Tensor components change under rotation-translation I am currently working on a research project in a non-physics field, where I would like to work on a very constrained 2nd order tensor (3x3, symmetric, traceless). The tensor represents probability of orientation of a rigid-body in space. 
I would like to express this tensor in two different reference frames, that are both rotated AND translated relative to each other. I am using a 4x4 augmented matrix to describe this rotation-translation transformation. Naming M this rotation-translation matrix and X the tensor, I try to make sense of:
X-Transpose(M).X'.M    
where X and X' are seen from different frames and M is the (non-orthogonal) 4x4 rotation-translation matrix. 
As I understand, I would first have to find a way to separate M into a rotation and a translation matrix. The change of frame by rotation may then be expressed by simple similarity transformation (?). However, I don't know how to deal with the translation part. 
I would assume the tensor components are written differently in rotated frames. However, would they be exactly identical if seen from translated frames? More specifically, how do the tensor components change under translation of the reference frame?
I would happily provide additional information in case the phrasing of my question is incomplete. 
 A: There's a trick which is often used in computergraphics to account for rotations + translations in one single matrix multiplication.
If $R$ is you rotation matrix and $\vec{t}$ is your translation vector you construct the following rotation-translation-matrix:
$$ M = \left( \matrix{. . .  \ \ |\ \ . \\
                      . R . \ \ |\ \ \vec{t} \\ 
                      ... \ \ |\ \ . \\
                      0 0 0 \ \ |\ \ 1} \right) $$
and a vector $\vec{r} = (x,y,z)^T$ which is to be rotated and translated is replaced by:
$$ \vec{r}\ ' =\left( \matrix{.\\\vec{r}\\.\\1} \right) = \left( \matrix{x\\y\\z\\1} \right) $$
Therefore you have:
$$ M \cdot \vec{r}\ ' = \left( \matrix{.\\R\cdot\vec{r} + \vec{t}\\.\\1} \right)$$
which in turn can be brought back to a 3D-vector. Now comes the tricky part. What does it mean to translate a tensor? Well, tensors operate on vector spaces and not on affine spaces and therefore a translation of a tensor isn't defined.
What you can do, if the tensor is expressible in terms of vector components, is to take the definition of your tensor and transform the vector components according to your rotation-translation-matrix $M$. There may or may not be a general expression which allows this transformation to be expressed as purely $M$ operating on the generalized tensor components.
