tensor rotation why does tensor rotation require multiplication by the rotation matrix twice, once from the right and once from the left by the inverse?
if $T$ is the tensor I wish to rotate and $R$ is the rotation matrix, why isn't $T'=RT$ but is $T=RTR^{-1}$?
I have seen and understood the construction of tensor transformation, but I am intuitively uncomfortable with it.
thank you
 A: Consider the following example
Let's say we have a certain vector v that we would like to first transform somehow and then rotate.(Lets say that the transformation we want to apply to v is a simple scaling, which is simpler to describe wrt to orthodox v i.e. before we rotate it) This leads us to having two options for the operations:

*

*First transform our vector v with Tv and then apply the rotation i. e RTv

*First apply the rotation and then transform the vector i.e. TRv

Doing this would yield 2 different results, in order to avoid this if we go down the 2nd pathway we would have to adjust our transformation matrix T by "rotating" it i.e. in order to get the same result from the first pathway RTv = b we would have to adjust our transformation T to T' s.t. T'Rv = b which leads to the simple description of how we should adjust T' i.e.
RT = T'R
multiply on the right with R^(-1) would yield the following relationship between the transformed T tensor (T') and the original one T
RTR^(-1) = T'
A: You are considering the transformation law of the tensors and this depends on the nature of the tensor. Vectors transform in a certain way and other objects transform in other ways. The transformation of the $T$ you are talking about can be understood as follows.
Consider rotating a vector $v$ by $R$
$$
v^{'}=Rv
$$
The operator $T$ maps $v$ to $Tv$.
In the rotated frame the rotated operator $T^{'}$ maps $v^{'}$ to $T^{'}v^{'}$
The mapping $v \to Tv$ can also be achieved via a different pathway
i.e. by transforming to the rotated frame and then back again.
Step 1. Rotate the vector $v$ to give $Rv$
Step 2. Apply $T^{'}$ to the rotated vector, giving $T^{'}Rv$
Step 3. Rotate back to the original frame. This needs $R^{-1}$, giving $R^{-1}T^{'}Rv$
This has shown
$$
Tv = R^{-1}T^{'}Rv
$$
from which follow
$$
T = R^{-1}T^{'}R
$$
and
$$
RTR^{-1} = T^{'}
$$
