quiz3-1806-S08-soln

# quiz3-1806-S08-soln - 18.06 Professor Strang Quiz 3 May 2...

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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 2-131 A. Ritter 2-085 2-1192 afr 2) M 2 4-149 A. Tievsky 2-492 3-4093 tievsky 3) M 3 2-131 A. Ritter 2-085 2-1192 afr 4) M 3 2-132 A. Tievsky 2-492 3-4093 tievsky 5) T 11 2-132 J. Yin 2-333 3-7826 jbyin 6) T 11 8-205 A. Pires 2-251 3-7566 arita 7) T 12 2-132 J. Yin 2-333 3-7826 jbyin 8) T 12 8-205 A. Pires 2-251 3-7566 arita 9) T 12 26-142 P. Buchak 2-093 3-1198 pmb 10) T 1 2-132 B. Lehmann 2-089 3-1195 lehmann 11) T 1 26-142 P. Buchak 2-093 3-1198 pmb 12) T 1 26-168 P. McNamara 2-314 4-1459 petermc 13) T 2 2-132 B. Lehmann 2-089 2-1195 lehmann 14) T 2 26-168 P. McNamara 2-314 4-1459 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. I’ll use the determinant test. A matrix A is positive definite when every one of the top-left 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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quiz3-1806-S08-soln - 18.06 Professor Strang Quiz 3 May 2...

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