Section4.3Diagonalization, similarity, and powers of a matrix
The first example we considered in this chapter was the matrix , which has eigenvectors and and associated eigenvalues and . In Subsection 4.1.2, we described how is, in some sense, equivalent to the diagonal matrix .
This equivalence is summarized by Figure 4.3.1. The diagonal matrix has the geometric effect of stretching vectors horizontally by a factor of and flipping vectors vertically. The matrix has the geometric effect of stretching vectors by a factor of in the direction and flipping them in the direction. That is, the geometric effect of is the same as that of when viewed in a basis of eigenvectors of .
Our goal in this section is to express this geometric observation in algebraic terms. In doing so, we will make precise the sense in which and are equivalent.
In this preview activity, we will review some familiar properties about matrix multiplication that appear in this section.
Remember that matrix-vector multiplication constructs linear combinations of the columns of the matrix. For instance, if , express the product in terms of and .
Next, remember how matrix-matrix multiplication is defined. Suppose that we have matrices and and that . How can we express the matrix product in terms of the columns of ?
When working with an matrix ,Subsection 4.1.2 demonstrated the value of having a basis of consisting of eigenvectors of . In fact, Proposition 4.2.9 tells us that if the eigenvalues of are real and distinct, then there is a such a basis. As we’ll see later, there are other conditions on that guarantee a basis of eigenvectors. For now, suffice it to say that we can find a basis of eigenvectors for many matrices. With this assumption, we will see how the matrix is equivalent to a diagonal matrix .
Suppose that is a matrix having eigenvectors and with associated eigenvalues and . Because the eigenvalues are real and distinct, we know by Proposition 4.2.9 that these eigenvectors form a basis of .
The results from the previous two parts of this activity demonstrate that . Using the fact that the eigenvectors and form a basis of , explain why is invertible and that we must have .
More generally, suppose that we have an matrix and that there is a basis of consisting of eigenvectors of with associated eigenvalues . If we use the eigenvectors to form the matrix
and the eigenvalues to form the diagonal matrix
and apply the same reasoning demonstrated in the activity, we find that and hence
If is an matrix and there is a basis of consisting of eigenvectors of having associated eigenvalues , then we can write where is the diagonal matrix whose diagonal entries are the eigenvalues of
This is the sense in which we mean that is equivalent to a diagonal matrix . The expression says that , expressed in the basis defined by the columns of , has the same geometric effect as , expressed in the standard basis .
By constructing Nul, we find a basis for consisting of the vector . Similarly, a basis for consists of the vector . This shows that we can construct a basis of consisting of eigenvectors of .
If we choose a different basis for the eigenspaces, we will also find a different matrix that diagonalizes . The point is that there are many ways in which can be written in the form .
In several earlier examples, we have been interested in computing powers of a given matrix. For instance, in Activity 4.1.3, we had the matrix and an initial vector , and we wanted to compute
.
In particular, we wanted to find and determine what happens as becomes very large. If a matrix is diagonalizable, writing can help us understand powers of more easily.
Suppose that is a matrix with eigenvector and associated eigenvalue ; that is, . By considering , explain why is also an eigenvector of with eigenvalue .
Let’s revisit Activity 4.1.3 where we had the matrix and the initial vector . We were interested in understanding the sequence of vectors , which means that .
We have been interested in diagonalizing a matrix because doing so relates a matrix to a simpler diagonal matrix . In particular, the effect of multiplying a vector by , viewed in the basis defined by the columns of , is the same as the effect of multiplying by in the standard basis.
While many matrices are diagonalizable, there are some that are not. For example, if a matrix has complex eigenvalues, it is not possible to find a basis of consisting of eigenvectors, which means that the matrix is not diagonalizable. In this case, however, we can still relate the matrix to a simpler form that explains the geometric effect this matrix has on vectors.
Notice that a matrix is diagonalizable if and only if it is similar to a diagonal matrix. In case a matrix has complex eigenvalues, we will find a simpler matrix that is similar to and note that has the same effect, when viewed in the basis defined by the columns of , as , when viewed in the standard basis.
The next activity shows that has a simple geometric effect on . First, however, we will use polar coordinates to rewrite . As shown in the figure, the point defines , the distance from the origin, and , the angle formed with the positive horizontal axis. We then have
Form the matrix using these values of and . Then rewrite the point in polar coordinates by identifying the values of and . Explain the geometric effect of multiplying vectors by .
We formed the matrix by choosing the eigenvalue . Suppose we had instead chosen . Form the matrix and use polar coordinates to describe the geometric effect of .
If the matrix has a complex eigenvalue , it turns out that is always similar to the matrix whose geometric effect on vectors can be described in terms of a rotation and a scaling. There is, in fact, a method for finding the matrix so that that we’ll see in Exercise 4.3.5.8. For now, we note that has the same geometric effect as , when viewed in the basis provided by the columns of . We will put this fact to use in the next section to understand certain dynamical systems.
Our goal in this section has been to use the eigenvalues and eigenvectors of a matrix to relate to a simpler matrix.
We said that is diagonalizable if we can write where is a diagonal matrix. The columns of consist of eigenvectors of and the diagonal entries of are the associated eigenvalues.
If is a matrix with complex eigenvalue , then is similar to . Writing the point in polar coordinates and , we see that rotates vectors through an angle and scales them by a factor of .
When is a matrix with a complex eigenvalue , we have said that there is a matrix such that where . In this exercise, we will learn how to find the matrix . As an example, we will consider the matrix .
Using the same eigenvalue, we will find an eigenvector where the entries of are complex numbers. As always, we will describe Nul by constructing the matrix and finding its reduced row echelon form. In doing so, we will necessarily need to use complex arithmetic.