What is "kronecker delta"

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just positive integers. The function is 1 if the variables are equal, and 0 otherwise:

$$\delta_{ij} = \begin{cases} 0 &\text{if } i \neq j, \\ 1 &\text{if } i=j. \end{cases}$$
where the Kronecker delta δ is a piecewise function of variables i and j. For example, δ = 0, whereas δ = 1.

The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above.

In linear algebra, the n × n identity matrix I has entries equal to the Kronecker delta:

(I) = δ 
where i and j take the values 1, 2, ..., n, and the inner product of vectors can be written as

$$\textstyle \boldsymbol{a}\cdot\boldsymbol{b} = \sum_{ij} a_{i}\delta_{ij}b_{j}.$$

The restriction to positive integers is common, but there is no reason it cannot have negative integers as well as positive, or any discrete rational numbers. If i and j above take rational values, then for example δ = 0 and δ = 0 but δ = 1 and δ = 1. This latter case is for convenience.