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Kronecker delta
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{{distinguish|text=the Dirac delta function, nor with the Kronecker symbol}}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:
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delta_{ij} = begin{cases}
1 &text{if } i=j. end{cases}where the Kronecker delta {{mvar|Î´ij}} is a piecewise function of variables {{mvar|i}} and {{mvar|j}}. For example, {{math|Î´1 2 {{=}} 0}}, whereas {{math|Î´3 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 {{math|n Ã— n}} identity matrix {{math|I}} has entries equal to the Kronecker delta:
I_{ij} = delta_{ij}
where {{mvar|i}} and {{mvar|j}} take the values {{math|1, 2, ..., n}}, and the inner product of vectors can be written as
mathbf{a}cdotmathbf{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 {{mvar|i}} and {{mvar|j}} above take rational values, then for example
begin{align} delta_{(-1)(-3)}&=0 &qquad delta_{(-2)(-2)}&=1 delta_{left(frac12right)left(-frac32right)}&=0 &qquad delta_{left(frac53right)left(frac53right)}&=1. end{align}
This latter case is for convenience.Properties
The following equations are satisfied:
begin{align}
sum_{j} delta_{ij} a_j &= a_i,sum_{i} a_idelta_{ij} &= a_j,sum_{k} delta_{ik}delta_{kj} &= delta_{ij}.end{align}Therefore, the matrix {{math|Î´}} can be considered as an identity matrix.Another useful representation is the following form:
delta_{nm} = frac{1}{N} sum_{k = 1}^N e^{2 pi i frac{k}{N}(n-m)}
This can be derived using the formula for the finite geometric series.Alternative notation
Using the Iverson bracket:
delta_{ij} = [i=j ].
Often, a single-argument notation {{mvar|Î´i}} is used, which is equivalent to setting {{math|j {{=}} 0}}:
delta_{i} = begin{cases}
1, & mbox{if } i=0 end{cases}In linear algebra, it can be thought of as a tensor, and is written {{mvar|Î´{{su|p=i|b=j}}}}. Sometimes the Kronecker delta is called the substitution tensor.JOURNAL, 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, 10.1175/1520-0426(1998)0152.0.CO;2, Digital signal processing
missing image!
- unit impulse.gif -
An impulse function
Similarly, in digital signal processing, the same concept is represented as a sequence or discrete function on {{math|â„¤}} (the integers):
- unit impulse.gif -
An impulse function
delta[n] = begin{cases} 0, & n ne 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 {{math|j âˆˆ â„¤}}:
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
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, {{math|Î´(t)}} generally indicates continuous time (Dirac), whereas arguments like {{mvar|i}}, {{mvar|j}}, {{mvar|k}}, {{mvar|l}}, {{mvar|m}}, and {{mvar|n}} are usually reserved for discrete time (Kronecker). Another common practice is to represent discrete sequences with square brackets; thus: {{math|Î´[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.{{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 }}.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 {{math|x {{=}} {x1, ..., xn}|}}, with corresponding probabilities {{math|p1, ..., pn}}, then the probability mass function {{math|p(x)}} of the distribution over {{math|x}} can be written, using the Kronecker delta, as
p(x) = sum_{i=1}^n p_i delta_{x x_i}.
Equivalently, the probability density function {{math|f(x)}} of the distribution can be written using the Dirac delta function as
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 Nyquistâ€“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{{mvar|Î´{{su|p=i|b=j}}}} with a covariant index {{mvar|j}} and contravariant index {{mvar|i}}:
delta^{i}_{j} = begin{cases} 0 & (i ne j), 1 & (i = j). end{cases}
This (1,1) tensor represents: - The identity mapping (or identity matrix), considered as a linear mapping {{math|V â†’ V}} or {{math|V{{sup|âˆ—}} â†’ V{{sup|âˆ—}}}}
- The trace or tensor contraction, considered as a mapping {{math|V{{sup|âˆ—}} âŠ— V â†’ K}}
- The map {{math|K â†’ V{{sup|âˆ—}} âŠ— V}}, representing scalar multiplication as a sum of outer products.
Definitions of generalized Kronecker delta
In terms of the indices:BOOK, Theodore, Frankel, The Geometry of Physics: An Introduction, 3rd, 2012, Cambridge University Press, 9781107602601, BOOK, D. C., Agarwal, Tensor Calculus and Riemannian Geometry, 22nd, 2007, Krishna Prakashan Media, {{ISBN missing}}
delta^{mu_1 dots mu_p }_{nu_1 dots nu_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}
delta^{mu_1 dots mu_p}_{nu_1 dots nu_p}
sum_{sigma in mathcal{S}_p} sgn(sigma), delta^{mu_1}_{nu_{sigma(1)}}cdotsdelta^{mu_p}_{nu_{sigma(p)}}
sum_{sigma in mathcal{S}_p} sgn(sigma), delta^{mu_{sigma(1)}}_{nu_1}cdotsdelta^{mu_{sigma(p)}}_{nu_p}.
