Double summation equation Could somebody use a simple example to explain this double summation equation? 
Thank you.
$$\sum_{j=1}^2 \sum_{i=1}^8 x_{i,j} = \sum_{i=1}^8 x_{i,1} + \sum_{i=1}^8 x_{i,2}$$
 A: $$
\sum_{j=1}^2 a_j \text{ means } a_1 + a_2.
$$
$$
\sum_{j=1}^2 \sum_{i=1}^8 x_{i,j} = \underbrace{\sum_{i=1}^8 x_{i,1}}_{\text{In this term, $j=1.$}} + \underbrace{\sum_{i=1}^8 x_{i,2}}_{\text{In this term, $j=2.$}}
$$
A: The summation represents the sum of the elements in a $2\times 8$ matrix.  Which can also be thought of as the sum of the elements in the first row plus the sum of the elements in the second row.
A: Take $x_{i,j} = ij\implies S(8,2) = x_{1,1}+x_{2,1}+\cdots+x_{8,1}+x_{1,2}+x_{2,2}+\cdots + x_{8,2}= 1+2+3+\cdots + 8+2+4+\cdots + 16 = ...$
A: Your equation is finding the sum of all elements in the following matrix:
$$x= \left[
    \begin{array}\
      x_{1,1}&x_{1,2}\\
      x_{2,1}&x_{2,2}\\
      x_{3,1}&x_{3,2}\\
      x_{4,1}&x_{4,2}\\
      x_{5,1}&x_{5,2}\\
      x_{6,1}&x_{6,2}\\
      x_{7,1}&x_{7,2}\\
      x_{8,1}&x_{8,2}\\
    \end{array}
\right] $$
Sigma notation allows us to simplify the following expression
$$x_{1,1}+x_{2,1}+x_{3,1}+x_{4,1}+x_{5,1}+x_{6,1}+x_{7,1}+x_{8,1}+x_{1,2}+x_{2,2}+x_{3,2}+x_{4,2}+x_{5,2}+x_{6,2}+x_{7,2}+x_{8,2}$$
To this
$$\sum_{i=1}^8x_{i,1}+\sum_{i=1}^8x_{i,2}$$
And then further to this
$$\sum_{j=1}^2\sum_{i=1}^8x_{i,j}$$
