Given a $3 \times 3$ rotation matrix $\mathbf{R}$ with an associated $3 \times 3$ covariance matrix $\mathbf{P}$ how do I compute the associated $4 \times 4$ covariance matrix $\mathbf{Q}$ of the corresponding quaternion $\mathbf{q}$?
The reason for this is that I am using a bunch of rotation matrices estimated by SO(3) Extended Kalman Filters in an Extended Information Filter.
However, I don't know a way of keeping multiple rotation matrices in the state $\mathbf{X}$ of the Extended Information Filter so that the information vector is given by $\mathbf{i} = \mathbf{P}^{-1}\mathbf{X}$. Instead, I want to convert the rotation matrices to quaternions and use the quaternion elements to represent each rotation matrix.
Thus, comes the problem with converting the covariance...