Kronecker symbol
The number defined by
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. When
, the Kronecker symbol
has
components, and the matrix
is the unit matrix. The Kronecker symbol was first used by L. Kronecker (1866).
The Kronecker symbol may be generalized, considering instead a set of quantities with
integer (upper and lower) indices,
,
, equal to
(or
) if the sequence
is an even (odd) permutation of the distinct indices
and zero otherwise. The numbers
(when
often denoted by
) are called the components of the Kronecker symbol. An affine tensor of type
whose components relative to some basis are equal to the components of the Kronecker symbol has the same components relative to any other basis.
The Kronecker symbol is convenient in various problems of tensor calculus. For example, the determinant
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is equal to the sum
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where the summation is performed over all permutations of the numbers
. The alternant of the tensor
is given by
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References
[1] | L. Kronecker, "Vorlesungen über die Theorie der Determinanten" , Leipzig (1903) |
Kronecker symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kronecker_symbol&oldid=35655