Could somebody use a simple example to explain this double summation equation?
Thank you.
$$\sum_{j=1}^2 \sum_{i=1}^8 x_{i,j} = \sum_{i=1}^8 x_{i,1} + \sum_{i=1}^8 x_{i,2}$$
Asked
Active
Viewed 329 times
4
-
By definition $\sum_{j=1}^2 a_j = a_1+a_2,$. Now let $a_j=\sum_{i=1}^8 x_{i,j},$. – dxiv Dec 30 '16 at 23:32
4 Answers
2
$$ \sum_{j=1}^2 a_j \text{ means } a_1 + a_2. $$ $$ \sum_{j=1}^2 \sum_{i=1}^8 x_{i,j} = \underbrace{\sum_{i=1}^8 x_{i,1}}_{\text{In this term, $j=1.$}} + \underbrace{\sum_{i=1}^8 x_{i,2}}_{\text{In this term, $j=2.$}} $$
-
This is also very nice illustration! Such things are horrible to me. – user697911 Dec 31 '16 at 00:29
0
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.
Doug M
- 57,877
0
Take $x_{i,j} = ij\implies S(8,2) = x_{1,1}+x_{2,1}+\cdots+x_{8,1}+x_{1,2}+x_{2,2}+\cdots + x_{8,2}= 1+2+3+\cdots + 8+2+4+\cdots + 16 = ...$
DeepSea
- 77,651
0
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 $$\sum_{j=1}^2\sum_{i=1}^8x_{i,j}$$
pseudoeuclidean
- 1,012
-
-
Does x1,1, x,1,2 have the values 1 & 2? I thought they are variables, not actual values. – user697911 Dec 31 '16 at 00:13
-
Your initial thought is right. $x_{1,1}$, $x_{1,2}$, (and so on) are all variables in this example. – pseudoeuclidean Dec 31 '16 at 00:17