Activity 4.2.3

In this activity, we will find the eigenvectors of a matrix as the null space of the matrix $A-\lambda I$.

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

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

3. Is it possible to form a basis of $\real^2$ consisting of eigenvectors of $A$?

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

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

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