# TBIL Activities for Understanding Linear Algebra

## Section4.1An introduction to eigenvalues and eigenvectors

### Activity4.1.0.1.

Suppose that $$A$$ is a $$2\times 2$$ matrix and that $$\vvec_1$$ and $$\vvec_2$$ are vectors such that
\begin{equation*} A\vvec_1 = 3 \vvec_1, \qquad A\vvec_2 = -\vvec_2\text{.} \end{equation*}
Use the linearity of matrix multiplication to express the following vectors in terms of $$\vvec_1$$ and $$\vvec_2\text{.}$$

#### (a)

$$A(4\vvec_1)\text{.}$$

#### (b)

$$A(\vvec_1 + \vvec_2)\text{.}$$

#### (c)

$$A(4\vvec_1 -3\vvec_2)\text{.}$$

#### (d)

$$A^2\vvec_1\text{.}$$

#### (e)

$$A^2(4\vvec_1 - 3\vvec_2)\text{.}$$

#### (f)

$$A^4\vvec_1\text{.}$$

### Definition4.1.0.1.

Given a square $$n\times n$$ matrix $$A\text{,}$$ we say that a nonzero vector $$\vvec$$ is an eigenvector of $$A$$ if there is a scalar $$\lambda$$ such that
\begin{equation*} A\vvec = \lambda \vvec\text{.} \end{equation*}
The scalar $$\lambda$$ is called the eigenvalue associated to the eigenvector $$\vvec\text{.}$$
This definition has an important geometric interpretation that we will investigate in the next few activities.

### Activity4.1.0.2.

Suppose that $$\vvec$$ is a nonzero vector and that $$\lambda$$ is a scalar.

#### (a)

What is the geometric relationship between $$\vvec$$ and $$\lambda\vvec\text{?}$$

#### (b)

Let's now consider the eigenvector condition: $$A\vvec = \lambda\vvec\text{.}$$ Here we have two vectors, $$\vvec$$ and $$A\vvec\text{.}$$ If $$A\vvec = \lambda\vvec\text{,}$$ what is the geometric relationship between $$\vvec$$ and $$A\vvec\text{?}$$

### Activity4.1.0.3.

Use the applet in Understanding Linear Algebra Activity 4.1.2, part c 15 .

#### (a)

Choose the matrix $$A= \left[\begin{array}{rr} 1\amp 2 \\ 2\amp 1 \\ \end{array}\right] \text{.}$$ Move the vector $$\vvec$$ so that the eigenvector condition holds. What is the eigenvector $$\vvec$$ and what is the associated eigenvalue?

#### (b)

By algebraically computing $$A\vvec\text{,}$$ verify that the eigenvector condition holds for the vector $$\vvec$$ that you found.

#### (c)

If you multiply the eigenvector $$\vvec$$ that you found by $$2\text{,}$$ do you still have an eigenvector? If so, what is the associated eigenvalue?

#### (d)

Are you able to find another eigenvector $$\vvec$$ that is not a scalar multiple of the first one that you found? If so, what is the eigenvector and what is the associated eigenvalue?

### Activity4.1.0.4.

Use the applet in Understanding Linear Algebra Activity 4.1.2, part c 16 .
Now consider the matrix $$A = \left[\begin{array}{rr} 2 \amp 1 \\ 0 \amp 2 \\ \end{array}\right] \text{.}$$ Use the applet to describe any eigenvectors and associated eigenvalues.

### Activity4.1.0.5.

Use the applet in Understanding Linear Algebra Activity 4.1.2, part c 17 .
Consider the matrix $$A = \left[\begin{array}{rr} 0 \amp -1 \\ 1 \amp 0 \\ \end{array}\right] \text{.}$$

#### (a)

Use the applet to find any eigenvectors and associated eigenvalues.

#### (b)

What geometric transformation does this matrix perform on vectors?

#### (c)

What does this tell you about the presence of eigenvectors for matrix $$A\text{?}$$
1. The matrix A has at least two eigenvectors.
2. The matrix A has at least one eigenvector.
3. The matrix A has no eigenvectors.
Now we will look at an application of eigenvalues and eigenvectors.

