I would like to know why,as a general rule and which under conditions, one says that tensor product is commutative unlike matricial product.
For example, for a product of 2 matrices, we get the element (i,j) by writing (with Einstein notations) :
$$C_{ij} = A_{ik} B_{kj}$$
and
$$D_{ij} = B_{ik} A_{kj}$$
So $C_{ij} \neq D_{ij}$
But with tensors, it seems that it doesn't matter to multiply the first tensor by the second or the contrary :
$$C_{ik}= A_{il}B_{lk} = B_{lk}A_{il}=B_{kl}A_{li}=D_{ki}$$
You can notice that I have supposed A and B as symetric tensors.
What are the rules for a tensor to respect for not consider the order in multiplications and get commutative product (unlike matricial product) ?
Taking symetric tensors is sufficient ? I don't think so since there are also antsymetric tensors (like Maxwell tensor).
If someone could give me a simple example highlighting this issue ?
Any clarifications is welcome, regards.