Kronecker's Delta
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Kronecker delta (named after
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, abstract algebra and logic, and criticized Georg Cantor's work on set theory. Heinrich Weber quoted Kronecker as having said, ...
) is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
of two
variables Variable may refer to: Computer science * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed Mathematics * Variable (mathematics), a symbol that represents a quantity in a mathemat ...
, usually just non-negative
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\text i=j. \end or with use of Iverson brackets: \delta_ = =j, For example, \delta_ = 0 because 1 \ne 2, whereas \delta_ = 1 because 3 = 3. The Kronecker delta appears naturally in many areas of mathematics, physics, engineering and computer science, as a means of compactly expressing its definition above. Generalized versions of the Kronecker delta have found applications in differential geometry and modern tensor calculus, particularly in formulations of gauge theory and topological field models. In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, the n\times n
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the obje ...
\mathbf has entries equal to the Kronecker delta: I_ = \delta_ where i and j take the values 1,2,\cdots,n, and the
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, ofte ...
of
vector Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...
s can be written as \mathbf\cdot\mathbf = \sum_^n a_\delta_b_ = \sum_^n a_ b_. Here the Euclidean vectors are defined as -tuples: \mathbf = (a_1, a_2, \dots, a_n) and \mathbf= (b_1, b_2, ..., b_n) and the last step is obtained by using the values of the Kronecker delta to reduce the summation over j. It is common for and to be restricted to a set of the form or , but the Kronecker delta can be defined on an arbitrary set.


Properties

The following equations are satisfied: \begin \sum_ \delta_ a_j &= a_i,\\ \sum_ a_i \delta_ &= a_j,\\ \sum_ \delta_\delta_ &= \delta_. \end Therefore, the matrix can be considered as an identity matrix. Another useful representation is the following form: \delta_ = \lim_\frac \sum_^N e^ This can be derived using the formula for the
geometric series In mathematics, a geometric series is a series (mathematics), series summing the terms of an infinite geometric sequence, in which the ratio of consecutive terms is constant. For example, 1/2 + 1/4 + 1/8 + 1/16 + ⋯, the series \tfrac12 + \tfrac1 ...
.


Alternative notation

Using the
Iverson bracket In mathematics, the Iverson bracket, named after Kenneth E. Iverson, is a notation that generalises the Kronecker delta, which is the Iverson bracket of the statement . It maps any statement to a function of the free variables in that statement. ...
: \delta_ = =j Often, a single-argument notation \delta_i is used, which is equivalent to setting j=0: \delta_ = \delta_ = \begin 0, & \text i \neq 0 \\ 1, & \text i = 0 \end In
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...
, it can be thought of as a
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, and is written \delta_j^i. Sometimes the Kronecker delta is called the substitution tensor.


Digital signal processing

In the study of
digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a ...
(DSP), the Kronecker delta function sometimes means the unit sample function \delta /math> , which represents a special case of the 2-dimensional Kronecker delta function \delta_ where the Kronecker indices include the number zero, and where one of the indices is zero: \delta \equiv \delta_ \equiv \delta_~~~\text -\infty Or more generally where: \delta -k\equiv \delta -n\equiv \delta_ \equiv \delta_\text -\infty For discrete-time signals, it is conventional to place a single integer index in square braces; in contrast the Kronecker delta, \delta_, can have any number of indexes. In
LTI system In system analysis, among other fields of study, a linear time-invariant (LTI) system is a system that produces an output signal from any input signal subject to the constraints of Linear system#Definition, linearity and Time-invariant system, ...
theory, the discrete unit sample function is typically used as an input to a discrete-time system for determining the
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse (). More generally, an impulse response is the reac ...
function of the system which characterizes the system for any general imput. In contrast, the typical purpose of the Kronecker delta function is for filtering terms from an
Einstein summation convention In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies s ...
. The discrete unit sample function is more simply defined as: \delta = \begin 1 & n = 0 \\ 0 & n \text\end In comparison, in continuous-time systems the
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
is often confused for both the Kronecker delta function and the unit sample function. The Dirac delta is defined as: \begin \int_^\delta(t)dt = 1 & \forall \varepsilon > 0 \\ \delta(t) = 0 & \forall t \neq 0\end Unlike the Kronecker delta function \delta_ and the unit sample function \delta /math>, the Dirac delta function \delta(t) does not have an integer index, it has a single continuous non-integer value . In continuous-time systems, the term "
unit impulse function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
" is used to refer to the
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
\delta(t) or, in discrete-time systems, the Kronecker delta function \delta /math>.


