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:

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


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


Numbers

The concept is most often seen when $R$ is one of the standard number systems, in which case the image is merely $\set {0, 1}$:


Let $\Gamma$ be a set.

Then $\delta_{\alpha \beta}: \Gamma \times \Gamma \to \set {0, 1}$ is the mapping on the cartesian square of $\Gamma$ defined as:

$\forall \tuple {\alpha, \beta} \in \Gamma \times \Gamma: \delta_{\alpha \beta} := \begin {cases} 1 & : \alpha = \beta \\ 0 & : \alpha \ne \beta \end {cases}$


Also denoted as

When used in the context of tensors, the notation can often be seen as ${\delta^i}_j$.


Also presented as

The Kronecker delta can be expressed using Iverson bracket notation as:

$\delta_{\alpha \beta} := \sqbrk {\alpha = \beta}$


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