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Kronecker delta (Definition)

The Kronecker delta $ \delta_{ij}$ is defined as having value 1 when $ i=j$ and 0 otherwise ($ i$ and $ j$ are integers). It may also be written as $ \delta^{ij}$ or $ \delta^i_j$. It is a special case of the generalized Kronecker delta symbol.

The delta symbol was first used in print by Kronecker in 1868[1].

Example.

The $ n \times n$ identity matrix $ I$ can be written in terms of the Kronecker delta as simply the matrix of the delta, $ I_{ij}=\delta_{ij}$, or simply $ I=(\delta_{ij})$.

Bibliography

1
N. Higham, Handbook of writing for the mathematical sciences, Society for Industrial and Applied Mathematics, 1998.



"Kronecker delta" is owned by akrowne.
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See Also: identity matrix, Levi-Civita permutation symbol
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Cross-references: matrix, terms, identity matrix, generalized Kronecker delta symbol, integers
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This is version 3 of Kronecker delta, born on 2002-01-04, modified 2006-10-28.
Object id is 1221, canonical name is KroneckerDelta.
Accessed 17524 times total.

Classification:
AMS MSC: 15A99 (Linear and multilinear algebra; matrix theory :: Miscellaneous topics)
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Subscripts by sgraves on 2006-07-28 10:31:58
Need the subscripts of the delta be integers? Take the case of an orthonormal set in a Hilbert space: U=\{u_\alpha|\alpha\in A\} for some index set A, with U orthonormal iff (u_\alpha,u_\beta)=\delta_{\alpha~\beta}.

~sjg
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