Kronecker delta
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:
- δij={0if i≠j,1if i=j.{displaystyle delta _{ij}={begin{cases}0&{text{if }}ineq j,\1&{text{if }}i=j.end{cases}}}
where the Kronecker delta δij is a piecewise function of variables i and j. For example, δ1 2 = 0, whereas δ3 3 = 1.
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:
- Iij=δij{displaystyle I_{ij}=delta _{ij}}
where i and j take the values 1, 2, ..., n, and the inner product of vectors can be written as
- a⋅b=∑i,j=1naiδijbj.{displaystyle mathbf {a} cdot mathbf {b} =sum _{i,j=1}^{n}a_{i}delta _{ij}b_{j}.}
The restriction to positive integers is common, but there is no reason it cannot have negative integers as well as positive, or any discrete rational numbers. If i and j above take rational values, then for example
- δ(−1)(−3)=0δ(−2)(−2)=1δ(12)(−32)=0δ(53)(53)=1.{displaystyle {begin{aligned}delta _{(-1)(-3)}&=0&qquad delta _{(-2)(-2)}&=1\delta _{left({frac {1}{2}}right)left(-{frac {3}{2}}right)}&=0&qquad delta _{left({frac {5}{3}}right)left({frac {5}{3}}right)}&=1.end{aligned}}}
This latter case is for convenience.
Contents
1 Properties
2 Alternative notation
3 Digital signal processing
4 Properties of the delta function
5 Relationship to the Dirac delta function
6 Generalizations
6.1 Definitions of the generalized Kronecker delta
6.2 Properties of the generalized Kronecker delta
7 Integral representations
8 The Kronecker comb
9 Kronecker integral
10 See also
11 References
Properties
The following equations are satisfied:
- ∑jδijaj=ai,∑iaiδij=aj,∑kδikδkj=δij.{displaystyle {begin{aligned}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{aligned}}}
Therefore, the matrix δ can be considered as an identity matrix.
Another useful representation is the following form:
- δnm=1N∑k=1Ne2πikN(n−m){displaystyle delta _{nm}={frac {1}{N}}sum _{k=1}^{N}e^{2pi i{frac {k}{N}}(n-m)}}
This can be derived using the formula for the finite geometric series.
Alternative notation
Using the Iverson bracket:
- δij=[i=j].{displaystyle delta _{ij}=[i=j].}
Often, a single-argument notation δi is used, which is equivalent to setting j = 0:
- δi={0,if i≠01,if i=0{displaystyle delta _{i}={begin{cases}0,&{mbox{if }}ineq 0\1,&{mbox{if }}i=0end{cases}}}
In linear algebra, it can be thought of as a tensor, and is written δi
j. Sometimes the Kronecker delta is called the substitution tensor.[1]
Digital signal processing
Similarly, in digital signal processing, the same concept is represented as a sequence or discrete function on ℤ (the integers):
- δ[n]={0,n≠01,n=0.{displaystyle delta [n]={begin{cases}0,&nneq 0\1,&n=0.end{cases}}}
The function is referred to as an impulse, or unit impulse. When it is the input to a discrete-time signal processing element, the output is called the impulse response of the element.
Properties of the delta function
The Kronecker delta has the so-called sifting property that for j ∈ ℤ:
- ∑i=−∞∞aiδij=aj.{displaystyle sum _{i=-infty }^{infty }a_{i}delta _{ij}=a_{j}.}
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
- ∫−∞∞δ(x−y)f(x)dx=f(y),{displaystyle int _{-infty }^{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, δ(t) 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: δ[n]. It is important to note that 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.[2]
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 of a distribution consists of points x = {x1, ..., xn}, with corresponding probabilities p1, ..., pn, then the probability mass function p(x) of the distribution over x can be written, using the Kronecker delta, as
- p(x)=∑i=1npiδxxi.{displaystyle p(x)=sum _{i=1}^{n}p_{i}delta _{xx_{i}}.}
