Kronecker delta

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: $${\displaystyle \delta _{ij}={\begin{cases}0&{\text{if }}i\neq j,\\1&{\text{if }}i=j.\end{cases}}}$$ or with use of Iverson brackets: $${\displaystyle \delta _{ij}=[i=j]\,}$$ For example, ${\displaystyle \delta _{12}=0}$ because ${\displaystyle 1\neq 2}$, whereas ${\displaystyle \delta _{33}=1}$ because ${\displaystyle 3=3}$.

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

In linear algebra, the ${\displaystyle n\times n}$ identity matrix ${\displaystyle \mathbf {I} }$ has entries equal to the Kronecker delta: $${\displaystyle I_{ij}=\delta _{ij}}$$ where ${\displaystyle i}$ and ${\displaystyle j}$ take the values ${\displaystyle 1,2,\cdots ,n}$, and the inner product of vectors can be written as $${\displaystyle \mathbf {a} \cdot \mathbf {b} =\sum _{i,j=1}^{n}a_{i}\delta _{ij}b_{j}=\sum _{i=1}^{n}a_{i}b_{i}.}$$ Here the Euclidean vectors are defined as n-tuples: ${\displaystyle \mathbf {a} =(a_{1},a_{2},\dots ,a_{n})}$ and ${\displaystyle \mathbf {b} =(b_{1},b_{2},...,b_{n})}$ and the last step is obtained by using the values of the Kronecker delta to reduce the summation over ${\displaystyle j}$.

It is common for i and j to be restricted to a set of the form {1, 2, ..., n} or {0, 1, ..., n − 1}, but the Kronecker delta can be defined on an arbitrary set.