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| {{Distinguish2|the [[Dirac delta function]], nor with the [[Kronecker symbol]]}}
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| In [[mathematics]], the '''Kronecker delta''' or '''Kronecker's delta''', named after [[Leopold Kronecker]], is a [[Function (mathematics)|function]] of two [[Variable (mathematics)|variables]], usually [[integer]]s. The function is 1 if the variables are equal, and 0 otherwise:
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| :<math>\delta_{ij} = \begin{cases}
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| 0 &\text{if } i \neq j \\
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| 1 &\text{if } i=j, \end{cases}</math>
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| where the Kronecker delta δ<sub>''ij''</sub> is a [[piecewise]] function of variables <math>i</math> and <math>j</math>. For example, δ<sub>1 2</sub> = 0, whereas δ<sub>3 3</sub> = 1.
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| In [[linear algebra]], the [[identity matrix]] can be written as
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| :<math>(\delta_{ij})_{i,j=1}^n\,</math>
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| and the [[inner product]] of [[Euclidean vector|vector]]s can be written as
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| :<math>\textstyle
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| \boldsymbol{a}\cdot\boldsymbol{b} = \sum_{ij} a_{i}\delta_{ij}b_{j}.
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| </math>
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| The Kronecker delta is used in many areas of mathematics, physics and engineering, primarily as an expedient to convey in a single equation what might otherwise take several lines of text.
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| == Properties ==
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| The following equations are satisfied:
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| :<math>\begin{align}
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| \sum_{j} \delta_{ij} a_j &= a_i,\\
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| \sum_{i} a_i\delta_{ij} &= a_j,\\
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| \sum_{k} \delta_{ik}\delta_{kj} &= \delta_{ij}.
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| \end{align}</math>
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| Therefore, δ<sub>ij</sub> can be considered as an identity matrix.
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| ==Alternative notation==
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| Using the [[Iverson bracket]]:
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| : <math>\delta_{ij} = [i=j ].\,</math>
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| Often, the notation <math>\delta_i</math> is used.
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| :<math>\delta_{i} = \begin{cases}
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| 0, & \mbox{if } i \ne 0 \\
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| 1, & \mbox{if } i=0 \end{cases}</math>
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| In [[linear algebra]], it can be thought of as a [[tensor]], and is written <math>\delta^i_j</math>.
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| Sometimes the Kronecker delta is called the substitution tensor.<ref name="Trowbridge">Trowbridge, 1998. Journal of Atmospheric and Oceanic Technology. V15, 1 p291</ref>
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| ==Digital signal processing==
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| [[Image:unit impulse.gif|thumb|right|An impulse function]]
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| Similarly, in [[digital signal processing]], the same concept is represented as a function on <math>\mathbb{Z}</math> (the [[integer]]s):
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| :<math>
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| \delta[n] = \begin{cases} 0, & n \ne 0 \\ 1, & n = 0.\end{cases}</math>
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| The function is referred to as an ''impulse'', or ''unit impulse''. And when it stimulates a signal processing element, the output is called the [[impulse response]] of the element.
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| ==Properties of the delta function==
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| <!-- Please do not "correct" sifting to shifting. The Kronecker delta acts as a sieve; that is, it *sifts*. -->
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| The Kronecker delta has the so-called ''sifting'' property that for <math>j\in\mathbb Z</math>:
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| :<math>\sum_{i=-\infty}^\infty a_i \delta_{ij} =a_j.</math>
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| and if the integers are viewed as a [[measure space]], endowed with the [[counting measure]], then this property coincides with the defining property of the [[Dirac delta function]]
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| :<math>\int_{-\infty}^\infty \delta(x-y)f(x) dx=f(y),</math>
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| 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, <math>\delta(t)\,</math> generally indicates continuous time (Dirac), whereas arguments like ''i'', ''j'', ''k'', ''l'', ''m'', and ''n'' are usually reserved for discrete time (Kronecker). Another common practice is to represent discrete sequences with square brackets; thus: <math>\delta[n]\,</math>. It is important to note that the Kronecker delta is not the result of directly sampling the Dirac delta function. | |
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| The Kronecker delta forms the multiplicative [[identity element]] of an [[incidence algebra]].<ref>{{citation | first1=Eugene | last1=Spiegel | first2=Christopher J. | last2=O'Donnell | title=Incidence algebras | publisher=Marcel Dekker | isbn=0-8247-0036-8 | year=1997 | series=Pure and Applied Mathematics | volume=206 }}.</ref>
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| ==Relationship to the [[Dirac delta function]]==
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| In [[probability theory]] and [[statistics]], the Kronecker delta and [[Dirac delta function]] can both be used to represent a [[discrete distribution]]. If the [[support (mathematics)|support]] of a distribution consists of points <math>\mathbf{x} = \{x_1,\dots,x_n\}</math>, with corresponding probabilities <math>p_1,\dots,p_n\,</math>, then the [[probability mass function]] <math>p(x)\,</math> of the distribution over <math>\mathbf{x}</math> can be written, using the Kronecker delta, as
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| :<math>p(x) = \sum_{i=1}^n p_i \delta_{x x_i}.</math>
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| Equivalently, the [[probability density function]] <math>f(x)\,</math> of the distribution can be written using the [[Dirac delta function]] as
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| :<math>f(x) = \sum_{i=1}^n p_i \delta(x-x_i).</math>
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| 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.
