Kronecker Delta

In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker, is a function of two variables, usually integers. The function is 1 if the variables are equal, and 0 otherwise:

$\delta_{ij} = \left\{\begin{matrix} 0, & \mbox{if } i \ne j \\ 1, & \mbox{if } i=j \end{matrix}\right.$

where Kronecker delta δ_ij is a piecewise function of variables and . For example, δ_{1 2} = 0, whereas δ_{3 3} = 1.

In linear algebra, the identity matrix can be written as, 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 Kronecker delta is used in many areas of mathematics.