The number defined by
. 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
is equal to the sum
where the summation is performed over all permutations of the numbers . The alternant of the tensor is given by
|||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=18809