View source for Kronecker delta

{{Short description|Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise}}
{{distinguish|text=the [[Dirac delta function]], [[Kronecker symbol]] or [[Kronecker integral]]}}

In [[mathematics]], the '''Kronecker delta''' (named after [[Leopold Kronecker]]) is a [[Function (mathematics)|function]] of two [[Variable (mathematics)|variables]], usually just non-negative [[integer]]s. The function is 1 if the variables are equal, and 0 otherwise:
<math display="block">\delta_{ij} = \begin{cases}
0 &\text{if } i \neq j,   \\
1 &\text{if } i=j.   \end{cases}</math>
or with use of [[Iverson bracket|Iverson brackets]]:
<math display="block">\delta_{ij} = [i=j]\,</math>
For example, <math>\delta_{12} = 0</math> because <math>1 \ne 2</math>, whereas <math>\delta_{33} = 1</math> because <math>3 = 3</math>.

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 <math>n\times n</math> [[identity matrix]] <math>\mathbf{I}</math> has entries equal to the Kronecker delta:
<math display="block"> I_{ij} = \delta_{ij} </math>
where <math>i</math> and <math>j</math> take the values <math>1,2,\cdots,n</math>, and the [[inner product]] of [[Euclidean vector|vector]]s can be written as
<math display="block"> \mathbf{a}\cdot\mathbf{b} = \sum_{i,j=1}^n a_{i}\delta_{ij}b_{j} = \sum_{i=1}^n a_{i} b_{i}.</math>
Here the [[Euclidean vector|Euclidean vectors]] are defined as {{mvar|n}}-tuples: <math> \mathbf{a} = (a_1, a_2, \dots, a_n)</math> and <math> \mathbf{b}= (b_1, b_2, ..., b_n) </math> and the last step is obtained by using the values of the Kronecker delta to reduce the summation over <math>j</math>.

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:
<math display="block">\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}</math>
Therefore, the matrix {{math|'''δ'''}} can be considered as an identity matrix.

Another useful representation is the following form:
<math display="block">\delta_{nm} = \lim_{N\to\infty}\frac{1}{N} \sum_{k = 1}^N e^{2 \pi i \frac{k}{N}(n-m)}</math>
This can be derived using the formula for the [[geometric series]].

==Alternative notation==
Using the [[Iverson bracket]]: <math display="block">\delta_{ij} = [i=j ].</math>

Often, a single-argument notation <math>\delta_i</math> is used, which is equivalent to setting <math>j=0</math>:
<math display="block">\delta_{i} = \delta_{i0} = \begin{cases}
0, & \text{if } i \neq 0  \\
1, & \text{if } i = 0
\end{cases}</math>

In [[linear algebra]], it can be thought of as a [[tensor]], and is written <math>\delta_j^i</math>. Sometimes the Kronecker delta is called the substitution tensor.<ref name="Trowbridge">{{cite journal |last=Trowbridge |first=J. H. |year=1998 |title=On a Technique for Measurement of Turbulent Shear Stress in the Presence of Surface Waves |journal=[[Journal of Atmospheric and Oceanic Technology]] |volume=15 |issue=1 |page=291 |doi=10.1175/1520-0426(1998)015<0290:OATFMO>2.0.CO;2 |bibcode=1998JAtOT..15..290T |doi-access=free }}</ref>

==Digital signal processing==
[[Image:unit impulse.gif|thumb|right|Unit sample function]]

In the study of [[digital signal processing]] (DSP), the unit sample function <math>\delta[n]</math> represents a special case of a 2-dimensional Kronecker delta function <math>\delta_{ij}</math> where the Kronecker indices include the number zero, and where one of the indices is zero.  In this case:
<math display="block">\delta[n] \equiv \delta_{n0} \equiv \delta_{0n}~~~\text{where} -\infty<n<\infty</math>

Or more generally where:
<math display="block">\delta[n-k] \equiv \delta[k-n] \equiv \delta_{nk} \equiv \delta_{kn}\text{where} -\infty<n<\infty, -\infty<k<\infty</math>

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 <math>\delta[n] \equiv \delta_{n0} \equiv \delta_{0n}</math> does not 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:
<math display="block">\delta[n] = \begin{cases} 1 & n = 0 \\ 0 & n \text{ is another integer}\end{cases}</math>

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:
<math display="block">\begin{cases} \int_{-\varepsilon}^{+\varepsilon}\delta(t)dt = 1 & \forall \varepsilon > 0 \\ \delta(t) = 0 & \forall t \neq 0\end{cases}</math>

Unlike the Kronecker delta function <math>\delta_{ij}</math> and the unit sample function <math>\delta[n]</math>, the Dirac delta function <math>\delta(t)</math> does not have an integer index, it has a single continuous non-integer value {{mvar|t}}.

