In this section, we define eigenvalues and eigenvectors. These form the most important facet of the structure theory of square matrices. As such, eigenvalues and eigenvectors tend to play a key role in the real-life applications of linear algebra.

Subsection5.1.1Eigenvalues and Eigenvectors

Here is the most important definition in this text.

Definition

Let be an matrix.

An eigenvector of is a nonzero vector in such that for some scalar

An eigenvalue of is a scalar such that the equation has a nontrivial solution.

If for we say that is the eigenvalue for and that is an eigenvector for

The German prefix “eigen” roughly translates to “self” or “own”. An eigenvector of is a vector that is taken to a multiple of itself by the matrix transformation which perhaps explains the terminology. On the other hand, “eigen” is often translated as “characteristic”; we may think of an eigenvector as describing an intrinsic, or characteristic, property of

Note

Eigenvalues and eigenvectors are only for square matrices.

Eigenvectors are by definition nonzero. Eigenvalues may be equal to zero.

We do not consider the zero vector to be an eigenvector: since for every scalar the associated eigenvalue would be undefined.

If someone hands you a matrix and a vector it is easy to check if is an eigenvector of simply multiply by and see if is a scalar multiple of On the other hand, given just the matrix it is not obvious at all how to find the eigenvectors. We will learn how to do this in Section 5.2.

To say that means that and are collinear with the origin. So, an eigenvector of is a nonzero vector such that and lie on the same line through the origin. In this case, is a scalar multiple of the eigenvalue is the scaling factor.

For matrices that arise as the standard matrix of a linear transformation, it is often best to draw a picture, then find the eigenvectors and eigenvalues geometrically by studying which vectors are not moved off of their line. For a transformation that is defined geometrically, it is not necessary even to compute its matrix to find the eigenvectors and eigenvalues.

Example(Reflection)

Here is an example of this. Let be the linear transformation that reflects over the line defined by and let be the matrix for We will find the eigenvalues and eigenvectors of without doing any computations.

This transformation is defined geometrically, so we draw a picture.

The vector is not an eigenvector, because is not collinear with and the origin.

The vector is not an eigenvector either.

The vector is an eigenvector because is collinear with and the origin. The vector has the same length as but the opposite direction, so the associated eigenvalue is

The vector is an eigenvector because is collinear with and the origin: indeed, is equal to This means that is an eigenvector with eigenvalue

It appears that all eigenvectors lie either on or on the line perpendicular to The vectors on have eigenvalue and the vectors perpendicular to have eigenvalue

We will now give five more examples of this nature

Here we mention one basic fact about eigenvectors.

Fact(Eigenvectors with distinct eigenvalues are linearly independent)

Let be eigenvectors of a matrix and suppose that the corresponding eigenvalues are distinct (all different from each other). Then is linearly independent.

Suppose that were linearly dependent. According to the increasing span criterion in Section 2.5, this means that for some the vector is in If we choose the first such then is linearly independent. Note that since

Since is in we can write

for some scalars Multiplying both sides of the above equation by gives

Subtracting times the first equation from the second gives

Since for this is an equation of linear dependence among which is impossible because those vectors are linearly independent. Therefore, must have been linearly independent after all.

When this says that if are eigenvectors with eigenvalues then is not a multiple of In fact, any nonzero multiple of is also an eigenvector with eigenvalue

As a consequence of the above fact, we have the following.

An matrix has at most eigenvalues.

Subsection5.1.2Eigenspaces

Suppose that is a square matrix. We already know how to check if a given vector is an eigenvector of and in that case to find the eigenvalue. Our next goal is to check if a given real number is an eigenvalue of and in that case to find all of the corresponding eigenvectors. Again this will be straightforward, but more involved. The only missing piece, then, will be to find the eigenvalues of this is the main content of Section 5.2.

Let be an matrix, and let be a scalar. The eigenvectors with eigenvalue if any, are the nonzero solutions of the equation We can rewrite this equation as follows:

Therefore, the eigenvectors of with eigenvalue if any, are the nontrivial solutions of the matrix equation i.e., the nonzero vectors in If this equation has no nontrivial solutions, then is not an eigenvector of

The above observation is important because it says that finding the eigenvectors for a given eigenvalue means solving a homogeneous system of equations. For instance, if

then an eigenvector with eigenvalue is a nontrivial solution of the matrix equation

This translates to the system of equations

This is the same as the homogeneous matrix equation

i.e.,

Definition

Let be an matrix, and let be an eigenvalue of The -eigenspace of is the solution set of i.e., the subspace

The -eigenspace is a subspace because it is the null space of a matrix, namely, the matrix This subspace consists of the zero vector and all eigenvectors of with eigenvalue

Note

Since a nonzero subspace is infinite, every eigenvalue has infinitely many eigenvectors. (For example, multiplying an eigenvector by a nonzero scalar gives another eigenvector.) On the other hand, there can be at most linearly independent eigenvectors of an matrix, since has dimension

is an eigenvalue of if and only if has a nontrivial solution, if and only if

In this case, finding a basis for the -eigenspace of means finding a basis for which can be done by finding the parametric vector form of the solutions of the homogeneous system of equations

The dimension of the -eigenspace of is equal to the number of free variables in the system of equations which is the number of columns of without pivots.

The eigenvectors with eigenvalue are the nonzero vectors in or equivalently, the nontrivial solutions of

We conclude with an observation about the -eigenspace of a matrix.

Fact

Let be an matrix.

The number is an eigenvalue of if and only if is not invertible.

In this case, the -eigenspace of is

Proof

We know that is an eigenvalue of if and only if is nonzero, which is equivalent to the noninvertibility of by the invertible matrix theorem in Section 3.6. In this case, the -eigenspace is by definition

Concretely, an eigenvector with eigenvalue is a nonzero vector such that i.e., such that These are exactly the nonzero vectors in the null space of