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