What does \sum\limits_{i\neq j}^N a_i a_j mean in summation notation?

[SergeantAngle]

Hi

I have a textbook which uses the notation:

$\sum\limits_{i\neq j}^N a_i a_j$

I can't find anywhere what this actually means. Is it equivalent to:

$\sum\limits_{i}^N \sum\limits_{j}^N a_i a_j$

where j can't equal i?

Thanks.

 

[1MileCrash]

It's just the sum of all $a_{i}a_{j}$ when i does not equal j.

You can put pretty much any condition you want in that space, though I see it abused more often for unions and intersections of sets.

 

[Mark44]

Hi

I have a textbook which uses the notation:

$\sum\limits_{i\neq j}^N a_i a_j$

I can't find anywhere what this actually means. Is it equivalent to:

$\sum\limits_{i}^N \sum\limits_{j}^N a_i a_j$

where j can't equal i?

Thanks.

Yes, pretty much. For example, if N = 3, and both indexes start at 1, then the summation expands to a₁a₂ + a₁a₃ + a₂a₁ + a₁a₃ + a₂a₃ + a₃a₂. The terms that are omitted are a₁², a₂², and a₃².

 

[SergeantAngle]

Okay, thanks.

 

[CellCoree]

Hi there,

Thank you for reaching out. The notation \sum\limits_{i\neq j}^N a_i a_j means the sum of all the terms a_i a_j where i and j are not equal, from i=1 to N. This is different from \sum\limits_{i}^N \sum\limits_{j}^N a_i a_j, which would include all possible combinations of a_i and a_j, including cases where i and j are equal. So, in the first notation, you are essentially excluding all terms where i and j are equal.

I hope this clarifies things for you. Let me know if you have any other questions. Keep up the good work with your studies!