##### Exercise5

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?

in-context