$A_{i,j}B_{i,j}$ is matrix dot-product in Einstein Notation?

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.

-
See en.wikipedia.org/wiki/Einstein_notation. I'm not very familiar with that notation, but I'm pretty sure some of those subscripts should be superscripts. –  Ilmari Karonen Sep 18 '11 at 18:24
I think they mean Frobenius inner product $A:B$. –  user13838 Sep 18 '11 at 18:29
percusse: you must be close, I saw that notation also. I will check it when I see the book again. –  hhh Sep 18 '11 at 19:26

Firstly, there's a bit of abuse-of-notation going on here; the convention for Einstein Summation is that 'upper' and 'lower' tensor indices can be contracted, but not two upper or two lower indices. Thus one can have, for instance, $A_{ij}B^{ij}$ or $A_{i}{}^{j}B^{i}{}_{j}$ or $A^{ij}B_{ij}$, but $A_{ij}B_{ij}$ doesn't make sense without some implicit raising/lowering of indices happening. When the indices are in the right places, then the 'Einstein convention' that they're talking about is simply that any index appearing twice in an expression - once 'on top' and once 'on the bottom' is to be summed over, and so cancelled out; for instance, with a matrix written in the form $A^i{}_j$, the matrix-vector product becomes simply $A^i{}_jv^j = w^i$. Notice how the $i$ index 'disappears' here by being summed over - this effectively says that each element of the vector $w$ is the dot product of $v$ with the corresponding row of the matrix $A$. When the metric is Euclidean (i.e., in 'normal' Newtonian mechanics) then indices are trivially raised and lowered, but putting them in the right spots can sometimes help with intuition (e.g., for matrix product: $A^i{}_jB^j{}_k = C^i{}_k$). Note that in this notation, the vector dot product is simply $v^iw_i$ - and this shows how the dot product is implicitly dependent on the metric, because the metric is needed to lower the index on $w$ (vectors are traditionally written with their index raised).