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Unformatted text preview: 18.06 Professor Strang Quiz 3 May 2, 2008 Grading 1 2 3 Your PRINTED name is: Please circle your recitation: 1) M 2 2131 A. Ritter 2085 21192 afr 2) M 2 4149 A. Tievsky 2492 34093 tievsky 3) M 3 2131 A. Ritter 2085 21192 afr 4) M 3 2132 A. Tievsky 2492 34093 tievsky 5) T 11 2132 J. Yin 2333 37826 jbyin 6) T 11 8205 A. Pires 2251 37566 arita 7) T 12 2132 J. Yin 2333 37826 jbyin 8) T 12 8205 A. Pires 2251 37566 arita 9) T 12 26142 P. Buchak 2093 31198 pmb 10) T 1 2132 B. Lehmann 2089 31195 lehmann 11) T 1 26142 P. Buchak 2093 31198 pmb 12) T 1 26168 P. McNamara 2314 41459 petermc 13) T 2 2132 B. Lehmann 2089 21195 lehmann 14) T 2 26168 P. McNamara 2314 41459 petermc 1 (40 pts.) The (real) matrix A is A = 1 1 2 1 x 3 2 3 6 . (a) What can you tell me about the eigenvectors of A ? What is the sum of its eigenvalues? (b) For which values of x is this matrix A positive definite? (c) For which values of x is A 2 positive definite? Why ? (d) If R is any rectangular matrix, prove from x T ( R T R ) x that R T R is positive semidefinite (or definite). What condition on R is the test for R T R to be positive definite? Solution (10+10+10+10 points) a) Since A is a symmetric matrix (no matter what x is), its eigenvectors may be chosen orthonormal (5 points). The sum of the eigenvalues is the same as the trace of A , that is, the sum of the diagonal entries: tr ( A ) = 7 + x . b) In this case, the easiest tests for positive definiteness are the pivot test and the determinant test. Ill use the determinant test. A matrix A is positive definite when every one of the topleft determinants is positive (3 points for correct defn.). In this case, the three determinants are 1 , x 1 , and det( A ) = 1(6 x 9) (6 6) + 2(3 2 x ) = 2 x 3 . (1) (6 points). All of these are positive precisely when x > 3 / 2 (1 point)....
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 Spring '08
 CHANILLO

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