Learn to interpret similar matrices geoemetrically.

Understand the relationship between the eigenvalues, eigenvectors, and characteristic polynomials of similar matrices.

Recipe: compute in terms of for

Picture: the geometry of similar matrices.

Vocabulary word:similarity.

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 5.5 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 5.4 and Section 5.5, we will show how to use eigenvalues and eigenvectors to find a simpler matrix that behaves like a given matrix.

Subsection5.3.1Similar Matrices

We begin with the algebraic definition of similarity.

Definition

Two matrices and are similar if there exists an invertible matrix such that

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 2.8 before reading this subsection.

By conditions 4 and 5 of the invertible matrix theorem in Section 5.1, 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 that

Since is the matrix with columns this says that Multiplying both sides by gives To summarize:

Let be an invertible matrix with columns and let a basis for Then for any in we have

This says that changes from the -coordinates to the usual coordinates, and changes from the usual coordinates to the -coordinates.

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 3.4.

Recipe: Computing in terms of

Suppose that where is an invertible matrix with columns Let a basis for Let be a vector in To compute one does the following:

Multiply by which changes to the -coordinates:

Multiply this by

Interpreting this vector as a -coordinate vector, we multiply it by to change back to the usual coordinates:

To summarize: if then and do the same thing, only in different coordinate systems.

The following example is the heart of this section.

Example

Consider the matrices

One can verify that see this example in Section 5.4. 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

We see from the -coordinate grid below that Therefore,

Multiplying by scales the coordinates:

Interpreting as a -coordinate vector, we multiply by to get
Of course, this vector lies at on the -coordinate grid.

Now let

We see from the -coordinate grid that Therefore,

Multiplying by scales the coordinates:

Interpreting as a -coordinate vector, we multiply by to get
This vector lies at on the -coordinate grid.

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.

Fact

Similar matrices have the same characteristic polynomial.

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 5.2, similar matrices also have the same trace and determinant.

Note

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.

Fact

Suppose that Then

The eigenvalues of / or / are the same.

Proof

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

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