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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?
If not, what has to be done differently?
the characteristic polynomial of: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.
... and the eigenvalues are [itex]\cos\theta \pm i \sin\theta[/itex]. Now, I wonder what that fact might have to do with "rotation"...the characteristic polynomial of:
[tex]\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}[/tex]
is:
[tex]x^2 - (2\cos\theta)x + 1[/tex]
What has that got to do with what I said?the characteristic polynomial of:
[tex]\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}[/tex]
is:
[tex]x^2 - (2\cos\theta)x + 1[/tex]
which has real solutions only when:
[tex]4\cos^2\theta - 4 \geq 0 \implies \cos\theta = \pm 1[/tex]
for angles that aren't integer multiples of pi, this will lead to complex eigenvalues.