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 δij, and treated as a notational shorthand rather than as a function.

Properties of the delta function

The Kronecker delta has the so-called sifting property that for jinmathbb Z:

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

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

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

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and in fact Dirac's delta was named after the Kronecker delta because of this analogous property.

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The Kronecker delta is used in many areas of mathematics. For example, in linear algebra, the identity matrix can be written as delta_{ij},

Related Topics:
Linear algebra - Identity matrix

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

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delta^j_i with a contravariant index j. This is a more accurate way to notate the identity matrix, considered as a linear mapping.

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