help in simplifying an easy but nasty expression

Question

I would like double check my work, I am trying to simplify the following summation, \begin{align} \sum_{\substack{(i,j) \in \mathcal{S}}} A_i v_i v_j \end{align} with the assumption that $$v_iv_j= c \ \ \ \forall i,j$$ where $\mathcal{S}= \{ (i,j) \in \mathbb{Z}^2: i \neq j, 1\leq i,j \leq n \}$, is the answer simply

$$\sum_{i=1}^n (n-1)A_i c $$

Answer 1

You are correct. One way to see this is to note that

$$\sum_{(i, j) \in \mathcal{S}}A_iv_iv_j = \sum_{i=1}^n\sum_{j\neq i}A_iv_iv_j.$$

Once you have this, we can manipulate the double sum as follows

$$\sum_{i=1}^n\sum_{j\neq i}A_iv_iv_j = \sum_{i=1}^nA_i\sum_{j\neq i}v_iv_j = \sum_{i=1}^nA_i\sum_{j\neq i}c = \sum_{i=1}^nA_i(n-1)c.$$