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Deveno
#4
Nov16-11, 11:27 AM
Sci Advisor
P: 906
Rotation matrix vs regular matrix

Quote Quote by chiro View Post
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.
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.