Better way to write the sum of multiple terms with cross-correlation

Question

how can I write this expression using summation?

$y^2=x_1^2 + x_2^2 + x_3^2 + x_4^2 + 2x_1x_2 + 2x_1x_3 + 2x_1x_4 + 2x_2x_3 + 2x_2x_4 + 2x_3x_4$

What do you mean by cross-correlation? – morrowmh Mar 23 '21 at 02:08 — morrowmh, Mar 23 '21 at 02:08
I don't know how to call the terms $2x_i x_j$ – Dealer Mar 23 '21 at 12:55 — Dealer, Mar 23 '21 at 12:55

score 2 · Accepted Answer · edited Mar 24 '21 at 18:24

2

Consider: $$y^2=\sum_{i=1}^4x_i^2+\sum_{1\leq i<j\leq 4}2x_ix_j=\left(\sum_{i=1}^4x_i\right)^2.$$

edited Mar 24 '21 at 18:24

amWhy

answered Mar 23 '21 at 02:05

morrowmh

2

Why not $$y^2= \left( \sum_{i=1}^4x_i \right)^2$$ ? Or even $$y= \sum_{i=1}^4x_i$$ ;) – PM 2Ring Mar 23 '21 at 02:20
Edited. Thanks for that haha. – morrowmh Mar 23 '21 at 02:23
Since I want to highlight also the terms $2x_i x_j$ the first expression is what I need. I think it can be also written as $\sum_{\substack{ i,j=1 \ i\neq j}}^4 x_i^2 +2x_i x_j$ – Dealer Mar 23 '21 at 12:57

1 Answers1