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# assign9 - the quadratic form subject to the constraint || x...

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MATH 235 Assignment 9 Quadratic Forms, Constrained Optimization and Singular Value Decomposition Not to be handed in. 1. An n × n symmetric matrix A with real entries is called positive definite if v T A v > 0 for all nonzero v R n . (a) Suppose U is an invertible n × n matrix with real entries. Prove that A = U T U is positive definite. (b) Let a,b,c R . Prove that A = bracketleftBigg a b b c bracketrightBigg is positive definite if and only if a > 0 and det A > 0. 2. (a) If A is symmetric and invertible, show that A 2 is positive definite. (b) If A = bracketleftBigg 3 2 2 1 bracketrightBigg , show that A 2 is positive definite while A is not positive definite. (c) If A is positive definite, show that A k is positive definite for all positive integers k . (d) If A k is positive definite where k is odd, show that A is positive definite. 3. For each of the quadratic forms below identify the type of conic it represents, and find
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Unformatted text preview: the quadratic form subject to the constraint || x || = 1 where x = ( x,y ) T . (a) 9 x 2-8 xy + 3 y 2 (b) 8 x 2 + 6 xy 4. Sketch the conic section given by 2 x 2-72 xy + 23 y 2 =-50 in the xy-plane. 5. Let Q ( x ) =-2 x 2 1-x 2 2 + 4 x 1 x 2 + 4 x 2 x 3 . ±ind a unit vector x in R 3 at which Q ( x ) is maximized subject to the constraint x T x = 1. 6. ±ind the singular value decomposition (SVD) for each of the following two matrices, A,B . In addition, use the singular vectors to Fnd bases for the ±our Subspaces. (Then do all the above for A T and B T .) (a) A = b 6 2-7 6 B (b) B = 0 1 1 1 1 0 1...
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