Notation Help: Understanding Summations

[holezch]

Homework Statement

Hi, I see something like this in my book:

\sum_{i,j} A_{i,j}... et c

does this mean that the index values of i and j are both accumulated at the same time? or that i or j gets accumulated first? I'm not sure

thanks

 

[Office_Shredder]

Generally that really means you have a double summation, that i and j are independent of each other and both range over the natural numbers (or whatever domain makes sense).

You can also think of it as summing over all pairs (i,j) in \mathbb{N}x\mathbb{N}.

Either way, implicitly stated is that the summation is independent of order (since none is given)

 

[CompuChip]

It is usually shorthand for
\sum_i \sum_j \cdots \sum_n A_{ij\cdots n},
i.e. a nested summation.

 

[HallsofIvy]

If, for example, i and j can both range from 1 to 3, that is
A_{11}+ A_{12}+ A_{13}+ A_{21}+ A_{22}+ A_{23}+ A_{31}+ A_{32}+ A_{33}.

That is, A_{ij} has 3(3)= 9 values and this is the sum of all of them. Since addition of numbers is commutative, the order does not matter so the order in which you take i or j does not matter.

More generally, if i ranges from 1 to m and j ranges from 1 to n, A_{ij} can have mn values and \sum_{i, j} A_{ij} is the sum of all of them.