Eigenvalues and Eigenvectors
One of the ways of thinking about n x n matrices is to consider them as defining transformations of Rn . Examples of the simplest transformations are those which stretch or shrink vectors in certain directions. For instance, the matrix
transforms the standard unit vector e 1 in R2 intoitself and the standard unit vector e 2 into 2e2. These two statements are clear from the form of the matrix. We want to consider less obvious cases where it again occurs that a square matrix sends some vector to a multiple of itself. (The vector 0 is always sent to itself, but that case is not particularly interesting.)
Def. Let A be an n x n matrix. A nonzero vector v in Rn is called an eigenvector of A if Av = lv for some scalar l. The scalar l is called the eigenvalue of A that corresponds to the eigenvector v.
Examples:
Determine whether a
given vector is an eigenvector ofa given matrix, and if so, what the
corresponding eigenvalue is.
For a given 2
x 2 matrix, determine all of the eigenvectorsand
eigenvalues.
Important Note
The scalar 0 may be an eigenvalue of a
square matrix A.
But the vector 0 cannot be
an eigenvector.
Examples
Notice that the definition of eigenvector allows us to reduce the question of finding eigenvectors corresponding to a given eigenvalue to the standard problem of finding the solutions of a homogeneous linear system.
Eigenvector Theorem
Let A be an n x n matrix,
with eigenvalue l.
The eigenvectors of A corresponding to
l are the nonzero solutions
of the linear system
(A -
lIn
)x
= 0.
Definition Let A be an n x n matrix with eigenvalue l. The nullspace of (A - lIn )is called the eigenspace of A corresponding to l. Notice that the nullspace consists exactly of the collection of all the eigenvectors corresponding to l and, additionally, the zero vector 0.
Examples
Examples: Finding a Basis for an Eigenspace
