In mathematics, the Kronecker delta or Kronecker's delta, named after Leopold Kronecker, is a
function of two variables, usually integers. The function is 1 if the variables are equal,
and 0 otherwise:
where the Kronecker delta δij is a piecewise function of variables and . For example, δ1
2 = 0, whereas δ3 3 = 1. In linear algebra, the identity matrix can
be written as
and the inner product of vectors can be written as
The Kronecker delta is used in many areas of mathematics, physics and engineering, primarily
as an expedient to convey in a single equation what might otherwise take several lines of
text.
Properties The following equations are satisfied:
Therefore, δij can be considered as an identity matrix.
Alternative notation Using the Iverson bracket:
Often, the notation is used.
In linear algebra, it can be thought of as a tensor, and is written . Sometimes the Kronecker
delta is called the substitution tensor. Digital signal processing
Similarly, in digital signal processing, the same concept is represented as a function
on :
The function is referred to as an impulse, or unit impulse. And when it stimulates a
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 :
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
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 that distinguishes the Kronecker
and Dirac "functions". And by convention, generally indicates continuous time, whereas
arguments like i, j, k, l, m, and n are usually reserved for discrete time. 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. The Kronecker delta forms the multiplicative
identity element of an incidence algebra. 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
Equivalently, the probability density function of the distribution can be written using the
Dirac delta function 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 per the Nyquist–Shannon sampling theorem, the resulting discrete-time
signal will be a Kronecker delta function. Generalizations of the Kronecker delta
If it is considered as a type tensor, the Kronecker tensor, it can be written with a
covariant index j and contravariant index i:
This tensor represents: The identity mapping, 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 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 In terms of the indices:
Let be the symmetric group of degree p, then:
Using anti-symmetrization:
In terms of an p × p determinant:
Using the Laplace expansion of determinant, it may be defined recursively:
where indicates an index that is omitted from the sequence.
When p = n, 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:
which are the generalized version of formulae written in the section 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
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:
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.
The Kronecker comb The Kronecker comb function with period N
is defined 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. Kronecker Integral
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:
See also Dirac measure
Indicator function Levi-Civita symbol
Unit function References
^ Trowbridge, 1998. Journal of Atmospheric and Oceanic Technology. V15, 1 p291
^ Spiegel, Eugene; O'Donnell, Christopher J., Incidence algebras, Pure and Applied Mathematics
206, Marcel Dekker, ISBN 0-8247-0036-8 . ^ http:people.physics.tamu.edugeom-group.pdf
Christopher Pope, Geometry and Group Theory ^ Theodore Frankel, The Geometry of Physics:
An Introduction 3rd edition, published by Cambridge University Press, ISBN 9781107602601
^ D. C. Agarwal, Tensor Calculus and Riemannian Geometry 22nd edition, published by Krishna
Prakashan Media ^ David Lovelock, Hanno Rund. 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 δμν = δμν for p = 1. ^ Sadri Hassani. Mathematical Methods: For
Students of Physics and Related Fields 2nd edition. Springer-Verlag. ISBN 978-0387095035.
^ Kaplan, Wilfred, Advanced Calculus, Pearson Education. Inc, p. 364, ISBN 0-201-79937-5