Kronecker delta function

The Kronecker delta function (represented with δ) is a piecewise function of two variables such that

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

It is most useful when used for integers. For example, the n × n identity matrix can be written as a matrix where each entry is the Kronecker delta of i and j. This is useful for defining an inner or dot product:

$\langle u,v \rangle = \sum_{ij} \delta_{ij} u_{i} v_{j}$