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| ==Generalizations of the Kronecker delta==
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| If it is considered as a type (1,1) [[tensor]], the Kronecker tensor, it can be written
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| <math>\delta^i_j</math> with a [[covariance and contravariance of vectors|covariant]] index ''j'' and [[Covariance and contravariance of vectors|contravariant]] index ''i'':
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| :<math>
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| \delta^{i}_{j} =
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| \begin{cases}
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| 0 & (i \ne j), \\
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| 1 & (i = j).
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| \end{cases}
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| </math>
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| This (1,1) tensor represents:
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| * The identity mapping (or identity matrix), considered as a [[linear mapping]] <math>V \to V</math> or <math>V^* \to V^*</math>
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| * The [[trace (linear algebra)|trace]] or [[tensor contraction]], considered as a mapping <math>V^* \otimes V \to K</math>
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| * The map <math>K \to V^* \otimes V</math>, representing scalar multiplication as a sum of [[outer product]]s.
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| {{anchor|generalized Kronecker delta}}The '''generalized Kronecker delta''' of order 2''p'' is a type (''p'',''p'') [[tensor]] that is a completely [[antisymmetric tensor|antisymmetric]] in its ''p'' upper indices, and also in its ''p'' lower indices.
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| === Definitions of generalized Kronecker delta ===
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| In terms of the indices:<ref>Theodore Frankel, ''The Geometry of Physics: An Introduction'' 3rd edition (2012), published by Cambridge University Press, ISBN 9781107602601</ref><ref>D. C. Agarwal, ''Tensor Calculus and Riemannian Geometry'' 22nd edition (2007), published by Krishna Prakashan Media</ref>
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| :<math>
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| \delta^{\mu_1 \dots \mu_p }_{\nu_1 \dots \nu_p} =
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| \begin{cases}
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| +1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are distinct integers and are an even permutation of } \mu_1 \dots \mu_p \\
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| -1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are distinct integers and are an odd permutation of } \mu_1 \dots \mu_p \\
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| \;\;0 & \quad \text{in all other cases}.\end{cases}
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| </math>
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| Let <math>\mathfrak{S}_p</math> be the [[symmetric group]] of degree p, then:
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| :<math>
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| \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
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| = \sum_{\sigma \in \mathfrak{S}_p} \sgn(\sigma)\, \delta^{\mu_1}_{\nu_{\sigma(1)}}\cdots\delta^{\mu_p}_{\nu_{\sigma(p)}}
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| = \sum_{\sigma \in \mathfrak{S}_p} \sgn(\sigma)\, \delta^{\mu_{\sigma(1)}}_{\nu_1}\cdots\delta^{\mu_{\sigma(p)}}_{\nu_p}.
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| </math>
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| Using [[Antisymmetric_tensor#Notation|anti-symmetrization]]:
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| :<math>
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| \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
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| = p! \delta^{\mu_1}_{\lbrack \nu_1} \dots \delta^{\mu_p}_{\nu_p \rbrack}
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| = p! \delta^{\lbrack \mu_1}_{\nu_1} \dots \delta^{\mu_p \rbrack}_{\nu_p}.
