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I know that in 2-dimension, a rotation matrix has no non-zero eigenvector. But in 3-d space, I can only imagine the rotation axis to be an eigenvector, and it looks like that the rotation angle can't be represent through eigenvectors or eigenvalues. So I guess in 3-dimension, any rotation matrix has and only has one eigenvector. Am I right?

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    What if the matrix rotates $180^{o}$? Consider 2D and 3D cases.2012-01-26

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