Suppose we have 3 vectors: $A_i$, $B_i$, $C_i$, where $i=1,2,3$. How to build all possible dependent 2nd rank tensors based on these vectors?
I believe, that at first we need to obtain all possible products like $A_iB_j$, $B_iC_j$, $A_iC_j$, $C_iC_j$, etc. Then tensor should be something like $T_{ij} = C_1A_iB_j+C_2B_iC_j+...$. Is that correct? Should all the product be linear independent? How to check linear independency? It seems, total amount of products should not be more than $3X3$, should it?
If $A,B,C$ are rank one tensors then $$A\otimes A\ ,\ B\otimes B\ ,\ C\otimes C$$ are rank two tensor tensors as well $$A\otimes B\ ,\ A\otimes C\ ,\ B\otimes A\ ,\ B\otimes C\ ,\ C\otimes A\ ,\ C\otimes B,$$ to begin, but also linear combination of them.