Transformation matrix on tensors

[kernelinho]

Hello.

I wasn't sure whether to post this here on in some of the physics sections.

I have a rank 2 tensor in one coordinate reference system [x1, x2, x3], the one where only the principal elements are non zero: R=[ a11 0 0; 0 a22 0; 0 0 a33 ].

I want the tensor R in some other orthogonal coordinate reference system. I have the transformation matrix U from the system [x1, x2, x3] to the second one [X1, X2, X3].

I know how to use U to transform vectors from one system to the other:

[V1; V2; V3]= U [v1; v2; v3]

But I don't know what operation to do to transform a tensor. I'm led to believe that it could be something like

[R(in Xi)] = U^-1 R(in xi) U

But I'm not sure whether this is right nor what's the rationale for it.

I would appreciate any help you could give me.

 

[HallsofIvy]

Same thing. Think of the tensor and transformation as matrices and multiply the matrices.

 

[slider142]

In addition, even if the tensor was not representable as a matrix (two dimensional rectangular array of numbers), you still have the functional representation, T(v) is a multilinear map from some space V to some space W, and you have linear maps C and C' from V to V' and W to W' respectively, where V' and W' are identical to V and W except for the coordinate maps for their elements. Then C'(T(C^-1v)) is T with the coordinate change applied, as T can only act on objects in V and transforms them to W objects. The C's take care of translating the objects from V' and into W', where if v is a multivector, the linear transformations are distributed appropriately. By linearity, we have C'TC^-1 as the proper tensor, as you have already surmised.