# Definition:Kronecker Delta

## 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

- 1868: Leopold Kronecker:
*Ueber bilineare Formen*(*J. reine angew. Math.***Vol. 68**: pp. 273 – 285) - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 28$. Linear Transformations