##### Activity4.2.3

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

1. 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 $$A\vvec = 3\vvec\text{.}$$

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

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

4. 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 $$A\text{?}$$

5. 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 $$A\text{?}$$

6. 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 $$A\text{.}$$

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