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Encyclopedia > 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, δ12 = 0, but δ33 = 1. It is written as the symbol δij, and treated as a notational shorthand rather than as a function. Mathematics is the study of quantity, structure, space and change. ... Leopold Kronecker Leopold Kronecker (December 7, 1823 - December 29, 1891) was a German mathematician and logician who argued that arithmetic and analysis must be founded on whole numbers, saying, God made the natural numbers; all else is the work of man (Bell 1986, p. ... 1823 was a common year starting on Wednesday (see link for calendar). ... 1891 was a common year starting on Thursday (see link for calendar). ... The integers consist of the positive natural numbers (1, 2, 3, â€¦), their negatives (âˆ’1, âˆ’2, âˆ’3, ...) and the number zero. ...

## Properties of the delta function GA_googleFillSlot("encyclopedia_square");

The Kronecker delta has the so-called shifting property that for :

This property is similar to one of the main properties of the Dirac delta function: The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere such that the total integral...

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 while if it is considered as a tensor, the Kronecker tensor, it can be written with a contravariant index j. This is a more accurate way to notate the identity matrix, considered as a linear mapping. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (or linear spaces), linear transformations, and systems of linear equations. ... In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and zeros elsewhere. ... In mathematics, a tensor is a certain kind of geometrical entity, or alternatively generalized quantity. The tensor concept includes the ideas of scalar, vector and linear operator. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are...

## Extensions of the delta function

In the same fashion, we may define an analogous, multi-dimensional function of many variables

This function takes the value 1 if and only if all the upper indices match the corresponding lower one, and the value zero otherwise.

Kronecker Delta is also a German Lager In mathematics, and in particular in tensor calculus, the Levi-Civita symbol, also called the permutation symbol, is defined as follows: It is named after Tullio Levi-Civita. ... The Dirac delta function, sometimes referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere such that the total integral... In mathematics, a Dirac measure is a measure δx on a set X that gives a given element x measure 1, so that δx({x}) = 1 and in general δx(Y) = 0 for any subset Y of X not containing x, δx(Z) = 1 for any subset Z containing x. ... In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is defined as follows where P is a proposition. ...

Results from FactBites:

 PlanetMath: Kronecker delta (123 words) It is a special case of the generalized Kronecker delta symbol. The delta symbol was first used in print by Kronecker in 1868[1]. This is version 3 of Kronecker delta, born on 2002-01-04, modified 2006-10-28.
 Leopold Kronecker - Wikipedia, the free encyclopedia (271 words) Kronecker was a student and lifelong friend of Ernst Kummer. Kronecker wrote 1845 his dissertation, at the University of Berlin, on number theory, giving special formulation to units in certain algebraic number fields. Kronecker also contributed to the concept of continuity, reconstructing the form of irrational numbers in real numbers.
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