Kronecker delta

In mathematics, the Kronecker delta is a symbol, written as δij, depending on two integral numbers i and j. The symbol designates the number 1 if i = j and 0 if i &ne; j:

\delta_{ij} = \begin{cases} 1 &\quad\hbox{if}\quad i = j \\ 0 &\quad\hbox{if}\quad i \ne j \\ \end{cases} $$ 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 &ne; 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

(v_i,\; v_j) = \delta_{ij}. $$ An example may be taken from Fourier analysis:

(e^{im\phi},\; e^{in\phi}) = \frac{1}{2\pi} \int_{0}^{2\pi}  e^{-im\phi}\, e^{in\phi} \mathrm{d}\phi = \delta_{mn}. $$

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

\sum_{i=1}^5 S_i \delta_{i,3} = S_1 \sdot0 + S_2 \sdot0 +S_3 \sdot1 +S_4 \sdot0 +S_5 \sdot0 = S_3, $$ that is, the element S3 is "filtered out" of the summation by &delta;i,3.

In general, (i and k integers)

\sum_{i=-\infty}^{\infty} S_{i}\delta_{ik} = S_k,\qquad i,k \in \mathbb{Z}. $$

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