How to unify a sigma in a sigma to one sigma?

Question

For example lets say you want to express ABC (with ABC being matrices). With A being m x r, B being n x r and C being r x z.

I want to express this with a sigma notation so I know: $(ABC)_{ij}$ = $\sum_{k=1}^r (AB)_{ik}C_{kj}$

=> $\sum_{k=1}^r \sum_{l=1}^n A_{il}B_{lk}C_{kj}$

Now if I want to bring this together it will become $\sum_{k=1,l=1}$ but to what? What will be above the sigma sign? Will it be till

Answer 1

You could write

$$\sum_{k=1}^r\sum_{l=1}^nA_{il}B_{lk}C_{kj}=\sum_{k,l=1}^{k=r,l=n}A_{il}B_{lk}C_{kj}=\sum_{k,l=1}^{r,n}A_{il}B_{lk}C_{kj}=\sum_{k,l}A_{il}B_{lk}C_{kj}$$

but for clarity, it is better to write the double sum once before reducing it. Let it also be mentioned that such informal notation is more often used when the upper limits are equal, i.e., $r=n$.

If $r=n$, you could drop the sigmas altogether with Einstein's summation convention:

$$\sum_{k=1}^r\sum_{l=1}^nA_{il}B_{lk}C_{kj}\equiv A_{il}B_{lk}C_{kj}.$$

Answer 2

There's no really good way to notate that as a single summation. If you completely insist, you could write it as $$ \sum_{(k,l)\in\{1,\ldots r\}\times\{1,\ldots,n\}} A_{il}B_{lk}C_{kj} $$ but it would be hard to say that makes the whole thing more readable than the double sum.