A
linear mapping
from to is described by a
-matrix
with respect to the
standard basis.
We consider the eigenvalues for some elementary examples. A
homothety
is given as , with a scaling factor
.
Every vector
is an
eigenvector
for the
eigenvalue
, and the eigenspace for this eigenvalue is the whole . Beside , there are no other eigenvalues, and all eigenspaces for
are . The identity only has the eigenvalue .
The
reflection
at the -axis is described by the matrix . The eigenspace for the eigenvalue is the -axis, the eigenspace for the eigenvalue is the -axis. A vector with
is not an eigenvector, since the equation
-
does not have a solution.
A
plane rotation
is described by a rotation matrix for the rotation angle
,
For
,
this is the identity, for
,
this is a half rotation, which is the reflection at the origin or the homothety with factor . For all other rotation angles, there is no line sent to itself, so that these rotations have no eigenvalue and no eigenvector
(and all eigenspaces are ).