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| </math>
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| In terms of an {{nowrap|''p'' × ''p''}} [[determinant]]:<ref>{{cite book |author=David Lovelock, Hanno Rund |title=Tensors, Differential Forms, and Variational Principles |publisher=Courier Dover Publications |year=1989 |isbn=0-486-65840-6 }}</ref>
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| :<math>
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| \delta^{\mu_1 \dots \mu_p }_{\nu_1 \dots \nu_p} =
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| \begin{vmatrix}
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| \delta^{\mu_1}_{\nu_1} & \cdots & \delta^{\mu_1}_{\nu_p} \\
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| \vdots & \ddots & \vdots \\
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| \delta^{\mu_p}_{\nu_1} & \cdots & \delta^{\mu_p}_{\nu_p}
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| \end{vmatrix}.
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| </math>
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| Using the [[Laplace expansion]] ([[Determinant#Laplace's formula and the adjugate matrix|Laplace's formula]]) of determinant, it may be defined [[Recursion|recursively]]:<ref>A recursive definition requires a first case, which may be taken as ''δ'' = 1 for ''p'' = 0, or alternatively ''δ''{{su|p=''μ''|b=''ν''}} = ''δ''{{su|p=''μ''|b=''ν''}} for ''p'' = 1 (generalized delta in terms of standard delta).</ref>
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| :<math>\begin{align} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
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| & = \sum_{k=1}^p (-1)^{p+k} \delta^{\mu_p}_{\nu_k} \delta^{\mu_1 \dots \mu_{k} \dots \check\mu_p}_{\nu_1 \dots \check\nu_k \dots \nu_{p}} \\
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| & = \delta^{\mu_p}_{\nu_p} \delta^{\mu_1 \dots \mu_{p-1}}_{\nu_1 \dots \nu_{p-1}} - \sum_{k=1}^{p-1} \delta^{\mu_p}_{\nu_k} \delta^{\mu_1 \dots \mu_{k-1} \; \mu_k \; \mu_{k+1} \dots \mu_{p-1}}_{\nu_1 \dots \;\nu_{k-1} \; \nu_p \; \nu_{k+1}\; \dots \nu_{p-1}},\end{align}</math>
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| where <math>\check{~}</math> indicates an index that is omitted from the sequence.
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| When {{nowrap|1=''p'' = ''n''}} (the dimension of the vector space), in terms of the [[Levi-Civita symbol]]:
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| :<math>
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| \delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n} = \varepsilon^{\mu_1 \dots \mu_n}\varepsilon_{\nu_1 \dots \nu_n}.
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| </math>
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| === Properties of generalized Kronecker delta ===
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| The generalized Kronecker delta may be used for [[Antisymmetric_tensor#Notation|anti-symmetrization]]:
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| :<math> \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a^{\nu_1 \dots \nu_p} = a^{\lbrack \mu_1 \dots \mu_p \rbrack} ,</math>
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| :<math> \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a_{\mu_1 \dots \mu_p} = a_{\lbrack \nu_1 \dots \nu_p \rbrack} .</math>
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| From the above equations and the properties of [[anti-symmetric tensor]], we can derive the properties of the generalized Kronecker delta:
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| :<math> \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a^{\lbrack \nu_1 \dots \nu_p \rbrack} = a^{\lbrack \mu_1 \dots \mu_p \rbrack} ,</math>
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| :<math> \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a_{\lbrack \mu_1 \dots \mu_p \rbrack} = a_{\lbrack \nu_1 \dots \nu_p \rbrack} ,</math>
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| :<math> \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} \delta^{\nu_1 \dots \nu_p}_{\rho_1 \dots \rho_p}
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| = \delta^{\mu_1 \dots \mu_p}_{\rho_1 \dots \rho_p} ,</math>
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| which are the generalized version of formulae written in the section [[#Properties|Properties]].
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| The last formula is equivalent to the [[ Cauchy–Binet formula]].
