# what does this summation mean?

I apologize if this question has been asked. I know several similar ones have been asked but I cannot find one answering this in particular.

I want to know what this summation means: $$\sum_{i,j=1}^{M}$$
Is this equivalent to $$\sum_{i=1}^M \sum_{j=1}^M$$
or does it mean that i and j will be incremented together at the same time?
If context is needed, the equation is coming from here equation 16 on the third page (page 392 of the journal).

• Yes, they are equivalent. Nov 18 '14 at 13:05
• It's the sum over the set of indices $(i,j)$ with $1\leqslant i,j\leqslant M$ which is equivalent to $\sum_{i=1}^M\sum_{j=1}^M$. Nov 18 '14 at 13:05
• $\sum_{i,j=1}^{M}\equiv \sum_{i=1}^M \sum_{j=1}^M$ with no doubt. Nov 18 '14 at 13:06

$$\sum_{i,j=1}^{M}a_{i,j}=\sum_{1\le i,j\le M}a_{i,j}=\sum_{i=1}^M\sum_{j=1}^Ma_{i,j}=\sum_{j=1}^M\sum_{i=1}^Ma_{i,j}$$ means that we sum the terms $a_{i,j}$ with all the possible values of $i$ and $j$ between $1$ and $M$.