Section4.1An introduction to eigenvalues and eigenvectors
This section introduces the concept of eigenvalues and eigenvectors and offers an example that motivates our interest in them. The point here is to develop an intuitive understanding of eigenvalues and eigenvectors and explain how they can be used to simplify some problems that we have previously encountered. In the rest of this chapter, we will develop this concept into a richer theory and illustrate its use with more meaningful examples.
Suppose that is a nonzero vector and that is a scalar. What is the geometric relationship between and ?
Let’s now consider the eigenvector condition: . Here we have two vectors, and . If , what is the geometric relationship between and ?
Instructions.
The sliders in the diagram below allow you to choose a matrix . The vector , shaded red, may be moved by clicking in the head of the vector. The vector is then shown in outline.
Figure4.1.3.A geometric interpretation of the eigenvalue-eigenvector condition .
Choose the matrix . Move the vector so that the eigenvector condition holds. What is the eigenvector and what is the associated eigenvalue?
By algebraically computing , verify that the eigenvector condition holds for the vector that you found.
If you multiply the eigenvector that you found by , do you still have an eigenvector? If so, what is the associated eigenvalue?
Are you able to find another eigenvector 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 . Use the diagram to describe any eigenvectors and associated eigenvalues.
Finally, consider the matrix . Use the diagram to describe any eigenvectors and associated eigenvalues. What geometric transformation does this matrix perform on vectors? How does this explain the presence of any eigenvectors?
Let’s consider the ideas we saw in the activity in some more depth. To be an eigenvector of , the vector must satisfy for some scalar . This means that and are scalar multiples of each other so they must lie on the same line.
Consider now the matrix . On the left of Figure 4.1.4, we see that is not an eigenvector of since the vectors and do not lie on the same line. On the right, however, we see that is an eigenvector. In fact, is obtained from by stretching by a factor of . Therefore, is an eigenvector of with eigenvalue .
It is not difficult to see that any multiple of is also an eigenvector of with eigenvalue . Indeed, we will see later that all the eigenvectors associated to a given eigenvalue form a subspace of .
The interactive diagram we used in the activity is meant to convey the fact that the eigenvectors of a matrix are special vectors. Most of the time, the vectors and appear visually unrelated. For certain vectors, however, and line up with one another. Something important is going on when that happens so we call attention to these vectors by calling them eigenvectors. For these vectors, the operation of multiplying by reduces to the much simpler operation of scalar multiplying by . The reason eigenvectors are important is because it is extremely convenient to be able to replace matrix multiplication by scalar multiplication.
Subsection4.1.2The usefulness of eigenvalues and eigenvectors
In the next section, we will introduce an algebraic technique for finding the eigenvalues and eigenvectors of a matrix. Before doing that, however, we would like to discuss why eigenvalues and eigenvectors are so useful.
Let’s continue looking at the example . We have seen that is an eigenvector with eigenvalue and is an eigenvector with eigenvalue . This means that and . By the linearity of matrix multiplication, we can determine what happens when we multiply a linear combination of and by :
We can draw an analogy with the more familiar example of the diagonal matrix . As we have seen, the matrix transformation defined by combines a horizontal stretching by a factor of 3 with a reflection across the horizontal axis, as is illustrated in Figure 4.1.6.
Figure4.1.7.The matrix has the same geometric effect as the diagonal matrix when expressed in the coordinate system defined by the basis of eigenvectors.
In a sense that will be made precise later, having a set of eigenvectors of that forms a basis of enables us to think of as being equivalent to a diagonal matrix . Of course, as the other examples in the previous activity show, it may not always be possible to form a basis from the eigenvectors of a matrix. For example, the only eigenvectors of the matrix , which represents a shear, have the form . In this example, we are not able to create a basis for consisting of eigenvectors of the matrix. This is also true for the matrix , which represents a rotation.
Suppose that we work for a car rental company that has two locations, and . 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 are returned to and 20% are returned to .
40% of the cars rented at location are returned to and 60% are returned to .
Suppose that there are 1000 cars at location and no cars at location on Monday morning. How many cars are there are locations and at the end of the day on Monday?
How many are at locations and at end of the day on Tuesday?
If we let and be the number of cars at locations and , respectively, at the end of day , we then have
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We can write the vector to reflect the number of cars at the two locations at the end of day , which says that
or where .
Suppose that
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Compute and to demonstrate that and are eigenvectors of . What are the associated eigenvalues and ?
We said that 1000 cars are initially at location and none at location . This means that the initial vector describing the number of cars is . Write as a linear combination of and .
Remember that and are eigenvectors of . Use the linearity of matrix multiplication to write the vector , describing the number of cars at the two locations at the end of the first day, as a linear combination of and .
Write the vector as a linear combination of and . Then write the next few vectors as linear combinations of and :
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What will happen to the number of cars at the two locations after a very long time? Explain how writing as a linear combination of eigenvectors helps you determine the long-term behavior.
This activity is important and motivates much of our work with eigenvalues and eigenvectors so it’s worth reviewing to make sure we have a clear understanding of the concepts.
Therefore, we will write the vector describing the initial distribution of cars as a linear combination of and ; that is, . To do, we form the augmented matrix and row reduce:
Multiplying a number by is the same as taking 20% of that number. As each day goes by, the second term is multiplied by so the coefficient of in the expression for will eventually become extremely small. We therefore see that the distribution of cars will stabilize at .
Notice how our understanding of the eigenvectors of the matrix allows us to replace matrix multiplication with the simpler operation of scalar multiplication. As a result, we can look far into the future without having to repeatedly perform matrix multiplication.
Furthermore, notice how this example relies on the fact that we can express the initial vector as a linear combination of eigenvectors. For this reason, we would like, when given an matrix, to be able to create a basis of that consists of its eigenvectors. We will frequently return to this question in later sections.