Notable properties

The Kronecker delta has the so-called ''sifting'' property that for j\in\mathbb: \sum_^\infty a_i \delta_ = a_j. and if the integers are viewed as a
measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that ...
, endowed with the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinit ...
, then this property coincides with the defining property of the
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
\int_^\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 /math>. The Kronecker delta is not the result of directly sampling the Dirac delta function. The Kronecker delta forms the multiplicative
identity element In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
of an
incidence algebra In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered set and commutative ring with unity. Subalgebra#Subalgebras_for_algebras_over_a_ring_or_field, Subalgebras c ...
.


Relationship to the Dirac delta function

In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
and
statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the Kronecker delta and
Dirac delta function In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
can both be used to represent a
discrete distribution In probability theory and statistics, a probability distribution is a function that gives the probabilities of occurrence of possible events for an experiment. It is a mathematical description of a random phenomenon in terms of its sample spac ...
. If the support of a distribution consists of points \mathbf = \, with corresponding probabilities p_1,\cdots,p_n, then the
probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
p(x) of the distribution over \mathbf can be written, using the Kronecker delta, as p(x) = \sum_^n p_i \delta_. Equivalently, the
probability density function In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), function whose value at any given sample (or point) in the sample space (the s ...
f(x) of the distribution can be written using the Dirac delta function as f(x) = \sum_^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 Nyquist–Shannon sampling theorem is an essential principle for digital signal processing linking the frequency range of a signal and the sample rate required to avoid a type of distortion called aliasing. The theorem states that the sample r ...
, the resulting discrete-time signal will be a Kronecker delta function.


Generalizations

If it is considered as a type (1,1)
tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
, the Kronecker tensor can be written \delta^i_j with a covariant index j and contravariant index i: \delta^_ = \begin 0 & (i \ne j), \\ 1 & (i = j). \end This tensor represents: * The identity mapping (or identity matrix), considered as a
linear mapping In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vec ...
V\to V or V^*\to V^* * The
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band), a Dutch progressive rock band * ''The Trace'' (album), by Nell Other uses in arts and entertainment * ...
or
tensor contraction In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by ...
, considered as a mapping V^* \otimes V\to K * The map K\to V^*\otimes V, representing
scalar multiplication In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector ...
as a sum of
outer product In linear algebra, the outer product of two coordinate vectors is the matrix whose entries are all products of an element in the first vector with an element in the second vector. If the two coordinate vectors have dimensions ''n'' and ''m'', the ...
s. The or multi-index Kronecker delta of order 2p is a type (p,p) tensor that is 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 \pm 1. The second version has nonzero components that are \pm 1/p!, with consequent changes scaling factors in formulae, such as the scaling factors of 1/p! in ' below disappearing.


Definitions of the generalized Kronecker delta

In terms of the indices, the generalized Kronecker delta is defined as: \delta^_ = \begin \phantom-1 & \quad \text \nu_1 \dots \nu_p \text \mu_1 \dots \mu_p \\ -1 & \quad \text \nu_1 \dots \nu_p \text \mu_1 \dots \mu_p \\ \phantom-0 & \quad \text. \end Let \mathrm_p be the
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
of degree p, then: \delta^_ = \sum_ \sgn(\sigma)\, \delta^_\cdots\delta^_ = \sum_ \sgn(\sigma)\, \delta^_\cdots\delta^_. Using anti-symmetrization: \delta^_ = p! \delta^_ \dots \delta^_ = p! \delta^_ \dots \delta^_. In terms of a p\times p
determinant In mathematics, the determinant is a Scalar (mathematics), scalar-valued function (mathematics), function of the entries of a square matrix. The determinant of a matrix is commonly denoted , , or . Its value characterizes some properties of the ...
: \delta^_ = \begin \delta^_ & \cdots & \delta^_ \\ \vdots & \ddots & \vdots \\ \delta^_ & \cdots & \delta^_ \end. Using the
Laplace expansion In linear algebra, the Laplace expansion, named after Pierre-Simon Laplace, also called cofactor expansion, is an expression of the determinant of an -matrix as a weighted sum of minors, which are the determinants of some - submatrices of . Spe ...
( Laplace's formula) of determinant, it may be defined recursively: \begin \delta^_ &= \sum_^p (-1)^ \delta^_ \delta^_ \\ &= \delta^_ \delta^_ - \sum_^ \delta^_ \delta^_, \end where the caron, \check, 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 In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers , for some ...
: \delta^_ = \varepsilon^\varepsilon_\,. More generally, for m=n-p, using the
Einstein summation convention In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies s ...
: \delta^_ = \tfrac \varepsilon^\varepsilon_\,.


