kronecker delta

Kronecker delta

$${\delta }_{ij}=\{\begin{array}{ll}1& \text{if }i=j\\ 0& \text{if }i\ne j\end{array}$$

The Kronecker delta is a function of two variables, usually integers, that is 1 if the variables are equal and 0 otherwise. It is often used in summations and matrix operations to simplify expressions.

Examples

$({\delta }_{ij}{)}_{i,j=1}^{n,n}$ represents an $n\times n$ identity matrix, where the diagonal elements are 1 (when $i=j$) and the off-diagonal elements are 0 (when $i\ne j$).
${\sum }_{i=1}^n{a}_i{\delta }_{ij}=a_j$ extracts the $j$-th element from the sequence $a_1,a_2,\dots ,a_n$.
$e_k=({\delta }_{ik}{)}_{i=1}^n\in {\mathbb{R}}^n$ is the standard basis/unit vector with a 1 in the $k$-th position and 0s elsewhere.