Kronecker delta

In mathematics, the Kronecker delta or '''Kronecker's delta''', named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. So, for example,

\delta_{12} = 0

, but

\delta_{33} = 1

. It is written as the symbol δ_ij, and treated as a notational shorthand rather than as a function. :

\delta_{ij} = \left\{\begin{matrix} 
1 & \mbox{if } i=j  \\ 
0 & \mbox{if } i \ne j \end{matrix}\right.

==Properties of the delta function== The Kronecker delta has the so-called ''sifting'' property that for

j\in\mathbb Z

: :

\sum_{i=-\infty}^\infty \delta_{ij} a_i=a_j.

This property is similar to one of the main properties of the Dirac delta function: :

\int_{-\infty}^\infty \delta(x-y)f(x) dx=f(y),

and in fact Dirac's delta was named after the Kronecker delta because of this analogous property. The Kronecker delta is used in many areas of mathematics. For example, in linear algebra, the identity matrix can be written as

\delta_{ij}\,

while if it is considered as a tensor, the Kronecker tensor, it can be written

\delta^j_i

with a contravariant index ''j''. This is a more accurate way to notate the identity matrix, considered as a linear mapping. ==Extensions of the delta function== In the same fashion, we may define an analogous, multi-dimensional function of many variables :

\delta^{j_1 j_2 ... j_n}_{i_1 i_2 ...i_n}:= \prod_{k=1}^n \delta_{i_k j_k}.

This function takes the value 1 if and only if all the upper indices match the corresponding lower one, and the value zero otherwise. ==See also== *Levi-Civita symbol *Dirac delta function *Dirac measure *Iverson bracket [http://www.kroneckerdelta.co.uk Kronecker Delta] is also a German Lager Mathematical notation Tensors

Kronecker delta

To claim as the article does that the Kronecker delta is a discrete analogue of the Dirac delta is very funny. The Dirac delta came afterwards, was named delta by analogy with Kronecker's and is not even an integral kernel, while the Kronecker delta is a ''bona fide'' matrix. — User:Miguel 22:12, 2004 Nov 21 (UTC) There's another definition of the generalized Kronecker delta that is defined by the sign of the permutation that maps the contravariant indices to the covariant ones. It's equal to and sometimes defined by a determinant of regular Kronecker deltas. Do a search on Planetmath for more info. User:Jason Quinn 19:57, 11 Mar 2005 (UTC)

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Kronecker_delta
Kronecker_delta
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