Kronecker delta

In algebra, the Kronecker delta is a notation $\delta _{ij}$ for a quantity depending on two subscripts i and j which is equal to one when i and j are equal and zero when they are unequal:

$\delta _{ij}={\begin{cases}1&\quad \mathrm {if} \quad i=j\\0&\quad \mathrm {if} \quad i\neq j.\end{cases}}$

If the subscripts are taken to vary from 1 to n then δ gives the entries of the n-by-n identity matrix. The invariance of this matrix under orthogonal change of coordinate makes δ a rank two tensor.

Kronecker deltas appear frequently in summations where they act as a "filter". To clarify this we consider a simple example

$\sum _{i=1}^{6}S_{i}\delta _{i,4}=S_{1}\cdot 0+S_{2}\cdot 0+S_{3}\cdot 0+S_{4}\cdot 1+S_{5}\cdot 0+S_{6}\cdot 0=S_{4},$

that is, the element S₄ is "sifted out" of the summation by δ_i,4.

In general, (i and a integers)

$\sum _{i=-\infty }^{\infty }S_{i}\delta _{ia}=S_{a},\qquad i,a\in \mathbb {Z} .$

The Kronecker delta is named after the German mathematician Leopold Kronecker (1823 – 1891). See Dirac delta function for a generalization of the Kronecker delta to real i and j.