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A7 - A> 0 and a> 0 b Prove that Q is negative definite...

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Math 235 Assignment 7 Due: Wednesday, June 30th 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 . b) Q ( x, y, z ) = 4 x 2 + 4 xy + 4 y 2 + 4 xz + 4 yz + 4 z 2 2. Let A = - 5 3 - 3 1 . Find an orthogonal matrix P and upper triangular matrix T such that P T AP = T . 3. Let Q ( x ) = x T Ax with A = a b b c and det A = 0. a) Prove that Q is positive definite if det
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Unformatted text preview: A > 0 and a > 0. b) Prove that Q is negative definite if det A > 0 and a < 0. c) Prove that Q is indefinite if det A < 0. 4. Let A be an invertible symmetric matrix. Prove that if the quadratic form ~x T A~x is positive definite, then so it the quadratic form ~x T A-1 ~x . 5. Let A be an n × n symmetric matrix and let ~x,~ y ∈ R n . Define < ~x,~ y > = ~x T A~ y . Prove that < ,> is an inner product on R n if and only if A is positive definite....
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