3d rotational transformation of inertia properties
The inertia tensor, \mathbf I, relates angular momentum \mathbf L to angular velocity \boldsymbol\omega:
\mathbf L = \mathbf I \boldsymbol\omega
When rotating a rigid body, the components of vectors like angular momentum and angular velocity change. A rotation matrix, \mathbf T, transforms a vector from one frame to another, such as:
\mathbf V_2 = \mathbf T \mathbf V_1
Applying this to angular momentum and angular velocity, I have:
\mathbf I_2 \boldsymbol\omega_2 = \mathbf T \mathbf I_1 \mathbf T^{-1} \boldsymbol\omega_2
Because rotation matrices are orthonormal matrices, \mathbf T^{-1} = \mathbf T^T, I can write the final transformation as:
\mathbf I_2 = \mathbf T \mathbf I_1 \mathbf T^T
This equation allows me to calculate the inertia tensor in a rotated coordinate system once I know the original inertia tensor, \mathbf I_1, and the rotation matrix, \mathbf T.
For more insights into this topic, you can find the details here.