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Unformatted text preview: CS 205a Fall 2011 Midterm 1 Please write your name on the top right of this page. The exam is closed book/closed notes and no calculators are allowed. You have 1 hour and 15 minutes to complete the exam. Multiple Choice (4 x 1 pt each) For each of the following questions, circle all answers which are correct. You must circle ALL of the answers for a given question correctly to receive credit. 1. Which of the following matrices are always diagonalizable (i.e. the matrices that always have a full set of eigenvectors)? (a) Symmetric matrices (b) Orthogonal matrices (c) Householder matrices (d) Upper triangular matrices Answer: a, b, c 2. Which of the following computations have operation counts of O ( n 3 )? Consider the cases for general n n matrices. (a) QR factorization with the GramSchmidt method (b) LU decomposition (c) Back substitution step in solving A~x = ~ b (d) Cholesky factorization Answer: a,b,d 3. k ~x k is best for evaluating the numerical solution to a system of equations A~x = ~ b because: (a) It is often the only obtainable norm (b) It illustrates the worst error for a solution to an individual equation (c) It provides information about the total error in the solution (d) It is always cheapest to compute Answer: b 4. A Householder matrix H = I 2 ~v~v T ~v T ~v (a) ~x R n , k ~x k 2 = k H~x k 2 (b) has eigenvalues equal to 1 with multiplicity n (c) is a projection matrix onto the hyperplane orthogonal to ~v (d) has a determinant of 1 (i.e. det ( H ) = 1) Answer: a, d 1 Eigenvalues and Eigenvectors (10 pts) Given a 2 2 matrix A = 7 2 3 2 , answer the following: 1. What are the eigenvalues of A ? (3 pts) det 7 2 3 2 = 0 (7 ) (2 ) 6 = 0 14 9 + 2 6 = 0 2 9 + 8 = 0 (  8) (  1) = 0 1 = 1 , 2 = 8 2. For each eigenvalue of A , find the corresponding eigenvectors. (3 pts) A~x = ~x 7 2 3 2 ~ v 1 = ~ v 1 7 2 3 2 ~ v 2 = 8 ~ v 2 7 ~ v 1 , 1 + 2 ~ v 1 , 2 = ~ v 1 , 1 3 ~ v 1 , 1 + 2 ~ v 1 , 2 = ~ v 1 , 2 7 ~ v 2 , 1 + 2...
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 Fall '07
 Fedkiw

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