In mathematics, the Kronecker delta is a symbol, written as δij, depending on two integral numbers i and j. The symbol designates the number 1 if i = j and 0 if i ≠ j:
The symbol is named after the German mathematician Leopold Kronecker (1823-1891).
The general element of an identity matrix can be written as a Kronecker delta: the diagonal elements (i = j) are one; the off-diagonal elements (i ≠ j) are zero.
In linear algebra and functional analysis the Kronecker delta appears to indicate orthonormality of certain sets of vectors (elements of a vector space). Indicating an inner product by brackets, two vectors are mutually orthogonal and individually normalized to unity if
An example may be taken from Fourier analysis:
Kronecker deltas appear frequently in summations where they act as a "filter". To clarify this we consider a simple example
that is, the element S3 is "filtered out" of the summation by δi,3.
In general, (i and k integers)
See Dirac delta function for a generalization of the Kronecker delta to real i and j.