I am trying to do a computation and I'm not very good with actually using tensors for computations.
I have some $(n\times n)$ matrices $A = [a_{ij}]$ and $B = [b_{ij}]$ and some kind of rank-3 tensor $T = [T_{ijk}]$, where again $i,j$ and $k$ go from $1$ to $n$. In the course of the computation I have various terms with an odd number of repeated indices. Is there any good way to think about them or express them?
e.g. I have an expression like $$ \sum_{k=1}^n a_{ik}T_{kkj}. $$ Somehow one would like this to be the $(i,j)^{th}$ entry of something, but is that actually the case or is it just some weird combination of entries? There are three $k$ indices in the sum and that seems to make it weird.
Similarly I have something elsewhere like $$ \sum_{k,l,m=1}^n a_{ik}a_{kl}b_{lm}T_{kmj}. $$ Again you would normally want this to be the $(i,j)^{th}$ entry of a product but it doesn't seem to quite work???
$ \def\d{{\vec\delta}} \def\qiq{\quad\implies\quad} $Take the standard Kronecker delta symbol $$\eqalign{ \delta_{ij} &= \begin{cases} {\tt1}\qquad {\rm if}\;i=j \\ 0\qquad {\rm otherwise} \\ \end{cases} }$$ and extend it to three indices $$\eqalign{ \d_{ijk} &= \begin{cases} {\tt1}\qquad {\rm if}\;i=j=k \\ 0\qquad {\rm otherwise} \\ \end{cases} }$$ This extended Kronecker delta can be used to restore the usual Einstein convention $$\eqalign{ B_{ij} &= A_{il}\d_{lkm}\d_{mpq}T_{pqj} \qiq B_{ij}=\sum_k A_{ik}T_{kkj} \\ }$$ This result can also be written without indices by using explicit single/double-dot products $$\eqalign{ B &= A\cdot\d\cdot\d:T \\ }$$