What is the difference between the summations using $i<j$ and $i\neq j$ in the formula below: $$\sigma^{2}(\boldsymbol{w})=\sum_{i} \tilde{w}_{i}^{2}+2 \sum_{i<j} \tilde{w}_{i} \tilde{w}_{j} \rho_{i, j}=\sum_{i} \tilde{w}_{i}^{2}+\rho(\boldsymbol{w}) \sum_{i \neq j} \tilde{w}_{i} \tilde{w}_{j}$$ Screenshot here.
- Are both summations operationally equivalent?
- If so, why break consistency and have two competing representations?
- Which summation is more correct, or which to use for which situations?
One of the non-best answers here seem to apply, but not sure how in my case.