MATH307 December 2010
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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
(a) What is ?
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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
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.
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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,
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