Rotation matrix vs regular matrix

1. Nov 15, 2011

dmoney123

Can you calculate eigenvalues and eigenvectors for rotation matrices the same way you would for a regular matrix?

If not, what has to be done differently?

2. Nov 15, 2011

Deveno

it don't see why not. the existence of real eigenvalues will depend on the angle of rotation (most angles will give complex eigenvalues).

3. Nov 15, 2011

chiro

The determinant of a rotation matrix should always be 1 (since it preserves length) so there should always be eigenvalues and eigenvectors that can be calculated given a rotation matrix.

4. Nov 16, 2011

Deveno

the characteristic polynomial of:

$$\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}$$

is:

$$x^2 - (2\cos\theta)x + 1$$

which has real solutions only when:

$$4\cos^2\theta - 4 \geq 0 \implies \cos\theta = \pm 1$$

for angles that aren't integer multiples of pi, this will lead to complex eigenvalues.

5. Nov 16, 2011

AlephZero

... and the eigenvalues are $\cos\theta \pm i \sin\theta$. Now, I wonder what that fact might have to do with "rotation"...

6. Nov 16, 2011

chiro

What has that got to do with what I said?