Skim-read an engineering book stopped to this assertion "matrix dot product is $\sum_{i,j \in I} A_{i,j} B_{i,j} := A_{i,j} B_{i,j}$ in Einstein Notation". Sorry but what does it really mean? I have never read about matrix dot-product, rather about column vector dot-product (but the book was generalizing dot-product to non-column or non-row matrixes $n \times p$ where $n \not =1$ and $p \not = 1$.
Does it mean:
$\sum_{i,j \in I} A_{i,j} B_{i,j} = \sum_{i \in I}\sum_{j \in I} A_{i,j} B_{i,j}$
or does it mean a sum of all possible dot products between $a_{i}$ and $b_{j}$ for all ${i}$ and $j$?
or does it mean the sum of dot products between corresponding $a_{i}$ and $b_{i}$ for all $i$?
[Context]
The foreign book also used the notation:
$\nabla \cdot \sum_{i\in I} v_{i,i} := v_{i,i}$
and called it Einstein notation as well in the context of "Continuum models" which I did not fully understand, apparently analyzing some continuous models.