Definition:Kronecker Delta

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Definition

Let $\Gamma$ be a set.

Let $R$ be a ring with unity whose zero is $0_R$ and whose unity is $1_R$.


Then $\delta_{\alpha \beta}: \Gamma \times \Gamma \to R$ is the mapping on the cartesian square of $\Gamma$ defined as:

$\forall \left({\alpha, \beta}\right) \in \Gamma \times \Gamma: \delta_{\alpha \beta} := \begin{cases} 1_R & :\alpha = \beta \\ 0_R & :\alpha \ne \beta \end{cases}$

This use of $\delta$ is known as the Kronecker delta notation or Kronecker delta convention.


It can be expressed in Iverson bracket notation as:

$\delta_{\alpha \beta} := \left[{\alpha = \beta}\right]$


Also see


Source of Name

This entry was named for Leopold Kronecker.


Historical Note

The Kronecker delta notation was invented by Leopold Kronecker in $1868$.


Sources