##### Activity4.1.3

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 $$Q\text{.}$$ 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 $$P\text{.}$$

1. 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?

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

3. 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 0.4 \\ 0.2 \amp 0.6 \\ \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*}

Compute $$A\vvec_1$$ and $$A\vvec_2$$ to demonstrate that $$\vvec_1$$ and $$\vvec_2$$ are eigenvectors of $$A\text{.}$$ What are the associated eigenvalues $$\lambda_1$$ and $$\lambda_2\text{?}$$

4. 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 $$\vvec_2\text{.}$$

5. 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 $$\vvec_2\text{.}$$

6. 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. $$\xvec_6 = A\xvec_5\text{.}$$

7. What will happen to the number of cars at the two locations after a very long time? Explain how writing $$\xvec_0$$ as a linear combination of eigenvectors helps you determine the long-term behavior.

in-context