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.
Here is the most important definition in this text.
Definition
Let be an matrix.
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
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 6.2.
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.
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.
Let be eigenvectors of a matrix and suppose that the corresponding eigenvalues are distinct (all different from each other). Then is linearly independent.
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.
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 6.2.
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 eigenvalue 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.,
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
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
Let be an matrix and let be a number.
We conclude with an observation about the -eigenspace of a matrix.
Let be an matrix.
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 4.6. 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
We now have two new ways of saying that a matrix is invertible, so we add them to the invertible matrix theorem.
Let be an matrix, and let be the matrix transformation The following statements are equivalent:
