We introduce the concepts of eigenvalues and eigenvectors of a matrix.

EIG-0010: Describing Eigenvalues and Eigenvectors Algebraically and Geometrically

At several places in this course it has been valuable to restrict ourselves to square matrices, and we do so again when discussing eigenvalues and eigenvectors.

In Theorem th:matrixtran of Module LTR-0010, we proved that any matrix induces a linear transformation from to itself. For our first few examples, let us consider the case .

Let . The following animation helps us to visualize the matrix transformation associated with .

Given a vector in , the vector is also in . For many vectors, will not be pointing in the same direction as . This is the case for any of the gray vectors in the animation, as we can see that points in a different direction than . But if we look at the red vectors (vectors parallel to ), we notice that they appear unchanged in magnitude and direction. Such vectors are sometimes called fixed vectors of .

Looking next at the blue vectors (vectors parallel to ), we observe that the magnitudes of the vectors are changed, but the direction in which the blue vectors point is unchanged by this linear transformation.

In Exploration init:eignintro we found that certain vectors do not change direction under the linear transformation induced by matrix . Such vectors are examples of eigenvectors of .

In general, any nonzero vector whose image under a matrix transformation is parallel to the original vector is called an eigenvector of the matrix that induced the transformation. The following definition captures this idea algebraically.

In Exploration init:eignintro we observed visually that vectors parallel to were eigenvectors, as they changed length but did not change direction under the linear transformation. To verify this algebraically, observe that all vectors parallel to can be written in the form , . We compute This shows that any non-zero scalar multiple of is an eigenvector of which has a corresponding eigenvalue of 3.

Fixed vectors of Exploration init:eignintro are also eigenvectors. For example, This shows that is a fixed vector and an eigenvector of which has a corresponding eigenvalue of .

A couple of finer points of Definition def:eigen require clarification.

  • The definition requires that eigenvectors be non-zero. Imagine what would happen if we allowed to be an eigenvector of . Clearly for all scalars . This means that every number would be an eigenvalue of every matrix. Because eigenvalues are supposed to capture certain information about the matrix, allowing every number to be an eigenvalue of every matrix would defeat the purpose.
  • Up to now we had talked about eigenvectors as vectors whose images under a matrix transformation are parallel to the original vectors. But the algebraic definition allows non-zero vectors that map to zero to be considered eigenvectors. (What would an eigenvalue of such an eigenvector be?) The zero vector has no direction, so we cannot say that the image of such an eigenvector is parallel to the original vector. Example ex:eigen will illustrate this point.

A natural question is this: does every square matrix have eigenvalues and eigenvectors? We will see in Module EIG-0020 that the answer to this question is “yes”, provided that we permit eigenvalues and entries of eigenvectors to be complex numbers. The next example is one that requires complex numbers.

We will continue to work with complex numbers as we study eigenvalues and eigenvectors.

Why All the Fuss About Eigenvalues and Eigenvectors?

Practice Problems

Let .
Show that is an eigenvector of . What is its corresponding eigenvalue?
Show that is an eigenvector of . What is its corresponding eigenvalue?
Show that is an eigenvector of . What is its corresponding eigenvalue?
Let . Note that takes any vector in and projects it onto the -axis, as we learned in Practice Problem of Module LTR-0020. Which vectors in would be eigenvectors, and what are the corresponding eigenvalues?
Returning to Example ex:eigsrotation, let . Show that is an eigenvector of . What is its corresponding eigenvalue?
Arguing geometrically, identify the linear transformation whose standard matrix has eigenvalues and .
Vertical Shear Horizontal Shear Counterclockwise Rotation through a angle Reflection About the line Horizontal Stretch Vertical Stretch
Let . Can you find an eigenvector and its corresponding eigenvalue? Can you find another “eigenpair”? Can you find all of the eigenvectors of ?
The rotation matrix in Example ex:eigsrotation has complex eigenvectors and eigenvalues. Think geometrically to find an example of a (non-identity) rotation matrix with real eigenvectors and eigenvalues.

Answer: Rotation through degrees.

Can an eigenvalue have multiple eigenvectors associated with it?
Yes No

Can an eigenvector have multiple eigenvalues associated with it?

Yes No


[Trefethen and Embree] Trefethen, Lloyd and Embree, Mark, Spectra and Pseudospectra, Princeton University Press, 2005, p. 5-6