hw4_solutions

hw4_solutions - CS 205a Fall 2010 Midterm 1 Please write...

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Unformatted text preview: CS 205a Fall 2010 Midterm 1 Please write your name on the top right of the first page. The exam is closed book, and no calculators are allowed. You have 1 hour and 15 minutes to complete the exam. Multiple Choice (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 can be said about a Householder matrix H = I- 2 vv T v T v (a) The condition number for H is 1 (b) It is a projection matrix onto the hyperplane orthogonal to v (c) It preserves the 2-norm of a vector (ie k x k 2 = k Hx k 2 for all x ) (d) One of the eigenvalues of H is 1 with multiplicity n- 1 Answer: a, c, d 2. Which of the following classes of matrices are positive semi-definite? (a) Permutation matrices (b) Projection matrices (c) Reflection matrices (d) Symmetric matrices Answer: b 3. Suppose that a square matrix A is ill-conditioned. Which of the following matrices could potentially have a better condition number? (a) cA , where c is a non-zero scalar (b) DA , where D is a nonsingular diagonal matrix (c) PA , where P is a permutation matrix (d) A- 1 Answer: b 4. Which of the following about the least squares solutions are true? (a) It satisfies Ax = b (b) It can be found by solving the normal equation (c) Its associated residual lies in the nullspace of A T (d) It lies in the column space of A Answer: b, c 1 Eigenvalues (10 pts) 1. Given that A = T- 1 BT , prove that A and B have the same eigenvalues. How do the eigenvectors of A and B relate? (2 pts) Answer: Let λ , q be an eigenvalue, eigenvector pair of A . Aq = λq ⇒ T- 1 BTq = λq ⇒ BTq = λTq ⇒ B ˜ q = λ ˜ q, where ˜ q = Tq . Thus λ is also an eiqenvalue of B . The corresponding eigenvector is ˜ q = Tq . 2. What are the eigenvalues of a projection matrix? (Just state. No proof required) (1 pts) Answer: 1 , 2 3. If A = A T , and x T Ax > , ∀ x 6 = 0, prove that the eigenvalues of A are all positive. (3 pts) Answer: Let λ , q be an eigenvalue, eigenvector pair of A ....
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This note was uploaded on 01/05/2012 for the course CS 205A taught by Professor Fedkiw during the Fall '07 term at Stanford.

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hw4_solutions - CS 205a Fall 2010 Midterm 1 Please write...

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