Contractions of the generalized Kronecker delta

Kronecker Delta contractions depend on the dimension of the space. For example, \delta^_ \delta^_ = (d-1) \delta^_ , where is the dimension of the space. From this relation the full contracted delta is obtained as \delta^_ \delta^_ = 2d(d-1) . The generalization of the preceding formulas is \delta^_ \delta^_ = n! \frac \delta^_ .


Properties of the generalized Kronecker delta

The generalized Kronecker delta may be used for anti-symmetrization: \begin \frac \delta^_ a^ &= a^ , \\ \frac \delta^_ a_ &= a_ . \end From the above equations and the properties of anti-symmetric tensors, we can derive the properties of the generalized Kronecker delta: \begin \frac \delta^_ a^ &= a^ , \\ \frac \delta^_ a_ &= a_ , \\ \frac \delta^_ \delta^_ &= \delta^_ , \end which are the generalized version of formulae written in '. The last formula is equivalent to the
Cauchy–Binet formula In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so th ...
. Reducing the order via summation of the indices may be expressed by the identity \delta^_ = \frac \delta^_. 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: \delta^_ = \frac\varepsilon^\varepsilon_. The 4D version of the last relation appears in Penrose's spinor approach to general relativity that he later generalized, while he was developing Aitken's diagrams, to become part of the technique of
Penrose graphical notation In mathematics and physics, Penrose graphical notation or tensor diagram notation is a (usually handwritten) visual depiction of multilinear functions or tensors proposed by Roger Penrose in 1971. A diagram in the notation consists of several sh ...
.
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, Philosophy of science, philosopher of science and Nobel Prize in Physics, Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics i ...
, "Applications of negative dimensional tensors," in ''Combinatorial Mathematics and its Applications'', Academic Press (1971).
Also, this relation is extensively used in
S-duality In theoretical physics, S-duality (short for strong–weak duality, or Sen duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theore ...
theories, especially when written in the language of
differential forms In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
and Hodge duals.


Integral representations

For any integers j and k, the Kronecker delta can be written as a complex
contour integral In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...
using a standard residue calculation. The integral is taken over the
unit circle In mathematics, a unit circle is a circle of unit radius—that is, a radius of 1. Frequently, especially in trigonometry, the unit circle is the circle of radius 1 centered at the origin (0, 0) in the Cartesian coordinate system in the Eucli ...
in the
complex plane In mathematics, the complex plane is the plane (geometry), plane formed by the complex numbers, with a Cartesian coordinate system such that the horizontal -axis, called the real axis, is formed by the real numbers, and the vertical -axis, call ...
, oriented counterclockwise. An equivalent representation of the integral arises by parameterizing the contour by an angle around the origin. \delta_ = \frac1 \oint_ z^ \,dz=\frac1 \int_0^ e^ \,d\varphi


The Kronecker comb

The Kronecker comb function with period N is defined (using DSP notation) as: \Delta_N \sum_^\infty \delta -kN where N\ne 0, k and n are integers. The Kronecker comb thus consists of an infinite series of unit impulses that are units apart, aligned so one of the impulses occurs at zero. It may be considered to be the discrete analog of the
Dirac comb In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function, periodic Function (mathematics), function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given perio ...
.


See also

*
Dirac measure In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element ''x'' or not. It is one way of formalizing the idea of the Dirac delta function, an important tool in physics and other technical fields. ...
*
Indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator functio ...
*
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function named after Oliver Heaviside, the value of which is zero for negative arguments and one for positive arguments. Differen ...
*
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers defined from the sign of a permutation of the natural numbers , for some ...
*
Minkowski metric In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model ...
* 't Hooft symbol *
Unit function In number theory, the unit function is a completely multiplicative function on the positive integers defined as: :\varepsilon(n) = \begin 1, & \mboxn=1 \\ 0, & \mboxn \neq 1 \end It is called the unit function because it is the identity element f ...
*
XNOR gate The XNOR gate (sometimes ENOR, EXNOR, NXOR, XAND and pronounced as exclusive NOR) is a digital logic gate whose function is the logical complement of the exclusive OR ( XOR) gate. It is equivalent to the logical connective (\leftrightarrow) fr ...


References

{{Tensors Mathematical notation Elementary special functions