . Solution: We have A = 5 3 3- 3 so A- λI = 5- λ 3 3- 3- λ . We have C ( λ ) = 5- λ 3 3- 3- λ = λ 2- 2 λ- 24 = ( λ- 6)( λ + 4) . Thus, the eigenvalues of A are 6 and- 4. For λ = 6 we get A- 6 I =- 1 3 3- 9 ∼ 1- 3 , so ~v 1 = 1 √ 10 3 1 . For λ =- 4 we get A + 4 I = 9 3 3 1 ∼ 3 1 0 0 , so ~v 2 = 1 √ 10 1- 3 . Hence, we can take P = 3 / √ 10 1 / √ 10 1 / √ 10- 3 / √ 10 to get D = 6- 4 . Then, using the change of variables ~x = P~ y = 3 / √ 10 1 / √ 10 1 / √ 10- 3 / √ 10 x 1 y 1 = (3 / √ 10) x 1 + (1 / √ 10) y 1 (1 / √ 10) x 1- (3 / √ 10) y 1 , we get Q ( ~x ) = 6 x 2 1- 4 y 2 1 . Since one of the eigenvalues of A are positive and the other is negative, it follows that Q ( x,y ) is indefinite....
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A7_soln - Math 235 Assignment 7 Solutions 1 For each...

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