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{{Short description|Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise}}{{distinguish|text=the Dirac delta function, nor with the Kronecker symbol}}In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:delta_{ij} = begin{cases}1 &text{if } i=j. end{cases}or with use of Iverson brackets:delta_{ij} = [i=j],For example, delta_{12} = 0 because 1 ne 2, whereas delta_{33} = 1 because 3 = 3.The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above.In linear algebra, the ntimes n identity matrix mathbf{I} has entries equal to the Kronecker delta: where i and j take the values 1,2,cdots,n, and the inner product of vectors can be written as Here the Euclidean vectors are defined as {{mvar|n}}-tuples: mathbf{a} = (a_1, a_2, dots, a_n) and mathbf{b}= (b_1, b_2, ..., b_n) and the last step is obtained by using the values of the Kronecker delta to reduce the summation over j.It is common for {{mvar|i}} and {{mvar|j}} to be restricted to a set of the form {{math|{{(}}1, 2, ..., n{{)}}}} or {{math|{{(}}0, 1, ..., n − 1{{)}}}}, but the Kronecker delta can be defined on an arbitrary set.

Properties

The following equations are satisfied:begin{align}sum_{j} delta_{ij} a_j &= a_i,sum_{i} a_i delta_{ij} &= a_j,sum_{k} delta_{ik}delta_{kj} &= delta_{ij}.end{align}Therefore, the matrix {{math|δ}} can be considered as an identity matrix.Another useful representation is the following form:delta_{nm} = lim_{Ntoinfty}frac{1}{N} sum_{k = 1}^N e^{2 pi i frac{k}{N}(n-m)}This can be derived using the formula for the geometric series.

Alternative notation

Using the Iverson bracket: delta_{ij} = [i=j ].Often, a single-argument notation delta_i is used, which is equivalent to setting j=0:delta_{i} = delta_{i0} = begin{cases}1, & text{if } i = 0end{cases}In linear algebra, it can be thought of as a tensor, and is written delta_j^i. Sometimes the Kronecker delta is called the substitution tensor.JOURNAL, Trowbridge, J. H., 1998, On a Technique for Measurement of Turbulent Shear Stress in the Presence of Surface Waves, Journal of Atmospheric and Oceanic Technology, 15, 1, 291, 10.1175/1520-0426(1998)0152.0.CO;2, 1998JAtOT..15..290T, free,

Digital signal processing

In the study of digital signal processing (DSP), the unit sample function delta[n] represents a special case of a 2-dimensional Kronecker delta function delta_{ij} where the Kronecker indices include the number zero, and where one of the indices is zero. In this case:delta[n] equiv delta_{n0} equiv delta_{0n}~~~text{where} -inftyThe Kronecker delta has the so-called sifting property that for .