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| Reducing the order via summation of the indices may be expressed by the identity<ref>
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| {{cite book |author=Sadri Hassani |title=Mathematical Methods: For Students of Physics and Related Fields 2nd edition|publisher=Springer-Verlag |year=2008 |isbn=978-0387095035 }}</ref>
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| :<math> \delta^{\mu_1 \dots \mu_s \, \mu_{s+1} \dots \mu_p}_{\nu_1 \dots \nu_s \, \mu_{s+1} \dots \mu_p} = \tfrac{(n-s)!}{(n-p)!} \delta^{\mu_1 \dots \mu_s}_{\nu_1 \dots \nu_s}.</math>
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| Using both the summation rule for the case {{nowrap|1=''p'' = ''n''}} and the relation with the Levi-Civita symbol,
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| [[Levi-Civita_symbol#n dimensions|the summation rule of the Levi-Civita symbol]] is derived:
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| :<math>
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| \delta^{\mu_1 \dots \mu_s}_{\nu_1 \dots \nu_s} = {1 \over (n-s)!}\,
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| \varepsilon^{\mu_1 \dots \mu_s \, \rho_{s+1} \dots \rho_n}\varepsilon_{\nu_1 \dots \nu_s \, \rho_{s+1} \dots \rho_n}.
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| </math>
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| ==Integral representations==
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| For any integer ''n'', using a standard [[Residue (complex analysis)|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.
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| :<math> \delta_{x,n} = \frac1{2\pi i} \oint_{|z|=1} z^{x-n-1} dz=\frac1{2\pi} \int_0^{2\pi} e^{i(x-n)\varphi} d\varphi</math>
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| ==The Kronecker comb==
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| The Kronecker comb function with period ''N'' is defined (using [[digital signal processing|DSP]] notation) as:
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| :<math>\Delta_N[n]=\sum_{k=-\infty}^\infty \delta[n-kN],</math>
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| where ''N'' and ''n'' are integers. The Kronecker comb thus consists of an infinite series of unit impulses ''N'' units apart, and includes the unit impulse at zero. It may be considered to be the discrete analog of the [[Dirac comb]].
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| ==Kronecker Integral<ref>{{Citation |title=Advanced Calculus | first=Wilfred |last=Kaplan |publisher=Pearson Education. Inc |year=2003 |isbn=0-201-79937-5 |page=364}}</ref>==
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| The Kronecker delta is also called degree of mapping of one surface into another. Suppose a mapping takes place from surface <math> S_{uvw} </math> to <math> S_{xyz} </math> that are boundaries of regions, <math> R_{uvw}</math> and <math> R_{xyz} </math> which is simply connected with one-to-one correspondence. In this framework, if s and t are parameters for <math> S_{uvw} </math>, and <math> S_{uvw} </math> to <math> S_{xyz} </math> are each oriented by the outer normal n:
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| :<math> u=u(s,t), v=v(s,t),w=w(s,t), </math>
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| while the normal has the direction of:
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| :<math>(u_{s} i +v_{s} j + w_{s} k) \times (u_{t}i +v_{t}j +w_{t}k).</math>
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| Let x=x(u,v,w),y=y(u,v,w),z=z(u,v,w) be defined and smooth in a domain containing <math>S_{uvw}</math>, and let these equations define the mapping of <math> S_{uvw}</math> into <math>S_{xyz}</math>. Then the degree <math> \delta </math> of mapping is <math>1/4\pi</math> times the solid angle of the image S of <math>S_{uvw}</math> with respect to the interior point of <math> S_{xyz}</math>, O. If O is the origin of the region, <math>R_{xyz}</math>, then the degree, <math>\delta</math> is given by the integral:
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| :<math>\delta=\frac{1}{4\pi}\iint_{R_{st}}\frac{\begin{vmatrix}x&y&z\\\dfrac{\partial x}{\partial s}&\dfrac{\partial y}{\partial s}&\dfrac{\partial z}{\partial s}\\\dfrac{\partial x}{\partial t}&\dfrac{\partial y}{\partial t}&\dfrac{\partial z}{\partial t}\end{vmatrix}}{(x^2+y^2+z^2)\sqrt{x^2+y^2+z^2}}dsdt.</math>
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| ==See also==
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| *[[Dirac measure]]
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| *[[Indicator function]]
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| *[[Levi-Civita symbol]]
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| *[[Unit function]]
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| ==References==
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| <references />
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| {{tensor}}
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| {{DEFAULTSORT:Kronecker Delta}}
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| [[Category:Mathematical notation]]
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| [[Category:Elementary special functions]]
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