Equivalently, the probability density function f(x) of the distribution can be written using the Dirac delta function as
- f(x)=∑i=1npiδ(x−xi).{displaystyle f(x)=sum _{i=1}^{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 (1,1) tensor, the Kronecker tensor can be written
δi
j with a covariant index j and contravariant index i:
- δji={0(i≠j),1(i=j).{displaystyle delta _{j}^{i}={begin{cases}0&(ineq j),\1&(i=j).end{cases}}}
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 2p is a type (p,p) tensor that is a 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]
Definitions of the generalized Kronecker delta
In terms of the indices:[4][5]
- δν1…νpμ1…μp={+1if ν1…νp are distinct integers and are an even permutation of μ1…μp−1if ν1…νp are distinct integers and are an odd permutation of μ1…μp0in all other cases.{displaystyle delta _{nu _{1}dots nu _{p}}^{mu _{1}dots mu _{p}}={begin{cases}+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}\;;0&quad {text{in all other cases}}.end{cases}}}
Let Sp be the symmetric group of degree p, then:
- δν1…νpμ1…μp=∑σ∈Spsgn(σ)δνσ(1)μ1⋯δνσ(p)μp=∑σ∈Spsgn(σ)δν1μσ(1)⋯δνpμσ(p).{displaystyle delta _{nu _{1}dots nu _{p}}^{mu _{1}dots mu _{p}}=sum _{sigma in mathrm {S} _{p}}operatorname {sgn}(sigma ),delta _{nu _{sigma (1)}}^{mu _{1}}cdots delta _{nu _{sigma (p)}}^{mu _{p}}=sum _{sigma in mathrm {S} _{p}}operatorname {sgn}(sigma ),delta _{nu _{1}}^{mu _{sigma (1)}}cdots delta _{nu _{p}}^{mu _{sigma (p)}}.}
Using anti-symmetrization:
- δν1…νpμ1…μp=p!δ[ν1μ1…δνp]μp=p!δν1[μ1…δνpμp].{displaystyle delta _{nu _{1}dots nu _{p}}^{mu _{1}dots mu _{p}}=p!delta _{lbrack nu _{1}}^{mu _{1}}dots delta _{nu _{p}rbrack }^{mu _{p}}=p!delta _{nu _{1}}^{lbrack mu _{1}}dots delta _{nu _{p}}^{mu _{p}rbrack }.}
In terms of a p × p determinant:[6]
- δν1…νpμ1…μp=|δν1μ1⋯δνpμ1⋮⋱⋮δν1μp⋯δνpμp|.{displaystyle delta _{nu _{1}dots nu _{p}}^{mu _{1}dots mu _{p}}={begin{vmatrix}delta _{nu _{1}}^{mu _{1}}&cdots &delta _{nu _{p}}^{mu _{1}}\vdots &ddots &vdots \delta _{nu _{1}}^{mu _{p}}&cdots &delta _{nu _{p}}^{mu _{p}}end{vmatrix}}.}
Using the Laplace expansion (Laplace's formula) of determinant, it may be defined recursively:[7]
- δν1…νpμ1…μp=∑k=1p(−1)p+kδνkμpδν1…νˇk…νpμ1…μk…μˇp=δνpμpδν1…νp−1μ1…μp−1−∑k=1p−1δνkμpδν1…νk−1νpνk+1…νp−1μ1…μk−1μkμk+1…μp−1,{displaystyle {begin{aligned}delta _{nu _{1}dots nu _{p}}^{mu _{1}dots mu _{p}}&=sum _{k=1}^{p}(-1)^{p+k}delta _{nu _{k}}^{mu _{p}}delta _{nu _{1}dots {check {nu }}_{k}dots nu _{p}}^{mu _{1}dots mu _{k}dots {check {mu }}_{p}}\&=delta _{nu _{p}}^{mu _{p}}delta _{nu _{1}dots nu _{p-1}}^{mu _{1}dots mu _{p-1}}-sum _{k=1}^{p-1}delta _{nu _{k}}^{mu _{p}}delta _{nu _{1}dots nu _{k-1},nu _{p},nu _{k+1}dots nu _{p-1}}^{mu _{1}dots mu _{k-1},mu _{k},mu _{k+1}dots mu _{p-1}},end{aligned}}}
where the caron, ˇ, indicates an index that is omitted from the sequence.
When p = n (the dimension of the vector space), in terms of the Levi-Civita symbol:
- δν1…νnμ1…μn=εμ1…μnεν1…νn.{displaystyle delta _{nu _{1}dots nu _{n}}^{mu _{1}dots mu _{n}}=varepsilon ^{mu _{1}dots mu _{n}}varepsilon _{nu _{1}dots nu _{n}}.}
Properties of the generalized Kronecker delta
The generalized Kronecker delta may be used for anti-symmetrization:
- 1p!δν1…νpμ1…μpaν1…νp=a[μ1…μp],1p!δν1…νpμ1…μpaμ1…μp=a[ν1…νp].{displaystyle {begin{aligned}{frac {1}{p!}}delta _{nu _{1}dots nu _{p}}^{mu _{1}dots mu _{p}}a^{nu _{1}dots nu _{p}}&=a^{lbrack mu _{1}dots mu _{p}rbrack },\{frac {1}{p!}}delta _{nu _{1}dots nu _{p}}^{mu _{1}dots mu _{p}}a_{mu _{1}dots mu _{p}}&=a_{lbrack nu _{1}dots nu _{p}rbrack }.end{aligned}}}
From the above equations and the properties of anti-symmetric tensors, we can derive the properties of the generalized Kronecker delta:
