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In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\text i=j. \end or with use of Iverson brackets: \delta_ = =j, where the Kronecker delta is a
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. P ...
function of variables and . For example, , whereas . 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 Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
has entries equal to the Kronecker delta: I_ = \delta_ where and take the values , and the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of vectors can be written as \mathbf\cdot\mathbf = \sum_^n a_\delta_b_ = \sum_^n a_ b_. Here the Euclidean vectors are defined as -tuples: \mathbf = (a_1, a_2, \dots, a_n) and \mathbf= (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 . The restriction to positive or non-negative integers is common, but in fact, the Kronecker delta can be defined on an arbitrary set.


Properties

The following equations are satisfied: \begin \sum_ \delta_ a_j &= a_i,\\ \sum_ a_i \delta_ &= a_j,\\ \sum_ \delta_\delta_ &= \delta_. \end Therefore, the matrix can be considered as an identity matrix. Another useful representation is the following form: \delta_ = \frac \sum_^N e^ In the limit N \to \infty . This can be derived using the formula for the geometric series.


Alternative notation

Using the
Iverson bracket In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement ...
: \delta_ = =j Often, a single-argument notation is used, which is equivalent to setting : \delta_ = \begin 0, & \text i \neq 0 \\ 1, & \text i = 0 \end In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrice ...
, it can be thought of as a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
, and is written . Sometimes the Kronecker delta is called the substitution tensor.


Digital signal processing

In the study of
digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
(DSP), the unit sample function \delta /math> represents a special case of a 2-dimensional Kronecker delta function \delta_ where the Kronecker indices include the number zero, and where one of the indices is zero. In this case: \delta \equiv \delta_ \equiv \delta_~~~\text -\infty Or more generally where: \delta -k\equiv \delta -n\equiv \delta_ \equiv \delta_\text -\infty 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 \delta \equiv \delta_ \equiv \delta_ 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 In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
. The discrete unit sample function is more simply defined as: \delta = \begin 1 & n = 0 \\ 0 & n \text\end In addition, the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
is often confused for both the Kronecker delta function and the unit sample function. The Dirac delta is defined as: \delta(t) = \begin \infty & t = 0 \\ 0 & t \text\end Unlike the Kronecker delta function \delta_ and the unit sample function \delta /math>, the Dirac delta function \delta(t) doesn't have a integer index, it has a single continuous non-integer value . To confuse matters more, the unit impulse function is sometimes used to refer to either the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
\delta(t), or the unit sample function \delta /math>.


Properties of the delta function

The Kronecker delta has the so-called ''sifting'' property that for : \sum_^\infty a_i \delta_ = a_j. and if the integers are viewed as a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
, endowed with the counting measure, then this property coincides with the defining property of the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
\int_^\infty \delta(x-y)f(x)\, dx=f(y), and in fact Dirac's delta was named after the Kronecker delta because of this analogous property. In signal processing it is usually the context (discrete or continuous time) that distinguishes the Kronecker and Dirac "functions". And by convention, generally indicates continuous time (Dirac), whereas arguments like , , , , , and are usually reserved for discrete time (Kronecker). Another common practice is to represent discrete sequences with square brackets; thus: . The Kronecker delta is not the result of directly sampling the Dirac delta function. The Kronecker delta forms the multiplicative identity element of an incidence algebra.


Relationship to the Dirac delta function

In
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
and
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
, the Kronecker delta and
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the enti ...
can both be used to represent a discrete distribution. If the
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
of a distribution consists of points , with corresponding probabilities , then the
probability mass function In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
of the distribution over can be written, using the Kronecker delta, as p(x) = \sum_^n p_i \delta_. Equivalently, the
probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) ca ...
of the distribution can be written using the Dirac delta function as f(x) = \sum_^n p_i \delta(x-x_i). 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.


Generalizations

If it is considered as a type
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensor ...
, the Kronecker tensor can be written with a covariant index and contravariant index : \delta^_ = \begin 0 & (i \ne j), \\ 1 & (i = j). \end This tensor represents: * The identity mapping (or identity matrix), considered as a linear mapping or * The trace or tensor contraction, considered as a mapping * The map , representing scalar multiplication as a sum of
outer product In linear algebra, the outer product of two coordinate vectors is a matrix. If the two vectors have dimensions ''n'' and ''m'', then their outer product is an ''n'' × ''m'' matrix. More generally, given two tensors (multidimensional arrays of nu ...
s. The or multi-index Kronecker delta of order is a type tensor that is completely antisymmetric in its upper indices, and also in its lower indices. Two definitions that differ by a factor of are in use. Below, the version is presented has nonzero components scaled to be . The second version has nonzero components that are , with consequent changes scaling factors in formulae, such as the scaling factors of in ' below disappearing.


Definitions of the generalized Kronecker delta

In terms of the indices, the generalized Kronecker delta is defined as: \delta^_ = \begin +1 & \quad \text \nu_1 \dots \nu_p \text \mu_1 \dots \mu_p \\ -1 & \quad \text \nu_1 \dots \nu_p \text \mu_1 \dots \mu_p \\ \;\;0 & \quad \text. \end Let be the symmetric group of degree , then: \delta^_ = \sum_ \sgn(\sigma)\, \delta^_\cdots\delta^_ = \sum_ \sgn(\sigma)\, \delta^_\cdots\delta^_. Using anti-symmetrization: \delta^_ = p! \delta^_ \dots \delta^_ = p! \delta^_ \dots \delta^_. In terms of a
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if a ...
: \delta^_ = \begin \delta^_ & \cdots & \delta^_ \\ \vdots & \ddots & \vdots \\ \delta^_ & \cdots & \delta^_ \end. Using the Laplace expansion ( Laplace's formula) of determinant, it may be defined recursively: \begin \delta^_ &= \sum_^p (-1)^ \delta^_ \delta^_ \\ &= \delta^_ \delta^_ - \sum_^ \delta^_ \delta^_, \end where the caron, , indicates an index that is omitted from the sequence. When (the dimension of the vector space), in terms of the Levi-Civita symbol: \delta^_ = \varepsilon^\varepsilon_.


Contractions of the generalized Kronecker delta

Kronecker Delta contractions depend on the dimension of the space. For example, \delta^_ \delta^_ = (d-1) \delta^_ , where is the dimension of the space. From this relation the full contracted delta is obtained as \delta^_ \delta^_ = 2d(d-1) \delta^_ . The generalization of the preceding formulas is \left(x^2+y^2+z^2\right)^ \begin x & y & z \\ \frac & \frac & \frac \\ \frac & \frac & \frac \end \, ds \, dt.


See also

* Dirac measure *
Indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
* Levi-Civita symbol *
't Hooft symbol The t Hooft symbol is a collection of numbers which allows one to express the generators of the SU(2) Lie algebra in terms of the generators of Lorentz algebra. The symbol is a blend between the Kronecker delta and the Levi-Civita symbol. It wa ...
*
Unit function In number theory, the unit function is a completely multiplicative function on the positive integers defined as: :\varepsilon(n) = \begin 1, & \mboxn=1 \\ 0, & \mboxn \neq 1 \end It is called the unit function because it is the identity element f ...
* XNOR gate


References

{{Tensors Mathematical notation Elementary special functions