Editing Rotation (section)

=== Axis of 2-dimensional rotations ===
{{further|Rotations in two dimensions}}

2-dimensional rotations, unlike the 3-dimensional ones, possess no axis of rotation, only a point about which the rotation occurs. This is equivalent, for linear transformations, with saying that there is no direction in the plane which is kept unchanged by a 2-dimensional rotation, except, of course, the identity.

The question of the existence of such a direction is the question of existence of an [[eigenvector]] for the matrix ''A'' representing the rotation. Every 2D rotation around the origin through an angle <math>\theta</math> in counterclockwise direction can be quite simply represented by the following [[rotation matrix|matrix]]:

:<math>A =
\begin{bmatrix}
  \cos \theta & -\sin \theta\\
  \sin \theta &  \cos \theta
\end{bmatrix}
</math>

A standard [[eigenvalue]] determination leads to the [[Characteristic polynomial|characteristic equation]]
: <math>\lambda^2 -2 \lambda \cos \theta + 1 = 0,</math>
which has
: <math>\cos \theta \pm i \sin \theta</math>
as its eigenvalues. Therefore, there is no real eigenvalue whenever <math>\cos \theta \neq \pm 1</math>, meaning that no real vector in the plane is kept unchanged by ''A''.