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.

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