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Kronecker Delta#

Definition: Kronecker Delta

Let \(S\) be a set and let \(F\) be a field.

The Kronecker delta is the function \(\delta: S \times S \to \{0, 1\} \subseteq F\) defined as

\[\delta (i, j) \overset{\text{def}}{=} \begin{cases}0 & \text{if} & i \ne j \\ 1 & \text{if} & i = j\end{cases}\]

for all \(i,j \in S\).

Notation

We write \(\delta_{ij}\) or \(\delta_{i,j}\) instead of \(\delta(i, j)\).