# Understanding the Permitted Use of Double Summation in Math

• IniquiTrance
In summary, the equation \sum_{j=1}^n \sum_{k=1}^n a_{ik}b_{kj} = \left(\sum_{j=1}^n b_{kj}\right) \left(\sum_{k=1}^n a_{ik}\right) = 1\cdot 1 = 1 is not permitted because summing over a variable (in this case, k) and then having a term outside the sum (b_{kj}) is not allowed. A correct way to rewrite this equation would be \sum_{j=1}^n\sum_{k=1}^n a_{ik}b{kj}= \sum_{k=
IniquiTrance
If I know that $\sum_{k=1}^n a_{ik} = 1$ and $\sum_{j=1}^n b_{kj} = 1$, why is the following permitted?

$$\sum_{j=1}^n \sum_{k=1}^n a_{ik}b_{kj} = \left(\sum_{j=1}^n b_{kj}\right) \left(\sum_{k=1}^n a_{ik}\right) = 1\cdot 1 = 1$$

Thanks!

It's not permitted. What you wrote makes no sense since you sum over k and one of the $b_{kj}$ is outside that sum. That is not allowed.

What you could do is:

$$\sum_{j=1}^n\sum_{k=1}^n a_{ik}b{kj}= \sum_{k=1}^n \sum_{j=1}^n a_{ik}b_{kj} = \sum_{k=1}^n \left(a_{ik} \sum_{j=1}^n b_{kj}\right)= \sum_{k=1}^n a_{ik}=1$$

Thank you very much. I knew I wasn't understanding something.

## 1. What is double summation?

Double summation is a mathematical operation that involves adding up two sets of numbers. It is written as a double sigma (∑∑) and is often used in statistics and calculus to represent the sum of a function or series.

## 2. How is double summation calculated?

To calculate double summation, you first need to determine the limits of the two sums (represented by the lower and upper indices of the sigma). Then, you evaluate the inner sum first, replacing the variable with the lower limit and adding up the values until you reach the upper limit. Finally, you take this value and plug it into the outer sum, repeating the same process. The result is the double summation value.

## 3. What is the difference between single and double summation?

The main difference between single and double summation is the number of variables involved. Single summation involves adding up a series of numbers using one variable, while double summation involves adding up a series of numbers using two variables. Additionally, double summation is often used to represent the sum of a function or series of functions, while single summation is used for simpler calculations.

## 4. What are some common applications of double summation?

Double summation is commonly used in statistics to calculate the variance and standard deviation of a dataset. It is also used in calculus to represent the area under a curve and to approximate integrals. Additionally, double summation is used in computer science and programming to perform mathematical operations on matrices and arrays.

## 5. Are there any rules or properties for double summation?

Yes, there are several rules and properties for double summation, including linearity, commutativity, and associativity. Linearity means that a constant can be factored out of the summation, commutativity means that the order of the summations can be changed, and associativity means that the terms can be grouped in different ways without changing the result. These properties are helpful when simplifying double summation expressions.

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