Science:Math Exam Resources/Courses/MATH307/December 2010/Question 07 (a)
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Question 07 (a) 

Suppose that is a symmetric matrix with eigenvalues 0, 1, 4, 5. Define a sequence of vectors by choosing at random, and then settings for . You then observe that _{n} converges to as . (a) What is ? 
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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

In the power iteration procedure, the convergent vector is the eigenvector corresponding to the largest eigenvalue of the matrix in question. In this question the matrix is 
Hint 2 

The eigenvalues of are the eigenvalues of A shifted by 3. For the inverse matrix, take the inverse of these eigenvalues. Find the eigenvalue of with the largest absolute value, as well as the corresponding eigenvalue of A. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Using the hints, we know that is the eigenvector associated to the largest eigenvalue of the matrix . The eigenvalues of are the eigenvalues of shifted by 3, with the same eigenvectors. We can see this by noting for each eigenvector of , we have Since has the eigenvalues , , and , the matrix has the eigenvalues , , and . Next, the eigenvalues of the inverse are the reciprocal of the original eigenvalues, with the same eigenvectors. Hence the eigenvalues of are , , and . We see that the dominating eigenvalue of is , and therefore, by the power method, is an eigenvector of with eigenvalue 1. This means that is also an eigenvector of with eigenvalue . In other words, 