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MIT18_06S10_exam3_s10

MIT18_06S10_exam3_s10 - question is about the matrix ⎤ ...

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18.06 Quiz 3 May 8, 2010 Professor Strang Your PRINTED name is: 1. Your recitation number is 2. 3. 1. (40 points) Suppose u is a unit vector in R n , so u T u = 1 . This problem is about the n by n symmetric matrix H = I 2 u u T . (a) Show directly that H 2 = I. Since H = H T , we now know that H is not only symmetric but also . (b) One eigenvector of H is u itself. Find the corresponding eigenvalue. (c) If v is any vector perpendicular to u, show that v is an eigenvector of H and find the eigenvalue. With all these eigenvectors v, that eigenvalue must be repeated how many times? Is H diagonalizable ? Why or why not? (d) Find the diagonal entries H 11 and H ii in terms of u 1 , . . . , u n . Add up H 11 + . . . + H nn and separately add up the eigenvalues of H.
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2. (30 points) Suppose A is a positive definite symmetric n by n matrix. (a) How do you know that A 1 is also positive definite? (We know A 1 is symmetric. I just had an e-mail from the International Monetary Fund with this question.) (b) Suppose Q is any orthogonal n by n matrix. How do you know that Q A Q T = Q A Q 1 is positive definite? Write down which test you are using. (c) Show that the block matrix A A B = A A is positive semidefinite . How do you know B is not positive definite?
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3. (30
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Unformatted text preview: question is about the matrix ⎤ ⎡ ⎢ ⎣ − 1 A = ⎥ ⎦ 4 . (a) Find its eigenvalues and eigenvectors. ⎤ ⎡ Write the vector u (0) = ⎢ ⎣ 2 ⎥ ⎦ as a combination of those eigenvectors. du (b) Solve the equation = Au starting with the same vector u (0) at time t = 0 . dt In other words: the solution u ( t ) is what combination of the eigenvectors of A ? (c) Find the 3 matrices in the Singular Value Decomposition A = U Σ V T in two steps. –First, compute V and Σ using the matrix A T A. –Second, ±nd the (orthonormal) columns of U. MIT OpenCourseWare http://ocw.mit.edu 18.06 Linear Algebra Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms ....
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