二重シグマ（二重和）の計算方法・性質

\begin{align} \sum_{i=1}^{n} \sum_{j=1}^{m} x_{ij} &= \sum_{i=1}^{n} \left(x_{i1} + \cdots + x_{im}  \right) \\ &= \sum_{i} x_{i1} + \cdots + \sum_{i} x_{in} \\ &= (x_{11} + \cdots x_{n1}) + \cdots  + (x_{1m} + \cdots x_{nm})\end{align}

$$\sum_{i=1}^{n} \sum_{j=1}^{m} x_{ij} = \sum_{j=1}^{m} \sum_{i=1}^{n} x_{ij}$$

$$\sum_{i=1}^{n} \sum_{j=1}^{m} x_{i} y_{j} = \sum_{i=1}^{n} x_{i} \sum_{j=1}^{n} y_{j}$$

$$\sum_{i=1}^{n} \sum_{j=1}^{m} x_{i} y_{ij} = \sum_{i=1}^{n} x_{i} \sum_{j=1}^{m} y_{ij}$$

$$\left(\sum_{i=1}^{m} x_{i}  \right)^{2} = \sum_{i=1}^{n} \sum_{j=1}^{n} x_{i} x_{j}$$