Science:Math Exam Resources/Courses/MATH307/April 2013/Question 03 (b)
Work in progress: this question page is incomplete, there might be mistakes in the material you are seeing here.
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q2 (a) • Q2 (b) • Q2 (c) • Q2 (d) • Q2 (e) • Q2 (f) • Q3 (a) • Q3 (b) • Q3 (c) • Q4 (a) • Q4 (b) • QS201 5(a) • QS201 5(b) • QS201 5(c) • QS201 6(a) • QS201 6(b) • QS201 6(c) • QS201 6(d) • QS201 7(a) • QS201 7(b) • QS201 7(c) • QS202 5(a) • QS202 5(b) • QS202 5(c) • QS202 5(d) • QS202 6(a) • QS202 6(b) • QS202 7(a) • QS202 7(b) • QS202 7(c) • QS202 7(d) • QS202 7(e) •
Question 03 (b)
Explain how to find the eigenvalue of A that is closest to 2 using the power method.
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?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
Given: A is a real symmetric matrix.
Assuming: The eigenvalue of A closest to 2 is not repeated.
The eigenvalues of are .
The eigenvectors of are the eigenvectors of .
1) Take a random vector .
3) Normalize x
4) If is not close enough to (ignoring sign flips), go to 2) ('close enough' being REALLY close)
x is now an approximation of the eigenvector of the eigenvalue closest to 2
5) (x was normalized earlier).