Summation with two running indices

Question

I don't understand the notation of the following summation.

$$ \sum_{i,j=1}^m \gamma_i \cdot \beta_{ij} \cdot \alpha_j$$

I first thought $ i, j $ would be increased simultaneously, but that would be no different than just using only one index. And furthermore there were no $ \beta_{ii} $ defined.

Am I right, to assume it is the same as

$$ \sum_{i=1}^m \sum_{j=1}^m \gamma_i \cdot \beta_{ij} \cdot \alpha_j$$

Answer 1

You can indeed start by $i$ or by $j$ as you prefer.

However, if you say $\beta_{i,i}$ is not defined, probably you should include the condition $j\neq i$, so like

$$ \sum_{\substack{j=1\\j\neq i}}^{m} \gamma_i \cdot \beta_{i,j} \cdot \alpha_j $$

using

\sum_{\substack{j=1\\j\neq i}}^{m} \gamma_i \cdot \beta_{i,j} \cdot \alpha_j