MATH307 December 2010
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q2 (e) • Q3 (a) • Q3 (b) • Q3 (c) • Q3 (d) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q5 (c) • Q6 (a) • Q6 (b) • Q6 (c) • Q6 (d) • Q6 (e) • Q6 (f) • Q7 (a) • Q7 (b) • Q7 (c) •
Question 07 (c)
Suppose that A is a symmetric 4x4 matrix with eigenvalues 0, 1, 4, 5. Define a sequence of vectors by choosing x0 at random, and then settings
for n = 1, 2, .... You then observe that xn converges to
(c) What vector does converge to?
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
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Take the limit on both sides of the definition of .
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Taking the limit n → ∞ on both sides yields
We used, from part (a), that is an eigenvector of with eigenvalue .
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