# Kronecker delta

In mathematics, the **Kronecker delta** is a symbol, written as *δ*_{ij}, depending on two integral numbers *i* and *j*. The symbol designates the number 1 if *i* = *j* and 0 if *i* ≠ *j*:

The symbol is named after the German mathematician Leopold Kronecker (1823-1891).

The general element of an identity matrix can be written as a Kronecker delta: the diagonal elements (*i* = *j*) are one; the off-diagonal elements (*i* ≠ *j*) are zero.

In linear algebra and functional analysis the Kronecker delta appears to indicate orthonormality of certain sets of vectors (elements of a vector space). Indicating an inner product by brackets, two vectors are mutually orthogonal and individually normalized to unity if

An example may be taken from Fourier analysis:

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

that is, the element *S*_{3} is "filtered out" of the summation by δ_{i,3}.

In general, (*i* and *k* integers)

See Dirac delta function for a generalization of the Kronecker delta to real *i* and *j*.