Exercise 5

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

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

2. Explain why this is happening and provide a contrast with how the power method usually works.

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