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
- Results about the Kronecker delta can be found here.
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