Exam1 - CBE 660 Exam 1 Wednesday October 7 2009 NAME...

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CBE 660 Exam 1 Wednesday, October 7, 2009 NAME: Instructions: The exam is closed book and closed notes. Write your name on the exam. Work any two of the three problems. You have 50 minutes. Hand in your exam as well as your solution at the end. Exercise 1. Norm and matrix rotation. 50 points Given the following A matrix A = 0 . 46287 0 . 11526 0 . 53244 0 . 34359 invoking [u,s,v]=svd(A) in Matlab or Octave produces u = -0.59540 -0.80343 -0.80343 0.59540 s = 0.78328 0.00000 0.00000 0.12469 v = -0.89798 -0.44004 -0.44004 0.89798 (a) What vector x of unit norm maximizes k Ax k ? How large is k Ax k for this x ? (b) What vector x of unit norm minimizes k Ax k ? How large is k Ax k for this x ? (c) What is the definition of k A k ? What is the value of k A k for this A ? (d) Denote the columns of v by v 1 and v 2 . Draw a sketch of the unit circle traced by x as it travels from x = v 1 to x = v 2 and the corresponding curve traced by Ax . (e) Let’s find an A , if one exists, that rotates all x R 2 counterclockwise by θ radians. What do you choose for the singular values σ 1 and σ 2 ? Choose v 1 = e 1 and v 2 = e 2 for the V matrix in which e i , i = 1 , 2 is the
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