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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:

or with use of Iverson brackets:

where the Kronecker delta δ

The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above.

In linear algebra, the *n* × *n* identity matrix **I** has entries equal to the Kronecker delta:

where i and j take the values 1, 2, ...,

Here the Euclidean vectors are defined as n-tuples: and and the last step is obtained by using the values of the Kronecker delta to reduce the summation over j.

The restriction to positive or non-negative integers is common, but in fact, the Kronecker delta can be defined on an arbitrary set.

The following equations are satisfied:

Another useful representation is the following form:

Using the Iverson bracket:

Often, a single-argument notation δ_{i} is used, which is equivalent to setting *j* = 0:

In linear algebra, it can be thought of as a tensor, and is written δ^{i}_{j}. Sometimes the Kronecker delta is called the substitution tensor.^{[1]}

In the study of digital signal processing (DSP), the unit sample function represents a special case of a 2-dimensional Kronecker delta function where the Kronecker indices include the number zero, and where one of the indices is zero. In this case:

Or more generally where:

However, this is only a special case. In tensor calculus, it is more common to number basis vectors in a particular dimension starting with index 1, rather than index 0. In this case, the relation doesn't exist, and in fact, the Kronecker delta function and the unit sample function are different functions that overlap in the specific case where the indices include the number 0, the number of indices is 2, and one of the indices has the value of zero.

While the discrete unit sample function and the Kronecker delta function use the same letter, they differ in the following ways. For the discrete unit sample function, it is more conventional to place a single integer index in square braces; in contrast the Kronecker delta can have any number of indexes. Further, the purpose of the discrete unit sample function is different from the Kronecker delta function. In DSP, the discrete unit sample function is typically used as an input function to a discrete system for discovering the system function of the system which will be produced as an output of the system. In contrast, the typical purpose of the Kronecker delta function is for filtering terms from an Einstein summation convention.

The discrete unit sample function is more simply defined as:

In addition, the Dirac delta function is often confused for both the Kronecker delta function and the unit sample function. The Dirac delta is defined as:

Unlike the Kronecker delta function and the unit sample function , the Dirac delta function doesn't have an integer index, it has a single continuous non-integer value t.

To confuse matters more, the unit impulse function is sometimes used to refer to either the Dirac delta function , or the unit sample function .

The Kronecker delta has the so-called *sifting* property that for *j* ∈ **Z**:

The Kronecker delta forms the multiplicative identity element of an incidence algebra.^{[2]}

In probability theory and statistics, the Kronecker delta and Dirac delta function can both be used to represent a discrete distribution. If the support of a distribution consists of points **x** = {*x*_{1}, ..., *x _{n}*}, with corresponding probabilities

Equivalently, the probability density function *f*(*x*) of the distribution can be written using the Dirac delta function as

Under certain conditions, the Kronecker delta can arise from sampling a Dirac delta function. For example, if a Dirac delta impulse occurs exactly at a sampling point and is ideally lowpass-filtered (with cutoff at the critical frequency) per the Nyquist–Shannon sampling theorem, the resulting discrete-time signal will be a Kronecker delta function.

If it is considered as a type (1,1) tensor, the Kronecker tensor can be written
*δ*^{i}_{j} with a covariant index j and contravariant index i:

This tensor represents:

- The identity mapping (or identity matrix), considered as a linear mapping
*V*→*V*or*V*^{∗}→*V*^{∗} - The trace or tensor contraction, considered as a mapping
*V*^{∗}⊗*V*→*K* - The map
*K*→*V*^{∗}⊗*V*, representing scalar multiplication as a sum of outer products.

The **generalized Kronecker delta** or **multi-index Kronecker delta** of order 2*p* is a type (*p*, *p*) tensor that is completely antisymmetric in its p upper indices, and also in its p lower indices.

