Kronecker delta function

A mathematical operation used to select or compare indices.

Mathematical definition

Classical version

$$ \delta_{ij} = \begin{cases} 1, & i=j,\\ 0, & i\ne j. \end{cases} $$

single-index variant

$$ \delta_n = \begin{cases} 1, & n=0,\\ 0, & n\ne 0. \end{cases} $$

Properties

Symmetry

$$ \delta_{ij} = \delta_{ji}. $$

Application

Single out the $i$-th term of a sum

$$ \sum_j \delta_{ij} a_j = a_i. $$

Identity matrix

$$ I_{ij} = \delta_{ij}. $$