# How to write two for loops in math notation?

I have a vector of numbers that looks like this:

[1, 2, 3, 4, 5]

For every number the vector, I would like to multiply each number by every other number and find the sum:

1*1 + 1*2 + 1*3 + 1*4 + 1*5 + 2*1 + 2*2 + 2*3 + 2*4 + 2*5 + ... + 5*5

How can I write this in math notation?

• $$(1+2+3+4+5)^2$$ Commented Jun 18, 2013 at 13:17
• If you want to keep a vector notation, $$\left(\begin{pmatrix} 1 \\ 2 \\ 3 \\ 4 \\ 5\end{pmatrix}^{\mathrm{T}}\begin{pmatrix} 1 \\ 1 \\ 1 \\ 1 \\ 1\end{pmatrix}\right)^2$$ or even, for general $x\in\mathbb{R}_+^n$, $\lVert x \rVert_1^2$. Commented Jun 18, 2013 at 13:22
• Thanks. This was a trivialized example, I'm actually looking for a way to denote this with more complex expressions. Commented Jun 18, 2013 at 13:23
• Just out of curiosity, why do you want more complex expressions? Generally in math you want an expression to be as simple as possible. Commented Jun 18, 2013 at 13:25
• What I mean is that the actual math I am doing for each pairwise combinations is more complex than simply squaring each number. Commented Jun 18, 2013 at 13:27

$\sum_{i=1}^5 \sum_{j=1}^5 ij$.
However, you can also give the array $v$ values $v_i$ other than the index $i$. Then you would write:
$\sum_{i=1}^5 \sum_{j=1}^5 v_iv_j$.