###### Activity 4.2.3

In this activity, we will find the eigenvectors of a matrix as the null space of the matrix \(A-\lambda I\text{.}\)

Let's begin with the matrix \(A = \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 1 \\ \end{array}\right] \text{.}\) We have seen that \(\lambda = 3\) is an eigenvalue. Form the matrix \(A-3I\) and find a basis for the eigenspace \(E_3 = \nul(A-3I)\text{.}\) What is the dimension of this eigenspace? For each of the basis vectors \(\vvec\text{,}\) verify that \(A\vvec = 3\vvec\text{.}\)

We also saw that \(\lambda = -1\) is an eigenvalue. Form the matrix \(A-(-1)I\) and find a basis for the eigenspace \(E_{-1}\text{.}\) What is the dimension of this eigenspace? For each of the basis vectors \(\vvec\text{,}\) verify that \(A\vvec = -\vvec\text{.}\)

Is it possible to form a basis of \(\real^2\) consisting of eigenvectors of \(A\text{?}\)

Now consider the matrix \(A = \left[\begin{array}{rr} 3 \amp 0 \\ 0 \amp 3 \\ \end{array}\right] \text{.}\) Write the characteristic equation for \(A\) and use it to find the eigenvalues of \(A\text{.}\) For each eigenvalue, find a basis for its eigenspace \(E_\lambda\text{.}\) Is it possible to form a basis of \(\real^2\) consisting of eigenvectors of \(A\text{?}\)

Next, consider the matrix \(A = \left[\begin{array}{rr} 2 \amp 1 \\ 0 \amp 2 \\ \end{array}\right] \text{.}\) Write the characteristic equation for \(A\) and use it to find the eigenvalues of \(A\text{.}\) For each eigenvalue, find a basis for its eigenspace \(E_\lambda\text{.}\) Is it possible to form a basis of \(\real^2\) consisting of eigenvectors of \(A\text{?}\)

Finally, find the eigenvalues and eigenvectors of the diagonal matrix \(A = \left[\begin{array}{rr} 4 \amp 0 \\ 0 \amp -1 \\ \end{array}\right] \text{.}\) Explain your result by considering the geometric effect of the matrix transformation defined by \(A\text{.}\)