What do you observe about the eigenvalues?

What property of the matrix makes this “coincidence” possible?

We explore the theory behind finding the eigenvalues and associated eigenvectors of a square matrix.

Let be an matrix. In Module EIG-0010 we learned that the eigenvectors and eigenvalues of are vectors and scalars that satisfy the equation

We listed a few reasons why we are interested in finding eigenvalues and eigenvectors, but we did not give any process for finding them. In this module we will focus on the process.

If a vector is an eigenvector satisfying Equation (def:eigen), then it also satisfies the following equations.

This shows that any eigenvector of is in the null space of the related matrix, . Since eigenvectors are non-zero vectors, this means that will have eigenvectors if and only if the null space of is nontrivial. The only way that can be nontrivial is if .

If the rank of an matrix is less than , then the matrix is singular. Since must be singular for any eigenvalue , Theorem th:detofsingularmatrix implies that is an eigenvalue of if and only if

In theory, then, to find the eigenvalues of , one can solve Equation (eqn:chareqn) for .

The equation
is called the *characteristic equation* of . The left-hand side of the equation is a
polynomial in and is called the *characteristic polynomial* of .

In Example ex:3x3eig, the factor appears twice in the characteristic polynomial. This
repeated factor gives rise to the eigenvalue . We say that has *algebraic multiplicity*
.

The three examples above are a bit contrived. It is not always possible to completely factor the characteristic polynomial. However, a fundamental fact from algebra is that every degree polynomial has roots (counting multiplicity) provided that we allow complex numbers. This is why sometimes eigenvalues and their corresponding eigenvectors involve complex numbers. The next example illustrates this point.

Let . Compute the eigenvalues of this matrix.

What do you observe about the eigenvalues?

What property of the matrix makes this “coincidence” possible?

The matrix in Exploration Problem init:3x3tri is a triangular matrix, and the property you observed holds in general.

- Proof
- See Practice Problem prob:eigtri.

One final note about eigenvalues. We began this section with the sentence, ”In
theory, then, to find the eigenvalues of , one can solve Equation (eqn:chareqn) for .”
In general, one does not attempt to compute eigenvalues by solving the
characteristic equation of a matrix, as there is no simple way to solve such an
equation for . Instead, one can often approximate the eigenvalues using *iterative
methods*.

Once we have computed an eigenvalue of an matrix , the next step is to compute the associated eigenvectors. In other words, we seek vectors such that , or equivalently,

For any given eigenvalue there are infinitely many eigenvectors associated with it. In fact, the eigenvectors associated with form a subspace of . (see Practice Problems prob:eigenspace1 and prob:eigenspace2) This motivates the following definition.

The set of all eigenvectors associated with a given eigenvalue of a matrix is known as
the *eigenspace* associated with that eigenvalue.

So given an eigenvalue , there is an associated eigenspace , and our goal is to find a basis of , for then any eigenvector will be a linear combination of the vectors in that basis. Moreover, we are trying to find a basis for the set of vectors that satisfy Equation eqn:nullspace, which means we seek a basis for . We have already learned how to compute a basis of a null space - see Module VSP-0040.

Let’s return to the examples we did in the first section of this module.

(Finding eigenvectors for Example ex:2x2eig )

Recall that has eigenvalues and . Compute a basis for the eigenspace associated with each of these eigenvalues.

Eigenvectors associated with the
eigenvalue are in the null space of . So we seek a basis for . We compute:

From this we see that the eigenspace associated with consists of vectors of the form . This means that is one possible basis for .

In a similar way, we compute a basis for , the subspace of all eigenvectors associated with the eigenvalue . Now we compute:

Vectors in the null space have the form This means that is one possible basis for the eigenspace .

(Finding eigenvectors for Example ex:2x2eig2) We know from Example ex:2x2eig2 that has eigenvalues and .
Compute a basis for the eigenspace associated with each of these eigenvalues.