### Activity4.1.0.6.

Suppose that we work for a car rental company that has two locations, $$P$$ and $$Q\text{.}$$ 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 to $$Q\text{.}$$
• 40% of the cars rented at location $$Q$$ are returned to $$Q$$ and 60% are returned to $$P\text{.}$$

#### (a)

Suppose that there are 1000 cars at location $$P$$ and no cars at location $$Q$$ on Monday morning. How many cars are there are locations $$P$$ and $$Q$$ at the end of the day on Monday?

#### (b)

How many are at locations $$P$$ and $$Q$$ at end of the day on Tuesday?

### Activity4.1.0.7.

Continue with the car company example.
If we let $$P_k$$ and $$Q_k$$ be the number of cars at locations $$P$$ and $$Q\text{,}$$ respectively, at the end of day $$k\text{,}$$ we then have
\begin{equation*} \begin{aligned} P_{k+1}\amp {}={} 0.8P_k + 0.6Q_k \\ Q_{k+1}\amp {}={} 0.2P_k + 0.4Q_k\text{.} \\ \end{aligned} \end{equation*}
We can write the vector $$\xvec_k = \twovec{P_k}{Q_k}$$ to reflect the number of cars at the two locations at the end of day $$k\text{,}$$ which says that
\begin{equation*} \begin{aligned} \xvec_{k+1} \amp {}={} \left[\begin{array}{rr} 0.8 \amp 0.6 \\ 0.2 \amp 0.4 \\ \end{array}\right] \xvec_k \\ \\ \text{or}\qquad \xvec_{k+1} \amp {}={} A\xvec_k \end{aligned} \end{equation*}
where $$A=\left[\begin{array}{rr}0.8 \amp 0.6 \\ 0.2 \amp 0.4 \end{array}\right]\text{.}$$
Suppose that
\begin{equation*} \vvec_1 = \twovec{3}{1}, \qquad \vvec_2 = \twovec{-1}{1}\text{.} \end{equation*}

#### (a)

Compute $$A\vvec_1$$ and $$A\vvec_2$$ to demonstrate that $$\vvec_1$$ and $$\vvec_2$$ are eigenvectors of $$A\text{.}$$

#### (b)

What are the associated eigenvalues $$\lambda_1$$ and $$\lambda_2\text{?}$$

### Activity4.1.0.8.

Continue with the car company example and
\begin{equation*} \vvec_1 = \twovec{3}{1}, \qquad \vvec_2 = \twovec{-1}{1} \end{equation*}
from the previous activity.

#### (a)

We said that 1000 cars are initially at location $$P$$ and none at location $$Q\text{.}$$ This means that the initial vector describing the number of cars is $$\xvec_0 = \twovec{1000}{0}\text{.}$$ Write $$\xvec_0$$ as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

#### (b)

Remember that $$\vvec_1$$ and $$\vvec_2$$ are eigenvectors of $$A\text{.}$$ Use the linearity of matrix multiplicaiton to write the vector $$\xvec_1 = A\xvec_0\text{,}$$ describing the number of cars at the two locations at the end of the first day, as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$

#### (c)

Write the vector $$\xvec_2 = A\xvec_1$$ as a linear combination of $$\vvec_1$$ and $$\vvec_2\text{.}$$ Then write the next few vectors as linear combinations of $$\vvec_1$$ and $$\vvec_2\text{:}$$
1. $$\xvec_3 = A\xvec_2\text{.}$$
2. $$\xvec_4 = A\xvec_3\text{.}$$
3. $$\xvec_5 = A\xvec_4\text{.}$$
4. $$\xvec_6 = A\xvec_5\text{.}$$

#### (d)

What will happen to the number of cars at the two locations after a very long time? Be able to explain how writing $$\xvec_0$$ as a linear combination of eigenvectors helps you determine the long-term behavior.
davidaustinm.github.io/ula/sec-eigen-intro.html#activity-41
davidaustinm.github.io/ula/sec-eigen-intro.html#activity-41
davidaustinm.github.io/ula/sec-eigen-intro.html#activity-41