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.