We have seen that the matrix \(A = \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 1 \\ \end{array}\right]\) has eigenvalues \(\lambda_1 = 3\) and \(\lambda_2=-1\) and associated eigenvectors \(\vvec_1 = \twovec{1}{1}\) and \(\vvec_2=\twovec{-1}{1}\text{.}\)

  1. Describe what happens when we apply the power method using the initial vector \(\xvec_0 = \twovec{1}{0}\text{.}\)

  2. Use your understanding of the eigenvalues and eigenvectors to explain this behavior.

  3. How can we modify the power method to give the dominant eigenvalue in this case?