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Rotation matrix vs regular matrix

  1. Nov 15, 2011 #1
    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. jcsd
  3. Nov 15, 2011 #2

    Deveno

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    it don't see why not. the existence of real eigenvalues will depend on the angle of rotation (most angles will give complex eigenvalues).
     
  4. Nov 15, 2011 #3

    chiro

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    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.
     
  5. Nov 16, 2011 #4

    Deveno

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    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.
     
  6. Nov 16, 2011 #5

    AlephZero

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    ... and the eigenvalues are [itex]\cos\theta \pm i \sin\theta[/itex]. Now, I wonder what that fact might have to do with "rotation"... :smile:
     
  7. Nov 16, 2011 #6

    chiro

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    What has that got to do with what I said?
     
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