Using anti-symmetrization:
delta^{mu_1 dots mu_p}_{nu_1 dots nu_p}
p! delta^{mu_1}_{lbrack nu_1} dots delta^{mu_p}_{nu_p rbrack}
p! delta^{lbrack mu_1}_{nu_1} dots delta^{mu_p rbrack}_{nu_p}.
In terms of a {{math|p Ã— p}} determinant:BOOK, David, Lovelock, Hanno, Rund, Tensors, Differential Forms, and Variational Principles, Courier Dover Publications, 1989, 0-486-65840-6,
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}.Using the Laplace expansion (Laplace's formula) of determinant, it may be defined recursively:A recursive definition requires a first case, which may be taken as {{math|1=Î´ = 1}} for {{math|1=p = 0}}, or alternatively {{math|1=Î´{{su|p=Î¼|b=Î½}} = Î´{{su|p=Î¼|b=Î½}}}} for {{math|1=p = 1}} (generalized delta in terms of standard delta).
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 checkmu_p}_{nu_1 dots checknu_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}where the caron, {{math|Ë‡}}, indicates an index that is omitted from the sequence.When {{math|1=p = n}} (the dimension of the vector space), in terms of the Levi-Civita symbol:
&= sum_{k=1}^p (-1)^{p+k} delta^{mu_p}_{nu_k} delta^{mu_1 dots mu_{k} dots checkmu_p}_{nu_1 dots checknu_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}},
delta^{mu_1 dots mu_n}_{nu_1 dots nu_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:
begin{align}
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} ,
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} . end{align}
From the above equations and the properties of anti-symmetric tensors, we can derive the properties of the generalized Kronecker delta:
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} . end{align}
begin{align}
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} ,
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} ,
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} &= delta^{mu_1 dots mu_p}_{rho_1 dots rho_p} ,
end{align}which are the generalized version of formulae written in the section Properties. The last formula is equivalent to the Cauchyâ€“Binet formula.Reducing the order via summation of the indices may be expressed by the identityBOOK, Sadri, Hassani, Mathematical Methods: For Students of Physics and Related Fields, 2nd, Springer-Verlag, 2008, 978-0-387-09503-5, 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} ,
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} &= delta^{mu_1 dots mu_p}_{rho_1 dots rho_p} ,
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}.
Using both the summation rule for the case {{math|1=p = n}} and the relation with the Levi-Civita symbol,the summation rule of the Levi-Civita symbol is derived:
delta^{mu_1 dots mu_s}_{nu_1 dots nu_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 {{mvar|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.
delta_{x,n} = frac1{2pi i} oint_{|z|=1} z^{x-n-1} ,dz=frac1{2pi} int_0^{2pi} e^{i(x-n)varphi} ,dvarphi
The Kronecker comb
The Kronecker comb function with period {{mvar|N}} is defined (using DSP notation) as:
Delta_N[n]=sum_{k=-infty}^infty delta[n-kN],
where {{mvar|N}} and {{mvar|n}} are integers. The Kronecker comb thus consists of an infinite series of unit impulses {{mvar|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.BOOK, Advanced Calculus, Wilfred, Kaplan, Pearson Education, 2003, 0-201-79937-5, 364,weblink Suppose a mapping takes place from surface {{mvar|Suvw}} to {{mvar|Sxyz}} that are boundaries of regions, {{mvar|Ruvw}} and {{mvar|Rxyz}} which is simply connected with one-to-one correspondence. In this framework, if {{mvar|s}} and {{mvar|t}} are parameters for {{mvar|Suvw}}, and {{mvar|Suvw}} to {{mvar|Suvw}} are each oriented by the outer normal {{math|n}}:
u=u(s,t), quad v=v(s,t), quad w=w(s,t),
while the normal has the direction of
(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 {{math|x {{=}} x(u,v,w)}}, {{math|y {{=}} y(u,v,w)}}, {{math|z {{=}} z(u,v,w)}} be defined and smooth in a domain containing {{mvar|Suvw}}, and let these equations define the mapping of {{mvar|Suvw}} onto {{mvar|Sxyz}}. Then the degree {{mvar|Î´}} of mapping is {{math|{{sfrac|1|4Ï€}}}} times the solid angle of the image {{mvar|S}} of {{mvar|Suvw}} with respect to the interior point of {{mvar|Sxyz}}, {{math|O}}. If {{math|O}} is the origin of the region, {{mvar|Rxyz}}, then the degree, {{mvar|Î´}} is given by the integral:
delta=frac{1}{4pi}iint_{R_{st}}left(x^2+y^2+z^2right)^{-frac32}
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
References
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