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Kronecker delta

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:

\delta_{ij} = \left\{\begin{matrix} 0, & \mbox{if } i \ne j \\ 1, & \mbox{if } i=j \end{matrix}\right.

where Kronecker delta δij is a piecewise function of variables i and j.

For example, δ1 2 = 0, whereas δ3 3 = 1.


Alternate notation

Using the Iverson bracket:

\delta_{ij} = [i=j ].\,

Often, the notation \delta_i is used.

\delta_{i} = \begin{cases} 0, & \mbox{if } i \ne 0 \\ 1, & \mbox{if } i=0 \end{cases}

In linear algebra, it can be thought of as a tensor, and is written \delta^i_j. Sometimes the Kronecker delta is called the substitution tensor.[1]

Digital signal processing

An impulse function
An impulse function
Similarly, in digital signal processing, the same concept is represented as a function on \mathbb{Z} (the integers):

\delta[n] = \begin{cases} 0, & n \ne 0 \\ 1, & n = 0.\end{cases}

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 j\in\mathbb Z:

\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, \delta(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:  \delta[n]\,. It is important to note that the Kronecker delta is not the result of directly sampling the Dirac delta function.

The Kronecker delta is used in many areas of mathematics.

Linear algebra

In linear algebra, the identity matrix can be written as (\delta_{ij})_{i,j=1}^n\,.

If it is considered as a tensor, the Kronecker tensor, it can be written \delta^i_j with a covariant index j and contravariant index i.

This (1,1) tensor represents:

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 \mathbf{x} = \{x_1,\dots,x_n\}, with corresponding probabilities p_1,\dots,p_n\,, then the probability mass function p(x)\, of the distribution over \mathbf{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 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 Nyquist Shannon sampling theorem, the resulting discrete-time signal will be a Kronecker delta function.

Generalizations of the Kronecker delta

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. This characterization defines it up to a scalar multiplier.

In terms of the indices:[2][3]

\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}

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}

In terms of an determinant:[4]

\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}

Equivalently, it could be defined by induction

\delta^{\mu \rho}_{\nu \sigma} = \delta^{\mu}_{\nu} \delta^{\rho}_{\sigma} - \delta^{\mu}_{\sigma} \delta^{\rho}_{\nu} \,
\delta^{\mu \rho_1 \rho_2}_{\nu \sigma_1 \sigma_2} = \delta^{\mu}_{\nu} \delta^{\rho_1 \rho_2}_{\sigma_1 \sigma_2} - \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2}_{\nu \sigma_2} + \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2}_{\sigma_2 \nu} \,
\delta^{\mu \rho_1 \rho_2 \rho_3}_{\nu \sigma_1 \sigma_2 \sigma_3} = \delta^{\mu}_{\nu} \delta^{\rho_1 \rho_2 \rho_3}_{\sigma_1 \sigma_2 \sigma_3} - \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2 \rho_3}_{\nu \sigma_2 \sigma_3} + \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2 \rho_3}_{\sigma_2 \nu \sigma_3} - \delta^{\mu}_{\sigma_1} \delta^{\rho_1 \rho_2 \rho_3}_{\sigma_2 \sigma_3 \nu} \,


In the particular case where (the dimension of the vector space), in terms of the Levi-Civita symbol:

\delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n} = \varepsilon^{\mu_1 \dots \mu_n}\varepsilon_{\nu_1 \dots \nu_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.

\delta_{x,n} = \frac1{2\pi i} \oint_{|z|=1} z^{x-n-1} dz=\frac1{2\pi} \int_0^{2\pi} e^{i(x-n)\varphi} d\varphi

The Kronecker comb

The Kronecker comb function with period N is defined (using digital notation) as:

\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[5]

The Kronecker delta is also called degree of mapping of one surface into another. Suppose a mapping takes place from surface S_{uvw} to S_{xyz} that are boundaries of regions, R_{uvw} and R_{xyz} which is simply connected with one-to-one correspondence. In this framework, if s and t are parameters for S_{uvw} , and S_{uvw} to S_{xyz} are each oriented by the outer normal n:

u=u(s,t), v=v(s,t),w=w(s,t),

while the normal has the direction of:

(u_{s} i +v_{s} j + w_{s} k) \times (u_{t}i +v_{t}j +w_{t}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 S_{uvw}, and let these equations define the mapping of S_{uvw} into S_{xyz}. Then the degree \delta of mapping is 1/4\pi times the solid angle of the image S of S_{uvw} with respect to the interior point of S_{xyz}, O. If O is the origin of the region, R_{xyz}, then the degree, \delta is given by the integral:

\delta = \frac{1}{4\pi}\iint_{R_{st}}\det\begin{bmatrix}x & y & z \\ \dfrac{\partial x}{\partial s} & \dfrac{\partial y}{\partial s} & \dfrac{\partial z}{\partial s} \\ \dfrac{\partial x}{\partial t} & \dfrac{\partial y}{\partial t} & \dfrac {\partial z}{\partial t} \end{bmatrix} \frac{1}{(x^{2}+y^{2}+z^{2})^{\frac{3}{2}}}dsdt.

See also


  1. Trowbridge, 1998. Journal of Atmospheric and Oceanic Technology. V15, 1 p291
  2. Theodore Frankel, The Geometry of Physics: An Introduction 3rd edition (2012), published by Cambridge University Press, ISBN 9781107602601
  3. D. C. Agarwal, Tensor Calculus and Riemannian Geometry 22nd edition (2007), published by Krishna Prakashan Media
  4. David Lovelock, Hanno Rund, Tensors, Differential Forms, and Variational Principles, Dover Publications

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