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 $\set {0, 1}$:

Also see

 * Definition:Levi-Civita Symbol