Some matrices are easy to understand. For instance, a diagonal matrix
just scales the coordinates of a vector: The purpose of most of the rest of this chapter is to understand complicated-looking matrices by analyzing to what extent they “behave like” simple matrices. For instance, the matrix
has eigenvalues and with corresponding eigenvectors and Notice that
Using instead of the usual coordinates makes “behave” like a diagonal matrix.
The other case of particular importance will be matrices that “behave” like a rotation matrix: indeed, this will be crucial for understanding Section 6.5 geometrically. See this important note.
In this section, we study in detail the situation when two matrices behave similarly with respect to different coordinate systems. In Section 6.4 and Section 6.5, we will show how to use eigenvalues and eigenvectors to find a simpler matrix that behaves like a given matrix.
We begin with the algebraic definition of similarity.
Two matrices and are similar if there exists an invertible matrix such that
As in the above example, one can show that is the only matrix that is similar to and likewise for any scalar multiple of
Similarity is unrelated to row equivalence. Any invertible matrix is row equivalent to but is the only matrix similar to For instance,
are row equivalent but not similar.
As suggested by its name, similarity is what is called an equivalence relation. This means that it satisfies the following properties.
Let and be matrices.
We conclude with an observation about similarity and powers of matrices.
Let Then for any we have
First note that
Next we have
The pattern is clear.
Similarity is a very interesting construction when viewed geometrically. We will see that, roughly, similar matrices do the same thing in different coordinate systems. The reader might want to review -coordinates and nonstandard coordinate grids in Section 3.5 and well as -matrices in Section 4.7 before reading this subsection.
Recall that (by conditions 4 and 5 of the invertible matrix theorem in Section 6.1) an matrix is invertible if and only if its columns form a basis for This means we can speak of the -coordinates of a vector in where is the basis of columns of Recall from Section 4.7 that this means
If the linear map has as standard matrix and is the matrix with columns given by the basis then the -matrix of is
so
In other words, two -matrices and are similar if and only they represent the same linear map but expressed in different bases.
Let's now illustrate this more concretely. Suppose that The above observation gives us another way of computing for a vector in Recall that so that multiplying by means first multiplying by then by then by See this example in Section 4.4.
Suppose that where is an invertible matrix with columns and let be the corresponding basis for Let be a vector in To compute one does the following:
To summarize: if then and do the same thing, only in different coordinate systems.
The following example is the heart of this section.
Consider the matrices
One can verify that see this example in Section 6.4. Let and the columns of and let a basis of
The matrix is diagonal: it scales the -direction by a factor of and the -direction by a factor of
To compute first we multiply by to find the -coordinates of then we multiply by then we multiply by again. For instance, let
Now let
To summarize:
To summarize and generalize the previous example:
Let
where is assumed invertible. Then:
Since similar matrices behave in the same way with respect to different coordinate systems, we should expect their eigenvalues and eigenvectors to be closely related.
Similar matrices have the same characteristic polynomial.
Suppose that where are matrices. We calculate
Therefore,
Here we have used the multiplicativity property in Section 5.1 and its corollary in Section 5.1.
Since the eigenvalues of a matrix are the roots of its characteristic polynomial, we have shown:
Similar matrices have the same eigenvalues.
By this theorem in Section 6.2, similar matrices also have the same trace and determinant. Both of these observations also follow from the fact that similar matrices represent the same linear endomorphism considered with respect to different bases, and the determinant, trace, and characteristic polynomial don't depend on the choice of basis.
The converse of the fact is false. Indeed, the matrices
both have characteristic polynomial but they are not similar, because the only matrix that is similar to is itself.
Given that similar matrices have the same eigenvalues, one might guess that they have the same eigenvectors as well. Upon reflection, this is not what one should expect: indeed, the eigenvectors should only match up after changing from one coordinate system to another. This is the content of the next fact, remembering that and change between the usual coordinates and the -coordinates.
Suppose that Then
The eigenvalues of / or / are the same.
Suppose that is an eigenvector of with eigenvalue so that Then
so that is an eigenvector of with eigenvalue Likewise if is an eigenvector of with eigenvalue then and we have
so that is an eigenvalue of with eigenvalue
If then takes the -eigenspace of to the -eigenspace of and takes the -eigenspace of to the -eigenspace of
We continue with the above example: let
so Let and the columns of Recall that:
This means that the -axis is the -eigenspace of and the -axis is the -eigenspace of likewise, the “ -axis” is the -eigenspace of and the “ -axis” is the -eigenspace of This is consistent with the fact, as multiplication by changes into and into
