I am trying to figure out whether or not the following identity is true:
$D_{kl}A_{ij}A_{kl}=A_{kl}D_{kl}A_{ij}$
Or what I think is qually:
$D:(A\otimes A)= (A:D)A$
The matrices are both symmetric $A=A^T, B=B^T$.
I tried to "prove" it by simply doing the calculus with 2x2 matrices but I always end up wit another (wrong) result. I would very much appreciate it if you could help me or give me some hints whether this is right or wrong.
When employing index notation, you are effectively just summing over repeated indices, so if the resulting matrix is $M$, then
$$ M_{ij} = \sum_{k=1}^n \sum_{l=1}^n D_{kl}A_{ij}A_{kl}= \sum_{k=1}^n \sum_{l=1}^nA_{kl}D_{kl}A_{ij} = \sum_{k=1}^n \sum_{l=1}^n A_{ij}D_{kl}A_{kl}= \cdots $$
The terms being multiplied inside the summation can be taken in any order.
Thus, given definitions for "$\otimes$" and "$:$" you can come up with different interpretations to an operation which is effectively just summing up the product of elements in two matrices/tensors in a certain order. All this provided the dimensions of the matrices allow for such an operation.