To confuse matters more, the [[unit impulse function]] is sometimes used to refer to either the [[Dirac delta function]] <math>\delta(t)</math>, or the unit sample function <math>\delta[n]</math>.

==Notable properties ==
<!-- Please do not "correct" sifting to shifting.  The Kronecker delta acts as a sieve; that is, it *sifts*. -->
The Kronecker delta has the so-called ''sifting'' property that for <math>j\in\mathbb{Z}</math>:
<math display="block">\sum_{i=-\infty}^\infty a_i \delta_{ij} = a_j.</math>
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]]
<math display="block">\int_{-\infty}^\infty \delta(x-y)f(x)\, dx=f(y),</math>
and in fact Dirac's delta was named after the Kronecker delta because of this analogous property.<ref>{{cite book |last1=Dirac |first1=Paul |title=The Principles of Quantum Mechanics (1st ed.) |date=1930 |publisher=Oxford University Press |isbn=9780198520115}}</ref>  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 <math>i</math>, <math>j</math>, <math>k</math>, <math>l</math>, <math>m</math>, and <math>n</math> are usually reserved for discrete time (Kronecker).  Another common practice is to represent discrete sequences with square brackets; thus: <math>\delta[n]</math>. 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]].<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 | url-access=registration | url=https://archive.org/details/incidencealgebra0000spie }}.</ref>

==Relationship to the Dirac delta function==
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,\cdots,x_n\}</math>, with corresponding probabilities <math>p_1,\cdots,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
<math display="block">p(x) = \sum_{i=1}^n p_i \delta_{x x_i}.</math>

Equivalently, the [[probability density function]] <math>f(x)</math> of the distribution can be written using the Dirac delta function as
<math display="block">f(x) = \sum_{i=1}^n p_i \delta(x-x_i).</math>

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 <math>(1,1)</math> [[tensor]], the Kronecker tensor can be written <math>\delta^i_j</math> with a [[covariance and contravariance of vectors|covariant]] index <math>j</math> and [[Covariance and contravariance of vectors|contravariant]] index <math>i</math>:
<math display="block">\delta^{i}_{j} = \begin{cases} 0 & (i \ne j), \\ 1 & (i = j). \end{cases}</math>

This tensor represents:
* The identity mapping (or identity matrix), considered as a [[linear mapping]] <math>V\to V</math> or <math>V^*\to V^*</math>
* The [[trace (linear algebra)|trace]] or [[tensor contraction]], considered as a mapping <math>V^* \otimes V\to K</math>
* The map <math>K\to V^*\otimes V</math>, representing scalar multiplication as a sum of [[outer product]]s.

The '''{{visible anchor|generalized Kronecker delta}}''' or '''multi-index Kronecker delta''' of order <math>2p</math> is a type <math>(p,p)</math> tensor that is completely [[antisymmetric tensor|antisymmetric]] in its <math>p</math> upper indices, and also in its <math>p</math> lower indices.

Two definitions that differ by a factor of <math>p!</math> are in use. Below, the version is presented has nonzero components scaled to be <math>\pm 1</math>. The second version has nonzero components that are <math>\pm 1/p!</math>, with consequent changes scaling factors in formulae, such as the scaling factors of <math>1/p!</math> in ''{{section link||Properties of the generalized Kronecker delta}}'' below disappearing<!--This is worded awkwardly-->.<ref>{{cite web| url=http://people.physics.tamu.edu/pope/geom-group.pdf| first=Christopher|last=Pope| date=2008| title=Geometry and Group Theory}}</ref>

=== Definitions of the generalized Kronecker delta ===
In terms of the indices, the generalized Kronecker delta is defined as:<ref>{{cite book|first=Theodore|last=Frankel|title=The Geometry of Physics: An Introduction|edition=3rd|date=2012|publisher=Cambridge University Press|isbn=9781107602601}}</ref><ref>{{cite book|first=D. C.|last=Agarwal|title=Tensor Calculus and Riemannian Geometry|edition=22nd|date=2007|publisher=Krishna Prakashan Media}}{{ISBN missing}}</ref>
<math display="block">\delta^{\mu_1 \dots \mu_p }_{\nu_1 \dots \nu_p} = \begin{cases}
\phantom-1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are distinct integers and are an even permutation of } \mu_1 \dots \mu_p \\
-1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are distinct integers and are an odd permutation of } \mu_1 \dots \mu_p \\
\phantom-0 & \quad \text{in all other cases}.
\end{cases}</math>

