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
-
![{\displaystyle {}(s,-t)=\lambda (s,t)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0587414dbbd0ae72ddc2c69a68c6adbd817c3c4)
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
).