What is the relationship between a quanternion and the eigenvalues and eigenvectors of equivalent 3D rotation matrix?
A relation in terms of $\lambda_r, \mathbf{v}_{\lambda_r}, \mathbf{v}_{\lambda_1}$ in the following: (Correct me about the details if I am wrong)
If I understand correctly, a 3D rotation matrix has an eigenvalue $\lambda_1=1$ and two conjugate eigenvalues $\lambda_r, \overline{\lambda_r}$.
- The axis of 3D rotation will be $\mathbf{v}_{\lambda_1}$ (The eigenvector corresponding to eigenvalue $\lambda_1=1$).
- The angle of 3D rotation will be $\operatorname{arctan2}\left(\operatorname{Im}(\lambda_r), \operatorname{Re}(\lambda_r)\right)$, where $\operatorname{arctan2}$ is the two-argument arctangent.
Can we write the quaternion's four coordinates for such 3D rotation in terms of $\lambda_r, \mathbf{v}_{\lambda_r}, \mathbf{v}_{\lambda_1}$?
No. First of all, any correspondence of quaternions with rotations depends on an arbitrary choice of three orthogonal spatial directions to correspond to the there kinds of imaginary units in the quaternions, so at the very least one would need to use that correspondence (rather than a nice formula) to go from spatial data like the axis to quaternions.
Second, even fixing the correspondence, there are two (opposite) unit quaternions for every rotation; no formula can uniformly make a choice between those two possibilities in a continuous way.
Third, given an axis and a characteristic polynomial of a rotation (so with one root $1$ and two other complex conjugate roots of absolute value $1$), there are two rotations with that axis and that characteristic polynomial (they turn in opposite directions). So one cannot even determine the rotation uniquely from the data you provide; this gives another problem similar to the second point.