Let <math>\mathrm{S}_p</math> be the [[symmetric group]] of degree <math>p</math>, then:
<math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
= \sum_{\sigma \in \mathrm{S}_p} \sgn(\sigma)\, \delta^{\mu_1}_{\nu_{\sigma(1)}}\cdots\delta^{\mu_p}_{\nu_{\sigma(p)}}
= \sum_{\sigma \in \mathrm{S}_p} \sgn(\sigma)\, \delta^{\mu_{\sigma(1)}}_{\nu_1}\cdots\delta^{\mu_{\sigma(p)}}_{\nu_p}. </math>

Using [[Antisymmetric tensor#Notation|anti-symmetrization]]:
<math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
= p! \delta^{\mu_1}_{[ \nu_1} \dots \delta^{\mu_p}_{\nu_p ]}
= p! \delta^{[ \mu_1}_{\nu_1} \dots \delta^{\mu_p ]}_{\nu_p}.</math>

In terms of a <math>p\times p</math> [[determinant]]:<ref>{{cite book |first1=David |last1=Lovelock |first2=Hanno |last2=Rund |title=Tensors, Differential Forms, and Variational Principles |publisher=Courier Dover Publications |year=1989 |isbn=0-486-65840-6 }}</ref>
<math display="block">\delta^{\mu_1 \dots \mu_p }_{\nu_1 \dots \nu_p} =
\begin{vmatrix}
\delta^{\mu_1}_{\nu_1} & \cdots & \delta^{\mu_1}_{\nu_p} \\
\vdots & \ddots & \vdots \\
\delta^{\mu_p}_{\nu_1} & \cdots & \delta^{\mu_p}_{\nu_p}
\end{vmatrix}.</math>

Using the [[Laplace expansion]] ([[Determinant#Laplace's expansion 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 {{math|1=''δ'' = 1}} for {{math|1=''p'' = 0}}, or alternatively {{math|1=''δ''{{su|p=''μ''|b=''ν''|lh=0.9em}} = ''δ''{{su|p=''μ''|b=''ν''|lh=0.9em}}}} for {{math|1=''p'' = 1}} (generalized delta in terms of standard delta).</ref>
<math display="block">\begin{align}
  \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
    &= \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} \\
    &= \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>
where the caron, <math>\check{}</math>, indicates an index that is omitted from the sequence.

When <math>p=n</math> (the dimension of the vector space), in terms of the [[Levi-Civita symbol]]:
<math display="block">\delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n} = \varepsilon^{\mu_1 \dots \mu_n}\varepsilon_{\nu_1 \dots \nu_n}\,.</math>
More generally, for <math>m=n-p</math>, using the [[Einstein summation convention]]:
<math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = \tfrac{1}{m!} \varepsilon^{\kappa_1 \dots \kappa_m \mu_1 \dots \mu_p}\varepsilon_{\kappa_1 \dots \kappa_m \nu_1 \dots \nu_p}\,.</math>

=== Contractions of the generalized Kronecker delta ===
Kronecker Delta contractions depend on the dimension of the space. For example,
<math display="block">\delta^{\nu_1}_{\mu_1} \delta^{\mu_1 \mu_2}_{\nu_1 \nu_2} = (d-1) \delta^{\mu_2}_{\nu_2} ,</math>
where {{mvar|d}} is the dimension of the space. From this relation the full contracted delta is obtained as
<math display="block">\delta^{\nu_1 \nu_2}_{\mu_1 \mu_2} \delta^{\mu_1 \mu_2}_{\nu_1 \nu_2} = 2d(d-1) .</math>
The generalization of the preceding formulas is{{cn|date=January 2023}}
<math display="block">\delta^{\nu_1 \dots \nu_n}_{\mu_1 \dots \mu_n} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = n! \frac{(d-p+n)!}{(d-p)!} \delta^{\mu_{n+1} \dots \mu_p}_{\nu_{n+1} \dots \nu_p} .</math>

=== Properties of the generalized Kronecker delta ===
The generalized Kronecker delta may be used for [[Antisymmetric tensor#Notation|anti-symmetrization]]:
<math display="block">\begin{align}
 \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a^{\nu_1 \dots \nu_p} &= a^{[ \mu_1 \dots \mu_p ]} , \\
 \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a_{\mu_1 \dots \mu_p} &= a_{[ \nu_1 \dots \nu_p ]} . \end{align}</math>

