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# HW3 - COT4501 Spring 2012 Homework III This assignment has...

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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. 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 . Problem 2 Consider the vector a as an n × 1 matrix. 1. Write out its QR factorization, showing the matrices Q and R explicitly. 2. What is the solution to the linear least squares problem a x b , where b is a given n -vector. 3. How do you interpret the above result geometrically? Problem 3 Consider the following matrix A A = 1 1 0 0 , where is a positive number smaller than mach in a given floating-point system. In class,

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• Spring '08
• Davis
• Singular value decomposition, Orthogonal matrix, Linear least squares, QR algorithm, Householder transformation

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