Finding eigenvectors numerically

Suppose that $A$ is a matrix having eigenvalues $-3$, $-0.2$, $1$, and $4$.

1. What are the eigenvalues of $A^{-1}$?

2. What are the eigenvalues of $A+7I$?

Use the commands power, inverse_power, and find_closest_eigenvalue to approximate the eigenvalues and associated eigenvectors of the following matrices.

1. $A= \left[\begin{array}{rr} -2 & -2 \\ -8 & -2 \\ \end{array}\right]$.

2. $A= \left[\begin{array}{rr} 0.6 & 0.7 \\ 0.5 & 0.2 \\ \end{array}\right]$.

3. $A= \left[\begin{array}{rrrr} 1.9 & -16.0 & -13.0 & 27.0 \\ -2.4 & 20.3 & 4.6 & -17.7 \\ -0.51 & -11.7 & -1.4 & 13.1 \\ -2.1 & 15.3 & 6.9 & -20.5 \\ \end{array}\right]$.

Consider the matrix $A = \left[\begin{array}{rr} 0 & -1 \\ -4 & 0 \\ \end{array}\right]$.

1. Describe what happens if we apply the power method and the inverse power method using the initial vector $\xvec_0 = \twovec{1}{0}$.

2. Find the eigenvalues of this matrix and explain this observed behavior.

3. How can we apply the techniques of this section to find the eigenvalues of $A$?

We have seen that the matrix $A = \left[\begin{array}{rr} 1 & 2 \\ 2 & 1 \\ \end{array}\right]$ has eigenvalues $\lambda_1 = 3$ and $\lambda_2=-1$ and associated eigenvectors $\vvec_1 = \twovec{1}{1}$ and $\vvec_2=\twovec{-1}{1}$.

1. Describe what happens when we apply the inverse power method using the initial vector $\xvec_0 = \twovec{1}{0}$.

2. Explain why this is happening and provide a contrast with how the power method usually works.

3. How can we modify the power method to give the dominant eigenvalue in this case?

Suppose that $A$ is a $2\times2$ matrix with eigenvalues $4$ and $-3$ and that $B$ is a $2\times2$ matrix with eigenvalues $4$ and $1$. If we apply the power method to find the dominant eigenvalue of these matrices to the same degree of accuracy, which matrix will require more steps in the algorithm? Explain your response.

Suppose that we apply the power method to the matrix $A$ with an initial vector $\xvec_0$ and find the eigenvalue $\lambda=3$ and eigenvector $\vvec$. Suppose that we then apply the power method again with a different initial vector and find the same eigenvalue $\lambda=3$ but a different eigenvector $\wvec$. What can we conclude about the matrix $A$ in this case?

The power method we have developed only works if the matrix has real eigenvalues. Suppose that $A$ is a $2\times2$ matrix that has a complex eigenvalue $\lambda = 2+3i$. What would happen if we apply the power method to $A$?

Consider the matrix $A=\left[\begin{array}{rr} 1 & 1 \\ 0 & 1 \\ \end{array}\right]$.

1. Find the eigenvalues and associated eigenvectors of $A$.

2. Make a prediction about what happens if we apply the power method and the inverse power method to find eigenvalues of $A$.

3. Verify your prediction using Sage.