Let’s begin
by finding a basis for the eigenspace , which is the subspace of consisting of eigenvectors
corresponding to the eigenvalue . We need to compute a basis for . We compute:

From this we see that an eigenvector in has the form . This means that is one possible basis for the eigenspace . By letting , we obtain an arguably nicer-looking basis: .

To compute a basis for , the subspace of all eigenvectors associated to the eigenvalue , we compute:

From this we find that is one possible basis for the eigenspace .

(Finding eigenvectors for Example ex:3x3eig) We know from Example ex:3x3eig that has
eigenvalues and . Compute a basis for the eigenspace associated to each of these
eigenvalues.

We first find a basis for the eigenspace . We need to compute a basis for . We compute:

Notice that there are two free variables. The eigenvectors in have the form

So one possible basis for the eigenspace is given by .

Next we find a basis for the eigenspace . We need to compute a basis for . We compute:

This time there is one free variable. The eigenvectors in have the form , so a possible basis for the eigenspace is given by .

(Finding eigenvectors for Example ex:3x3_complex_eig) We know from Example ex:3x3_complex_eig that has eigenvalues ,
, and . Compute a basis for the eigenspace associated with each eigenvalue.

We first
find a basis for the eigenspace . We need to compute a basis for . We compute:

From this we see that for any eigenvector in we have and , but is a free variable. So one possible basis for the eigenspace is given by Next we find a basis for the eigenspace . We need to compute a basis for . We compute:

There is one free variable. Setting , we get and . From this we see that eigenvectors in have the form , so a possible basis for the eigenspace is given by . We ask you in Practice Problem prob:3x3_complex_ev to show that is a basis for .

In this exercise we will prove that the eigenvectors associated with an eigenvalue of
an matrix form a subspace of .

Complete Example ex:3x3_complex_ev by showing that a basis for is given by , where is the
eigenspace associated with the eigenvalue of the matrix .

Prove Theorem th:eigtri. (HINT: Proceed by induction on the dimension n. For the
inductive step, compute by expanding along the first column (or row) if is upper
(lower) triangular.)

The following set of problems deals with geometric interpretation of eigenvalues and
eigenvectors, as well as linear transformations of the plane. Please use EIG-0010 and
LTR-0070 for reference.

Recall that a vertical stretch/compression of the plane is a
linear transformation whose standard matrix is
Find the eigenvalues of . Find a basis for the eigenspace corresponding to each
eigenvalue.

Answer: A basis for is and a basis for is

Sketch several vectors in each eigenspace and use geometry to explain why the eigenvectors you sketched make sense.

Recall that a horizontal shear of the plane is a linear transformation whose standard
matrix is
Find the eigenvalue of .

Answer:

Find a basis for the eigenspace corresponding to .

Answer: A basis for is

Sketch several vectors in the eigenspace and use geometry to explain why the eigenvectors you sketched make sense.

Recall that a counterclockwise rotation of the plane through angle is a linear
transformation whose standard matrix is
Verify that the eigenvalues of are
Explain why is real number if and only if is a multiple of . (Compare this to
Practice Problem prob:rotmatrixrealeig1 of EIG-0010.)

Suppose is a muliple of . Then the eigenspaces corresponding to the two eigenvalues are the same. Which of the following describes the eigenspace?

All vectors in . All vectors along the -axis. All vectors along the -axis. All vectors along the line .

Recall that a reflection of the plane about the line is a linear transformation whose
standard matrix is
Verify that the eigenvalues of are
Find a basis for eigenspaces and . (For simplicity, assume that .)

Answer: A basis for is and a basis for is

Choose the best description of .

All vectors in . All vectors with “slope” . All
vectors with “slope” . All vectors with “slope” .

Choose the best description of .

All vectors along the line . All vectors parallel to
the -axis. All vectors parallel to the -axis. All vectors perpendicular to the line .

Use geometry to explain why the eigenspaces you found make sense.

Practice Problem prob:3x3fromKuttler1 is adopted from Problem 7.1.11 of Ken Kuttler’s A First Course in Linear Algebra. (CC-BY)

Ken Kuttler, A First Course in Linear Algebra, Lyryx 2017, Open Edition, p. 361.