- 1p!δν1…νpμ1…μpa[ν1…νp]=a[μ1…μp],1p!δν1…νpμ1…μpa[μ1…μp]=a[ν1…νp],1p!δν1…νpμ1…μpδρ1…ρpν1…νp=δρ1…ρpμ1…μp,{displaystyle {begin{aligned}{frac {1}{p!}}delta _{nu _{1}dots nu _{p}}^{mu _{1}dots mu _{p}}a^{lbrack nu _{1}dots nu _{p}rbrack }&=a^{lbrack mu _{1}dots mu _{p}rbrack },\{frac {1}{p!}}delta _{nu _{1}dots nu _{p}}^{mu _{1}dots mu _{p}}a_{lbrack mu _{1}dots mu _{p}rbrack }&=a_{lbrack nu _{1}dots nu _{p}rbrack },\{frac {1}{p!}}delta _{nu _{1}dots nu _{p}}^{mu _{1}dots mu _{p}}delta _{rho _{1}dots rho _{p}}^{nu _{1}dots nu _{p}}&=delta _{rho _{1}dots rho _{p}}^{mu _{1}dots mu _{p}},end{aligned}}}
which are the generalized version of formulae written in § 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[8]
- δν1…νsμs+1…μpμ1…μsμs+1…μp=(n−s)!(n−p)!δν1…νsμ1…μs.{displaystyle delta _{nu _{1}dots nu _{s},mu _{s+1}dots mu _{p}}^{mu _{1}dots mu _{s},mu _{s+1}dots mu _{p}}={frac {(n-s)!}{(n-p)!}}delta _{nu _{1}dots nu _{s}}^{mu _{1}dots mu _{s}}.}
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:
- δν1…νsμ1…μs=1(n−s)!εμ1…μsρs+1…ρnεν1…νsρs+1…ρn.{displaystyle delta _{nu _{1}dots nu _{s}}^{mu _{1}dots mu _{s}}={frac {1}{(n-s)!}}varepsilon ^{mu _{1}dots mu _{s},rho _{s+1}dots rho _{n}}varepsilon _{nu _{1}dots nu _{s},rho _{s+1}dots rho _{n}}.}
Integral representations
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.
- δx,n=12πi∮|z|=1zx−n−1dz=12π∫02πei(x−n)φdφ{displaystyle delta _{x,n}={frac {1}{2pi i}}oint _{|z|=1}z^{x-n-1},dz={frac {1}{2pi }}int _{0}^{2pi }e^{i(x-n)varphi },dvarphi }
The Kronecker comb
The Kronecker comb function with period N is defined (using DSP notation) as:
- ΔN[n]=∑k=−∞∞δ[n−kN],{displaystyle Delta _{N}[n]=sum _{k=-infty }^{infty }delta [n-kN],}
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.
Kronecker integral
The Kronecker delta is also called degree of mapping of one surface into another.[9] Suppose a mapping takes place from surface Suvw to Sxyz that are boundaries of regions, Ruvw and Rxyz which is simply connected with one-to-one correspondence. In this framework, if s and t are parameters for Suvw, and Suvw to Suvw are each oriented by the outer normal n:
- u=u(s,t),v=v(s,t),w=w(s,t),{displaystyle u=u(s,t),quad v=v(s,t),quad w=w(s,t),}
while the normal has the direction of
- (usi+vsj+wsk)×(uti+vtj+wtk).{displaystyle (u_{s}mathbf {i} +v_{s}mathbf {j} +w_{s}mathbf {k} )times (u_{t}mathbf {i} +v_{t}mathbf {j} +w_{t}mathbf {k} ).}
Let x = x(u,v,w), y = y(u,v,w), z = z(u,v,w) be defined and smooth in a domain containing Suvw, and let these equations define the mapping of Suvw onto Sxyz. Then the degree δ of mapping is 1/4π times the solid angle of the image S of Suvw with respect to the interior point of Sxyz, O. If O is the origin of the region, Rxyz, then the degree, δ is given by the integral:
- δ=14π∬Rst(x2+y2+z2)−32|xyz∂x∂s∂y∂s∂z∂s∂x∂t∂y∂t∂z∂t|dsdt.{displaystyle delta ={frac {1}{4pi }}iint _{R_{st}}left(x^{2}+y^{2}+z^{2}right)^{-{frac {3}{2}}}{begin{vmatrix}x&y&z\{frac {partial x}{partial s}}&{frac {partial y}{partial s}}&{frac {partial z}{partial s}}\{frac {partial x}{partial t}}&{frac {partial y}{partial t}}&{frac {partial z}{partial t}}end{vmatrix}},ds,dt.}
See also
- Dirac measure
- Indicator function
- Levi-Civita symbol
- Unit function
- XNOR gate
References
^ 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. doi:10.1175/1520-0426(1998)015<0290:OATFMO>2.0.CO;2..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
^ Spiegel, Eugene; O'Donnell, Christopher J. (1997), Incidence Algebras, Pure and Applied Mathematics, 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 δμ
ν = δμ
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.
^ Kaplan, Wilfred (2003). Advanced Calculus. Pearson Education. p. 364. ISBN 0-201-79937-5.