20105ee103_1_hw4_sols

# 20105ee103_1_hw4_sols - L. Vandenberghe 10/21/10 EE103...

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Unformatted text preview: L. Vandenberghe 10/21/10 EE103 Homework 4 solutions 1. Exercise 4.5. We can write f ( u 1 , u 2 ) as f ( u 1 , u 2 ) = u 2 1 p 11 + 2 u 1 u 2 p 12 + u 2 2 p 22 + u 1 q 1 + u 2 q 2 + r. For fixed u 1 and u 2 , this is a linear function of the unknowns p 11 , p 12 , p 22 , q 1 , q 2 , r . For example, f (0 , 1) = 6 means p 22 + q 2 + r = 6 . We therefore obtain the following set of equations: 0 0 1 1 1 1 0 0 1 0 1 1 2 1 1 1 1 1 2 1 − 1 − 1 1 1 4 4 1 2 1 4 4 1 2 1 1 p 11 p 12 p 22 q 1 q 2 r = 6 6 3 7 2 6 . >> A = [0 1 1 1; 1 1 1; 1 2 1 1 1 1; 1 2 1 -1 -1 1; 1 4 4 1 2 1; 4 4 1 2 1 1] >> b = [6; 6; 3; 7; 2; 6 ]; >> x = A\b x = 3.0000-2.0000 1.0000-2.0000 0.0000 5.0000 We can check the answer as follows. 1 >> P = [x(1), x(2); x(2), x(3)]; >> q = [x(4); x(5)]; >> r = x(6); >> u = [0;1]; u’*P*u + q’*u + r ans = 6 >> u = [1;0]; u’*P*u + q’*u + r ans = 6.0000 >> u = [1;1]; u’*P*u + q’*u + r ans = 3.0000 >> u = [-1;-1]; u’*P*u + q’*u + r ans = 7.0000 >> u = [1;2]; u’*P*u + q’*u + r ans = 2 >> u = [2;1]; u’*P*u + q’*u + r ans = 6 2. Exercise 6.15. (a) The matrix is positive definite if and only if x T ( I + auu T ) x > 0 for all nonzero x ....
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## This note was uploaded on 02/03/2011 for the course EE 103 taught by Professor Vandenberghe,lieven during the Spring '08 term at UCLA.

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20105ee103_1_hw4_sols - L. Vandenberghe 10/21/10 EE103...

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