Editing 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 Kronecker delta function sometimes means the unit sample function <math>\delta[n]</math> , which represents a special case of the 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:
<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>

For discrete-time signals, it is conventional to place a single integer index in square braces; in contrast the Kronecker delta, <math>\delta_{ij}</math>, can have any number of indexes. In [[LTI system]] theory, the discrete unit sample function is typically used as an input to a discrete-time system for determining the [[impulse response]] function of the system which characterizes the system for any general imput.  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 comparison, in [[Discrete_time_and_continuous_time|continuous-time systems]] 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}}.

In continuous-time systems, the term "[[unit impulse function]]" is used to refer to the [[Dirac delta function]] <math>\delta(t)</math> or, in discrete-time systems, the Kronecker delta 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]]