notation clarifications

Question

In many articles, I see notations of the form

$$ \sum\limits_{i,j = 1..n} \mathbf{X}_{ik}\mathbf{Y}_{kj} $$

and $$ \sum\limits_{i= 1..n \\ j=1..n} \mathbf{X}_{ik}\mathbf{Y}_{kj} $$

What is the difference in the two summations ?. Is the first one sums like $[i=1,j=1], [i=2,j=2], [i=3,j=3] ...[i=n-1, j=n-1], [i=n, j=n]$, and second one sums like $[i=1,j=1], [i=1,j=2], [i=1,j=3] ... [i=n,j=1], [i=n, j=2] ... [i=n, j=n]$

Answer 1

It is a simplification of the notation for a double sum: $$ \sum_{i,j=1..n} = \sum\limits_{i= 1..n \\ j=1..n} = \sum_{i=1}^n\sum_{j=1}^n = \sum_{j=1}^n\sum_{i=1}^n $$ You have finite sums and the notation claims that you have the association law, so the order is irrelevant.

Remark: You first assumption

[i=1,j=1], [i=2,j=2], [i=3,j=3] ...[i=n-1, j=n-1], [i=n, j=n]

is something totally different, because there $i$ and $j$ are always the same. So $$ X_{1k}X_{k1}+X_{2k}X_{k2}+\ldots+X_{nk}X_{kn}=\sum_{i=1..n} X_{ik}X_{kj}. $$ But if you have more than one variable for the sum, each has to go through the given set of numbers independently.

Answer 2

I believe that both have the same meaning and the first and second both are short forms for what should have been $$ \sum_{i=1}^n(\sum_{j=1}^nX_{kj}) X_{ik}$$

If the first was like $[i=1,j=1],[i=2,j=2],[i=3,j=3]... $ only one variable could have been used instead.