Determining The Axis
Given a rotation matrix R, a vector u parallel to the rotation axis must satisfy
since the rotation of around the rotation axis must result in . The equation above may be solved for which is unique up to a scalar factor.
Further, the equation may be rewritten
which shows that is the null space of . Viewed another way, is an eigenvector of R corresponding to the eigenvalue (every rotation matrix must have this eigenvalue).
Read more about this topic: Rotation Matrix, In Three Dimensions, Conversion From and To Axis-angle
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