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

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 \(A\vvec\text{?}\)

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?

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.

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 \(A\text{?}\)

The matrix A has at least two eigenvectors.

The matrix A has at least one eigenvector.

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

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

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 \(\vvec_2\text{:}\)

\(\xvec_3 = A\xvec_2\text{.}\)

\(\xvec_4 = A\xvec_3\text{.}\)

\(\xvec_5 = A\xvec_4\text{.}\)

\(\xvec_6 = A\xvec_5\text{.}\)

(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.