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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

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2 Answers

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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 $$ RTR^{-1} = T^{'} $$

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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:

1. First transform our vector v with Tv and then apply the rotation i. e RTv
2. 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'

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