From the above equations and the properties of [[anti-symmetric tensor]]s, we can derive the properties of the generalized Kronecker delta:
<math display="block">\begin{align}
 \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a^{[ \nu_1 \dots \nu_p ]} &= a^{[ \mu_1 \dots \mu_p ]} , \\
 \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a_{[ \mu_1 \dots \mu_p ]} &= a_{[ \nu_1 \dots \nu_p ]} , \\
 \frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} \delta^{\nu_1 \dots \nu_p}_{\kappa_1 \dots \kappa_p} &= \delta^{\mu_1 \dots \mu_p}_{\kappa_1 \dots \kappa_p} ,
\end{align}</math>
which are the generalized version of formulae written in ''{{section link||Properties}}''. The last formula is equivalent to the [[Cauchy–Binet formula]].

Reducing the order via summation of the indices may be expressed by the identity<ref>{{cite book |first=Sadri |last=Hassani |title=Mathematical Methods: For Students of Physics and Related Fields |edition=2nd |publisher=Springer-Verlag |year=2008 |isbn=978-0-387-09503-5 }}</ref>
<math display="block"> \delta^{\mu_1 \dots \mu_s \, \mu_{s+1} \dots \mu_p}_{\nu_1 \dots \nu_s \, \mu_{s+1} \dots \mu_p} = \frac{(n-s)!}{(n-p)!} \delta^{\mu_1 \dots \mu_s}_{\nu_1 \dots \nu_s}.</math>

Using both the summation rule for the case <math>p=n</math> and the relation with the Levi-Civita symbol, [[Levi-Civita symbol#n dimensions|the summation rule of the Levi-Civita symbol]] is derived:
<math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = \frac{1}{(n-p)!}\varepsilon^{\mu_1 \dots \mu_p \, \kappa_{p+1} \dots \kappa_n}\varepsilon_{\nu_1 \dots \nu_p \, \kappa_{p+1} \dots \kappa_n}.</math>
The 4D version of the last relation appears in Penrose's [[Mathematics of general relativity#Spinor formalism|spinor approach to general relativity]]<ref>{{Cite journal|last=Penrose|first=Roger|date=June 1960|title=A spinor approach to general relativity |url=https://linkinghub.elsevier.com/retrieve/pii/000349166090021X|journal=Annals of Physics|language=en| volume=10|issue=2 |pages=171–201|doi=10.1016/0003-4916(60)90021-X|bibcode=1960AnPhy..10..171P}}</ref> that he later generalized, while he was developing Aitken's diagrams,<ref>{{Cite book|last=Aitken|first=Alexander Craig|title=Determinants and Matrices|publisher=Oliver and Boyd|year=1958|location=UK}}</ref> to become part of the technique of [[Penrose graphical notation]].<ref>[[Roger Penrose]], "Applications of negative dimensional tensors," in ''Combinatorial Mathematics and its Applications'', Academic Press (1971).</ref> Also, this relation is extensively used in [[S-duality]] theories, especially when written in the language of [[Differential form|differential forms]] and [[Hodge star operator#Duality|Hodge duals]].

==Integral representations==
For any integers <math>j</math> and <math>k</math>, the Kronecker delta can be written as a complex [[contour integral]] using a standard [[residue (complex analysis)|residue]] calculation. The integral is taken over the unit circle in the complex plane, oriented counterclockwise. An equivalent representation of the integral arises by parameterizing the contour by an angle around the origin.
<math display="block"> \delta_{jk} = \frac1{2\pi i} \oint_{|z|=1} z^{j-k-1} \,dz=\frac1{2\pi} \int_0^{2\pi} e^{i(j-k)\varphi} \,d\varphi</math>

==The Kronecker comb==
The Kronecker comb function with period <math>N</math> is defined (using [[digital signal processing|DSP]] notation) as:
<math display="block">\Delta_N[n]=\sum_{k=-\infty}^\infty \delta[n-kN],</math>
where <math>N\ne 0</math> and <math>n</math> are integers. The Kronecker comb thus consists of an infinite series of unit impulses that are {{mvar|N}} units apart, aligned so one of the impulses occurs at zero. It may be considered to be the discrete analog of the [[Dirac comb]].

==See also==
*[[Dirac measure]]
*[[Indicator function]]
*[[Heaviside step function]]
*[[Levi-Civita symbol]]
*[[Minkowski space#Standard basis|Minkowski metric]]
*[['t Hooft symbol]]
*[[Unit function]]
*[[XNOR gate]]

==References==
<references />

{{Tensors}}

[[Category:Mathematical notation]]
[[Category:Elementary special functions]]