Definition:Kronecker Delta

From ProofWiki
Jump to navigation Jump to search


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.

It can be expressed in Iverson bracket notation as:

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

Also denoted as

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

Also see

Source of Name

This entry was named for Leopold Kronecker.

Historical Note

The Kronecker delta notation was invented by Leopold Kronecker in $1868$.