Kronecker delta

From Citizendium, the Citizens' Compendium

Jump to: navigation, search


This article is a stub and thus not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
 
http://en.citizendium.org/wiki?title=Kronecker_delta&printable=yes
This is a draft article, under development and not meant to be cited but you can help to improve it. These unapproved articles are subject to a disclaimer.

In algebra, the Kronecker delta is a notation \delta_{ij} for a quantity depending on two subscripts i and j which is equal to one when i and j are equal and zero when they are unequal:

\delta_{ij} = \begin{cases} 1 &\quad\mathrm{if} \quad i = j \\ 0 &\quad\mathrm{if} \quad i \ne j. \end{cases}

If the subscripts are taken to vary from 1 to n then δ gives the entries of the n-by-n identity matrix. The invariance of this matrix under orthogonal change of coordinate makes δ a rank two tensor.

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

\sum_{i=1}^6 S_i \delta_{i,4} = S_1 \sdot0 + S_2 \sdot0 +S_3 \sdot0 +S_4 \sdot1 +S_5 \sdot0 +S_6 \sdot0 = S_4,

that is, the element S4 is "sifted out" of the summation by δi,4.

In general, (i and a integers)

\sum_{i=-\infty}^{\infty} S_{i}\delta_{ia} = S_a,\qquad i,a \in \mathbb{Z}.

The Kronecker delta is named after the German mathematician Leopold Kronecker (1823 – 1891). See Dirac delta function for a generalization of the Kronecker delta to real i and j.

Retrieved from "http://reid.citizendium.org/wiki/Kronecker_delta"
Views
Personal tools