# Definition:Kronecker Delta

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## 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**.

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 $1868$.

## Sources

- 1868: Leopold Kronecker:
*Ueber bilineare Formen*(*J. reine angew. Math.***Vol. 68**: pp. 273 – 285) - 1961: I.N. Sneddon:
*Fourier Series*... (previous) ... (next): Chapter One: $\S 2$. Fourier Series - 1965: Seth Warner:
*Modern Algebra*... (previous) ... (next): $\S 28$ - 1980: A.J.M. Spencer:
*Continuum Mechanics*... (previous) ... (next): $2.1$: Matrices: $(2.5)$ - 1992: Donald E. Knuth:
*Two Notes on Notation*(*Amer. Math. Monthly***Vol. 99**: pp. 403 – 422) www.jstor.org/stable/2325085 - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.3$: Sums and Products: $(19)$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Kronecker delta** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Kronecker delta** - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Kronecker delta**