##### Preview Activity5.2.1

Let's recall some earlier observations about eigenvalues and eigenvectors.

1. How are the eigenvalues and associated eigenvectors of $$A$$ related to those of $$A^{-1}\text{?}$$

2. How are the eigenvalues and associated eigenvectors of $$A$$ related to those of $$A-3I\text{?}$$

3. If $$\lambda$$ is an eigenvalue of $$A\text{,}$$ what can we say about the pivot positions of $$A-\lambda I\text{?}$$

4. Suppose that $$A = \left[\begin{array}{rr} 0.8 \amp 0.4 \\ 0.2 \amp 0.6 \\ \end{array}\right] \text{.}$$ Explain how we know that $$1$$ is an eigenvalue of $$A$$ and then explain why the following Sage computation is incorrect.

5. Suppose that $$\xvec_0 = \twovec{1}{0}\text{,}$$ and we define a sequence $$\xvec_{k+1} = A\xvec_k\text{;}$$ in other words, $$\xvec_{k} = A^k\xvec_0\text{.}$$ What happens to $$\xvec_k$$ as $$k$$ grows increasingly large?

6. Explain how the eigenvalues of $$A$$ are responsible for the behavior noted in the previous question.

in-context