In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker, is a function of two variables, usually just positive integers. The function is 1 if the variables are equal, and 0 otherwise:
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.
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 = 0 and δ+1/2, −3/2 = 0 but δ−2,−2 = 1 and δ5/3, 5/3 = 1. This latter case is ultimately for convenience.
- 1 Properties
- 2 Alternative notation
- 3 Digital signal processing
- 4 Properties of the delta function
- 5 Relationship to the Dirac delta function
- 6 Generalizations of the Kronecker delta
- 7 Integral representations
- 8 The Kronecker comb
- 9 Kronecker integral
- 10 See also
- 11 References
The following equations are satisfied:
Therefore, δ can be considered as an identity matrix.
Using the Iverson bracket:
Often, the notation is used.
Digital signal processing
Properties of the delta function
The Kronecker delta has the so-called sifting property that for :
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, 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: . It is important to note that the Kronecker delta is not the result of directly sampling the Dirac delta function.
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 , with corresponding probabilities , then the probability mass function of the distribution over can be written, using the Kronecker delta, as
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 of the Kronecker delta
This (1,1) tensor represents:
- The identity mapping (or identity matrix), considered as a linear mapping or
- The trace or tensor contraction, considered as a mapping
- The map , representing scalar multiplication as a sum of outer products.
The generalized 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!, which results in the explicit scaling factors in § Properties of generalized Kronecker delta below disappearing.
Definitions of generalized Kronecker delta
Let be the symmetric group of degree p, then:
where 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:
Properties of generalized Kronecker delta
The generalized Kronecker delta may be used for anti-symmetrization:
From the above equations and the properties of anti-symmetric tensor, we can derive the properties of the generalized Kronecker delta:
Reducing the order via summation of the indices may be expressed by the identity
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:
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.
The Kronecker comb
The Kronecker comb function with period N is defined (using DSP notation) as:
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.
The Kronecker delta is also called degree of mapping of one surface into another. Suppose a mapping takes place from surface to that are boundaries of regions, and which is simply connected with one-to-one correspondence. In this framework, if s and t are parameters for , and to are each oriented by the outer normal n:
while the normal has the direction of:
Let x=x(u,v,w),y=y(u,v,w),z=z(u,v,w) be defined and smooth in a domain containing , and let these equations define the mapping of into . Then the degree of mapping is times the solid angle of the image S of with respect to the interior point of , O. If O is the origin of the region, , then the degree, is given by the integral:
- Trowbridge, 1998. Journal of Atmospheric and Oceanic Technology. V15, 1 p291
- Spiegel, Eugene; O'Donnell, Christopher J. (1997), Incidence algebras, Pure and Applied Mathematics 206, Marcel Dekker, ISBN 0-8247-0036-8.
- http://people.physics.tamu.edu/pope/geom-group.pdf Christopher Pope (2008), Geometry and Group Theory
- Theodore Frankel, The Geometry of Physics: An Introduction 3rd edition (2012), published by Cambridge University Press, ISBN 9781107602601
- D. C. Agarwal, Tensor Calculus and Riemannian Geometry 22nd edition (2007), published by Krishna Prakashan Media
- David Lovelock, Hanno Rund (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).
- Sadri Hassani (2008). Mathematical Methods: For Students of Physics and Related Fields 2nd edition. Springer-Verlag. ISBN 978-0387095035.
- Kaplan, Wilfred (2003), Advanced Calculus, Pearson Education. Inc, p. 364, ISBN 0-201-79937-5