Two definitions that differ by a factor of *p*! are in use. Below, the version is presented has nonzero components scaled to be ±1. The second version has nonzero components that are ±1/*p*!, with consequent changes scaling factors in formulae, such as the scaling factors of 1/*p*! in *§ Properties of the generalized Kronecker delta* below disappearing.^{[3]}

In terms of the indices, the generalized Kronecker delta is defined as:^{[4]}^{[5]}

Let S_{p} be the symmetric group of degree p, then:

Using anti-symmetrization:

In terms of a *p* × *p* determinant:^{[6]}

Using the Laplace expansion (Laplace's formula) of determinant, it may be defined recursively:^{[7]}

When *p* = *n* (the dimension of the vector space), in terms of the Levi-Civita symbol:

Kronecker Delta contractions depend on the dimension of the space. For example,

The generalized Kronecker delta may be used for anti-symmetrization:

From the above equations and the properties of anti-symmetric tensors, we can derive the properties of the generalized Kronecker delta:

Reducing the order via summation of the indices may be expressed by the identity^{[8]}

Using both the summation rule for the case *p* = *n* and the relation with the Levi-Civita symbol,
the summation rule of the Levi-Civita symbol is derived:

For any integer n, using a standard residue calculation we can write an integral representation for the Kronecker delta as the integral below, where the contour of the integral goes counterclockwise around zero. This representation is also equivalent to a definite integral by a rotation in the complex plane.

The Kronecker comb function with period N is defined (using DSP notation) as:

The Kronecker delta is also called degree of mapping of one surface into another.^{[12]} Suppose a mapping takes place from surface S_{uvw} to S_{xyz} that are boundaries of regions, R_{uvw} and R_{xyz} which is simply connected with one-to-one correspondence. In this framework, if s and t are parameters for S_{uvw}, and S_{uvw} to S_{uvw} are each oriented by the outer normal **n**:

Let *x* = *x*(*u*, *v*, *w*), *y* = *y*(*u*, *v*, *w*), *z* = *z*(*u*, *v*, *w*) be defined and smooth in a domain containing S_{uvw}, and let these equations define the mapping of S_{uvw} onto S_{xyz}. Then the degree δ of mapping is 1/4π times the solid angle of the image S of S_{uvw} with respect to the interior point of S_{xyz}, *O*. If *O* is the origin of the region, R_{xyz}, then the degree, δ is given by the integral:

**^**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. Bibcode:1998JAtOT..15..290T. doi:10.1175/1520-0426(1998)015<0290:OATFMO>2.0.CO;2.**^**Spiegel, Eugene; O'Donnell, Christopher J. (1997),*Incidence Algebras*, Pure and Applied Mathematics, vol. 206, Marcel Dekker, ISBN 0-8247-0036-8.**^**Pope, Christopher (2008). "Geometry and Group Theory" (PDF).**^**Frankel, Theodore (2012).*The Geometry of Physics: An Introduction*(3rd ed.). Cambridge University Press. ISBN 9781107602601.**^**Agarwal, D. C. (2007).*Tensor Calculus and Riemannian Geometry*(22nd ed.). Krishna Prakashan Media.^{[ISBN missing]}**^**Lovelock, David; Rund, Hanno (1989).*Tensors, Differential Forms, and Variational Principles*. Courier Dover Publications. ISBN 0-486-65840-6.**^**A recursive definition requires a first case, which may be taken as*δ*= 1 for*p*= 0, or alternatively*δ*^{μ}_{ν}=*δ*^{μ}_{ν}for*p*= 1 (generalized delta in terms of standard delta).**^**Hassani, Sadri (2008).*Mathematical Methods: For Students of Physics and Related Fields*(2nd ed.). Springer-Verlag. ISBN 978-0-387-09503-5.**^**Penrose, Roger (June 1960). "A spinor approach to general relativity".*Annals of Physics*.**10**(2): 171–201. Bibcode:1960AnPhy..10..171P. doi:10.1016/0003-4916(60)90021-X.**^**Aitken, Alexander Craig (1958).*Determinants and Matrices*. UK: Oliver and Boyd.**^**Roger Penrose, "Applications of negative dimensional tensors," in*Combinatorial Mathematics and its Applications*, Academic Press (1971).**^**Kaplan, Wilfred (2003).*Advanced Calculus*. Pearson Education. p. 364. ISBN 0-201-79937-5.