Activity 4.1.2
This definition has an important geometric interpretation that we will investigate here.
Suppose that \(\vvec\) is a nonzero vector and that \(\lambda\) is a scalar. What is the geometric relationship between \(\vvec\) and \(\lambda\vvec\text{?}\)
Let's now consider the eigenvector condition: \(A\vvec = \lambda\vvec\text{.}\) Here we have two vectors, \(\vvec\) and \(A\vvec\text{.}\) If \(A\vvec = \lambda\vvec\text{,}\) what is the geometric relationship between \(\vvec\) and \(A\vvec\text{?}\)

EIGENVECTORS
Choose the matrix \(A= \left[\begin{array}{rr} 1\amp 2 \\ 2\amp 1 \\ \end{array}\right] \text{.}\) Move the vector \(\vvec\) so that the eigenvector condition holds. What is the eigenvector \(\vvec\) and what is the associated eigenvalue?
By algebraically computing \(A\vvec\text{,}\) verify that the eigenvector condition holds for the vector \(\vvec\) that you found.
If you multiply the eigenvector \(\vvec\) that you found by \(2\text{,}\) do you still have an eigenvector? If so, what is the associated eigenvector?
Are you able to find another eigenvector \(\vvec\) that is not a scalar multiple of the first one that you found? If so, what is the eigenvector and what is the associated eigenvalue?
Now consider the matrix \(A = \left[\begin{array}{rr} 2 \amp 1 \\ 0 \amp 2 \\ \end{array}\right] \text{.}\) Use the diagram to describe any eigenvectors and associated eigenvalues.
Finally, consider the matrix \(A = \left[\begin{array}{rr} 0 \amp 1 \\ 1 \amp 0 \\ \end{array}\right] \text{.}\) Use the diagram to describe any eigenvectors and associated eigenvalues. What geometric transformation does this matrix perform on vectors? How does this explain the presence of any eigenvectors?