# Kronecker delta

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In algebra, the Kronecker delta is a notation ${\displaystyle \delta _{ij}}$ 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:

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

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

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

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

In general, (i and a integers)

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

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.