Kronecker delta function: Difference between revisions
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The Kronecker delta function is a mathematical operation used to select or compare indices. | |||
== Mathematical definition == | |||
=== Classical version === | |||
$$ | |||
\delta_{ij} = | |||
\begin{cases} | |||
1, & i=j,\\ | |||
0, & i\ne j. | |||
\end{cases} | |||
$$ | |||
=== single-index variant === | |||
$$ | |||
\delta_n = | |||
\begin{cases} | |||
1, & n=0,\\ | |||
0, & n\ne 0. | |||
\end{cases} | |||
$$ | |||
== Properties == | |||
=== Symmetry === | |||
$$ | |||
\delta_{ij} = \delta_{ji}. | |||
$$ | |||
== Application == | |||
=== Single out the $i$-th term of a sum === | |||
$$ | |||
\sum_j \delta_{ij} a_j = a_i. | |||
$$ | |||
=== Identity matrix === | |||
$$ | |||
I_{ij} = \delta_{ij}. | |||
$$ | |||
Revision as of 10:41, 1 October 2026
The Kronecker delta function is a mathematical operation used to select or compare indices.
Mathematical definition
Classical version
$$ \delta_{ij} = \begin{cases} 1, & i=j,\\ 0, & i\ne j. \end{cases} $$
single-index variant
$$ \delta_n = \begin{cases} 1, & n=0,\\ 0, & n\ne 0. \end{cases} $$
Properties
Symmetry
$$ \delta_{ij} = \delta_{ji}. $$
Application
Single out the $i$-th term of a sum
$$ \sum_j \delta_{ij} a_j = a_i. $$
Identity matrix
$$ I_{ij} = \delta_{ij}. $$