An introduction to eigenvalues and eigenvectors

Let's consider an example that illustrates how we can put these ideas to use.

Suppose that we work for a car rental company that has two locations, $P$ and $Q$. When a customer rents a car at one location, they have the option to return it to either location at the end of the day. After doing some market research, we determine:

• 80% of the cars rented at location $P$ are returned to $P$ and 20% are returned…

Consider the matrix and vectors .

1. Show that $\vvec_1$ and $\vvec_2$ are eigenvectors of $A$ and find their associated eigenvalues.

2. Express the vector $\xvec = \twovec{-4}{-1}$ as a linear combination of $\vvec_1$ and $\vvec_2$.

3. Use this expression to compute $A\xvec$, $A^2\xvec$, and $A^{-1}\xvec$ as a linear combination of eigenvectors.

Consider the matrix and vectors

1. Show that the vectors $\vvec_1$, $\vvec_2$, and $\vvec_3$ are eigenvectors of $A$ and find their associated eigenvalues.

2. Express the vector $\xvec = \threevec{0}{-3}{-4}$ as a linear combination of the eigenvectors.

3. Use this expression to compute $A\xvec$, $A^2\xvec$, and $A^{-1}\xvec$ as a linear combination of eigenvectors.

Suppose that $A$ is an $n\times n$ matrix.

1. Explain why $\lambda = 0$ is an eigenvalue of $A$ if and only if there is a nonzero solution to the homogeneous equation $A\xvec = 0$.

2. Explain why $A$ is not invertible if and only if $\lambda=0$ is an eigenvalue.

3. If $\vvec$ is an eigenvector of $A$ having associated eigenvalue $\lambda$, explain why $\vvec$ is also an eigenvector of $A^2$ with associated eigenvalue $\lambda^2$.

4. If $A$ is invertible and $\vvec$ is eigenvector of $A$ having associated e…

For the following matrices, find the eigenvectors and associated eigenvalues by thinking geometrically about the corresponding matrix transformation.

1. $\left[\begin{array}{rr} 3 & 0 \\ 0 & 3 \\ \end{array}\right]$.

2. $\left[\begin{array}{rr} -2 & 0 \\ 0 & 4 \\ \end{array}\right]$.

3. What are the eigenvectors and associated eigenvalues of the identity matrix?

4. What are the eigenvectors and associated eigenvalues of a diagonal matrix with distinct diagonal entries?

Suppose that $A$ is a $2\times2$ matrix having eigenvectors and associated eigenvalues $\lambda_1=2$ and $\lambda_2=-3$.

1. If $\xvec=\twovec{5}{0}$, find the vector $A^4\xvec$.

2. Find the vectors $A\twovec10$ and $A\twovec01$.

3. What is the matrix $A$?

Determine whether the following statements are true or false and provide a justification for your response.

1. The eigenvalues of a diagonal matrix are equal to the entries on the diagonal.

2. If $A\vvec=\lambda\vvec$, then $A^2\vvec=\lambda\vvec$ as well.

3. Every vector is an eigenvector of the identity matrix.

4. If $\lambda=0$ is an eigenvalue of $A$, then $A$ is invertible.

5. For every $n\times n$ matrix $A$, it is possible to find a basis of $\real^n$ consisting of eigenvectors of $A$.

Suppose that $A$ is an $n\times n$ matrix.

1. Assuming that $\vvec$ is an eigenvector of $A$ whose associated eigenvalue is nonzero, explain why $\vvec$ is in $\col(A)$.

2. Assuming that $\vvec$ is an eigenvector of $A$ whose associated eigenvalue is zero, explain why $\vvec$ is in $\nul(A)$.

3. Consider the two special matrices below and find their eigenvectors and associated eigenvalues.

For each of the following matrix transformations, describe the eigenvalues and eigenvectors of the corresponding matrix $A$.

1. A reflection in $\real^2$ in the line $y=x$.

2. A $180^\circ$ rotation in $\real^2$.

3. A $180^\circ$ rotation in $\real^3$ about the $y$-axis.

4. A $90^\circ$ rotation in $\real^3$ about the $x$-axis.

Suppose we have two species, $P$ and $Q$, where species $P$ preys on $Q$. Their populations, in millions, in year $k$ are denoted by $P_k$ and $Q_k$ and satisfy . We will keep track of the populations in year $k$ using the vector $\xvec_k=\twovec{P_k}{Q_k}$ so that .

1. Show that $\vvec_1=\twovec{1}{3}$ and $\vvec_2=\twovec{2}{1}$ are eigenvectors of $A$ and find their associated eigenvalues.

2. Suppose that the initial populations are described by the vector $\xvec_0 = \twovec{38}{44}$. Express $\xvec_0$ as a linear combination…