PS6 - Mathematic 104 Fall 2010 Assignment#6 Due Wednesday...

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Mathematic 104, Fall 2010: Assignment #6 Due: Wednesday, November 17th Instructions: Please ensure that your answers are legible. Also make sure that all steps are shown – even for problems consisting of a numerical answer. Bonus problems cover advanced material and, while good practice, are not required and will not be graded. Problem #1. Consider the matrix A = 1 2 0 2 a) By hand compute the SVD of A . To do this it is useful to recall that all vectors in x with || x || 2 = 1 are of the form x = cos θ sin θ . b) Using the SVD determine the rank one matrix B that best approximates A in the Frobenius norm. c) Compare how well B approximates A in the Frobenius norm with how well the rank one matrices A 1 = 1 0 0 0 and A 2 = 0 2 0 2 approximate A in the Frobenius norm. Problem #2. Excercise 4.4 of Lecture 4 of Trefethen-Bau. Problem #3. Let A 1 , A 2 C m × m suppose that the left singular vectors of A 1 are { u 1 1 , . . . , u 1 m } and the right singular vectors are { v 1 1 , . . . , v 1 m } while the left singular vectors of A 2 are { u 2 1 , . . . , u 2 m } and the right
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