Suppose there are two vectors $x$ and $y$ in $\mathbb{R}^n$. I have stumbled across the following quantity: $$ \sum_{i \neq j} (\alpha x_i - \beta y_i)(\alpha x_j - \beta y_j), $$ where $\alpha, \beta$ are positive and satisfy $\alpha+ \beta=1$.
The above quantity can also be written as $$ \sum_{i \neq j}\left( \alpha^2 x_i x_j + \beta^2 y_i y_j - \alpha \beta (x_i y_j + x_j y_i) \right). $$
Are there any geometric interpretations for this quantity or any of the subparts in the last representation?