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# A7 - T such that P T AP = T 3 Let Q ~x = ~x T A~x with A =...

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Math 235 Assignment 7 Due: Wednesday, Nov 10th 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 . b) Q ( x,y,z ) = 7 x 2 + 8 xy + y 2 + 8 xz - 16 yz + z 2 2. Let A = - 1 3 - 1 0 1 2 0 2 1 . Find an orthogonal matrix P and upper triangular matrix
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Unformatted text preview: T such that P T AP = T . 3. Let Q ( ~x ) = ~x T A~x with A = ± a b b c ² and det A 6 = 0. a) Prove that Q is positive deﬁnite if det 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 and B be symmetric n × n matrices whose eigenvalues are all positive. Show that the eigenvalues of A + B are all positive....
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