Activity4.1.2

This definition has an important geometric interpretation that we will investigate here.

1. Suppose that $$\vvec$$ is a nonzero vector and that $$\lambda$$ is a scalar. What is the geometric relationship between $$\vvec$$ and $$\lambda\vvec\text{?}$$

2. 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 $$A\vvec\text{?}$$

3. 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?

4. By algebraically computing $$A\vvec\text{,}$$ verify that the eigenvector condition holds for the vector $$\vvec$$ that you found.

5. 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?

6. 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?

7. 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.

8. 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?

in-context