The Characteristic Equation

Let be an matrix. In Describing Eigenvalues and Eigenvectors Algebraically and Geometrically 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 section we will focus on a process which can be used for small matrices. For larger matrices, the best methods we have are iterative methods, and we will explore some of these in The Power Method and the Dominant Eigenvalue.

For an matrix, we will see that the eigenvalues are the roots of a polynomial called the characteristic polynomial. So finding eigenvalues is equivalent to solving a polynomial equation of degree . Finding the corresponding eigenvectors turns out to be a matter of computing the null space of a matrix, as the following exploration demonstrates.

If a vector is an eigenvector satisfying Equation (def:eigen), then clearly it also satisfies 0 .

It seems natural at this point to try to factor. We would love to “factor out” . Here is the procedure:

The middle step was necessary before factoring because we cannot subtract a scalar from an matrix is a Greek letter .

This shows that any eigenvector of is in the row spacecolumn spacenull 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 , we see that is an eigenvalue of if and only if

In theory, Exploration exp:slowdown offers us a way to find eigenvalues. To find the eigenvalues of , one can solve Equation (eqn:chareqn) for .

Eigenvalues

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

The three examples above are a bit contrived. It is not always possible to completely factor the characteristic polynomial using only real numbers. 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. (List your answers in an increasing order.)

What do you observe about the eigenvalues?

The eigenvalues are the diagonal entries of the matrix.

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

is a triangular matrix.

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 this polynomial equation for . Instead, one can often approximate the eigenvalues using iterative methods. We will explore some of these techniques in The Power Method and the Dominant Eigenvalue.

Eigenvectors

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 .

Proof
See Practice Problems prob:eigenspace1 and prob:eigenspace2.

This motivates the following definition.

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 Subspaces Associated with Matrices.

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

We conclude this section by establishing the significance of a matrix having an eigenvalue of zero.

Proof
A square matrix is singular if and only if .(see th:detofsingularmatrix). But if and only if , which is true if and only if zero is an eigenvalue of .

Practice Problems

Problems prob:eigenspace1-prob:eigenspace2 In this exercise we will prove that the eigenvectors associated with an eigenvalue of an matrix form a subspace of .

Let and be eigenvectors of associated with . Show that is also an eigenvector of associated with . (This shows that the set of eigenvectors of associated with is closed under addition).
Show that the set of eigenvectors of associated with is closed under scalar multiplication.

Problems prob:eigenspace3-prob:eigenspace4 Compute the eigenvalues of the given matrix and find the corresponding eigenspaces.

Answer: (List the eigenvalues in an increasing order.)

A basis for is . A basis for is .

Answer:

A basis for is . A basis for is .

Let . Compute a basis for each of the eigenspaces of this matrix, , , and .

Problems prob:3x3fromKuttler1-prob:3x3fromKuttler2 Let .

Compute the eigenvalues of this matrix.
One of the eigenvalues of is -3.

Answer:

(List your answers in an increasing order.)

Compute a basis for each of the eigenspaces of this matrix, , , and .

Answer: A basis for is , a basis for is ,

and a basis for is .

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.)

Problems prob:eigvectorstransfr2_1-prob:eigvectorstransfr2_3 The following set of problems deals with geometric interpretation of eigenvalues and eigenvectors, as well as linear transformations of the plane. Please use Describing Eigenvalues and Eigenvectors Algebraically and Geometrically and Geometric Transformations of the Plane 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 Describing Eigenvalues and Eigenvectors Algebraically and Geometrically.)

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 and .

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.

Exercise Source

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.