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Unformatted text preview: COT4501 Spring 2012 Homework III This assignment has six problems and they are equally weighted. The assignment is due in class on Tuesday, February 21, 2012. There are four regular problems and two computer prob- lems (using MATLAB). For the computer problems, turn in your results (e.g., graphs, plots, simple analysis and so on) and also a printout of your (MATLAB) code. Problem 1 1. Show that if the vector v 6 = 0 , then the matrix H = I- 2 vv > v > v is orthogonal and symmetric. Solution: H is symmetric: H > = I >- 2 ( vv > ) > v > v = I- 2 vv > v > v = H . We have used the formula that for two matrices A,B , ( AB ) > = B > A > . 2. Let a be any nonzero vector. If v = a- α e 1 , where α = ±k a k 2 , and H = I- 2 vv > v > v , show that Ha = α e 1 . Solution: By direct computation: v > v = ( a- α e 1 ) > ( a- α e 1 ) = ( a ) > a- 2 α a > e 1 + α 2 e > 1 e 1 = 2 α 2- 2 α a > e 1 , and 2 vv > a = 2( a- α e 1 )( a- α e 1 ) > a = 2( a- α e 1 )( α 2- α a > e 1 ) . Using the two results above, we have Ha = a- 2 vv > a v > v = a- 2( a- α e 1 )( α 2- α a > e 1 ) 2 α 2- 2 α a > e 1 = α e 1 . Problem 2 Consider the vector a as an n × 1 matrix. 1. Write out its QR factorization, showing the matrices Q and R explicitly. Solution: Q is simply the vector (one column matrix) a / k a k 2 , and R = k a k 2 . 1 2. What is the solution to the linear least squares problem a x ’ b , where b is a given n-vector. Solution: Let u = a / k a k 2 , the Q .. The least squares minimizes the cost function k a x- b k 2 2 = k QR x- b k 2 2 = k R x- Q > b k 2 2 . The solution is simply given as (using Q and R in Part 1) ˆ x = Q > b R = u > b k a k 2 = a > b k a k 2 2 ....
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This note was uploaded on 04/05/2012 for the course COT 4501 taught by Professor Davis during the Spring '08 term at University of Florida.
- Spring '08