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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, the Kronecker delta (named after
Leopold Kronecker Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

Leopold Kronecker
) is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
of two variables, usually just non-negative
integer An integer (from the Latin Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Re ...
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, where the Kronecker delta is a
piecewise In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
function of variables and . For example, , whereas . 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 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 (mat ...
, the
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

identity matrix
has entries equal to the Kronecker delta: : I_ = \delta_ where and take the values , and the
inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...
of
vector Vector may refer to: Biology *Vector (epidemiology) In epidemiology Epidemiology is the study and analysis of the distribution (who, when, and where), patterns and risk factor, determinants of health and disease conditions in defined pop ...
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, ..., 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 . The restriction to positive or non-negative integers is common, but in fact, 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_ = \frac \sum_^N e^ This can be derived using the formula for the finite geometric series.


Alternative notation

Using the
Iverson bracketIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
: : \delta_ = =j Often, a single-argument notation is used, which is equivalent to setting : :\delta_ = \begin 0, & \mbox i \ne 0 \\ 1, & \mbox 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 (mat ...
, it can be thought of as a
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

tensor
, and is written . 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 Digital data, in information theory and information systems, is information represented as a string of discrete symbols each of which can take on one of only a finite number of ...
(DSP), the unit sample function \delta /math> represents a special case of a 2-dimensional Kronecker delta function \delta_ where the kronecker indices include the number zero, and where one of the indices is zero. In this case: :\delta \equiv \delta_ \equiv \delta_~~~\text -\infty Or more generally where: :\delta -k\equiv \delta -n\equiv \delta_ \equiv \delta_\text -\infty However, this is only a very special case. In Tensor calculus, it is more common to number basis vectors in a particular dimension starting with index 1, rather than index 0. In this case, the relation \delta \equiv \delta_ \equiv \delta_ doesn't exist, and in fact, the Kronecker delta function and the unit sample function are really different functions that by chance overlap in one specific case where the indices include the number 0, the number of indices is 2, and one of the indices has the value of zero. While the discrete unit sample function and the Kronecker delta function use the same letter, they differ in the following ways. For the discrete unit sample function it is more conventional to place a single integer index in square braces, in contrast the Kronecker delta can have any number of indexes. Further, the purpose of the discrete unit sample function is different from the Kronecker delta function. In DSP, the discrete unit sample function is typically used as an input function to a discrete system for discovering the system function of the system which will be produced as an output of the system. In contrast, the typical purpose of the Kronecker delta function is for filtering terms from an
Einstein summation convention In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The discrete unit sample function is more simply defined as: :\delta = \begin 1 & n = 0 \\ 0 & n \text\end In addition, DSP has a function called the
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
, that is often confused for both the Kronecker delta function and the unit sample function. The Dirac Delta is defined as: :\delta(t) = \begin \infty & t = 0 \\ 0 & t \text\end Unlike the Kronecker delta function \delta_ and the unit sample function \delta /math>, the Dirac Delta function \delta(t) doesn't have a integer index, it has a single continuous non-integer value t. To confuse matters more, the unit impulse function is sometimes used to refer to either the
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
\delta(t), or the unit sample function \delta /math>.


Properties of the delta function

The Kronecker delta has the so-called ''sifting'' property that for : :\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 Measure is a fundamental concept of mathematics. Measures provide a mathematical abstraction for common notions like mass, distance/length, area, volume, probability of events, and — after si ...
, endowed with the
counting measure In mathematics, specifically measure theory, the counting measure is an intuitive way to put a Measure (mathematics), measure on any Set (mathematics), set – the "size" of a subset is taken to be the number of elements in the subset if the subset ...
, then this property coincides with the defining property of the
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
:\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, generally indicates continuous time (Dirac), whereas arguments like , , , , , and are usually reserved for discrete time (Kronecker). Another common practice is to represent discrete sequences with square brackets; thus: . The Kronecker delta is not the result of directly sampling the Dirac delta function. The Kronecker delta forms the multiplicative
identity element In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
of an
incidence algebraIn order theory Order theory is a branch of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...
.


Relationship to the Dirac delta function

In
probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...
and
statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sens ...

statistics
, the Kronecker delta and
Dirac delta function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
can both be used to represent a
discrete distribution In probability theory and statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is ...
. If the
support Support may refer to: Business and finance * Support (technical analysis) In stock market technical analysis, support and resistance are certain predetermined levels of the price of a security (finance), security at which it is thought that th ...
of a distribution consists of points , with corresponding probabilities , then the
probability mass function In probability Probability is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which th ...
of the distribution over can be written, using the Kronecker delta, as :p(x) = \sum_^n p_i \delta_. Equivalently, the
probability density function In probability theory Probability theory is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...
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 a theorem in the field of signal processing Signal processing is an electrical engineering Electrical engineering is an engineering discipline concerned with the study, design, and application of ...
, the resulting discrete-time signal will be a Kronecker delta function.


Generalizations

If it is considered as a type
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

tensor
, the Kronecker tensor can be written with a covariant index and contravariant index : :\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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
or * The
trace Trace may refer to: Arts and entertainment Music * ''Trace'' (Son Volt album), 1995 * ''Trace'' (Died Pretty album), 1993 * Trace (band) Trace was a Netherlands, Dutch progressive rock trio founded by Rick van der Linden in 1974 after leavin ...
or
tensor contraction In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the natural pairing of a finite-dimensional vector space and its dual vector space, dual. In components, it is expressed as a sum of products of scalar compo ...
, considered as a mapping * The map , representing scalar multiplication as a sum of
outer product 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 t ...
s. The or multi-index Kronecker delta of order is a type tensor that is completely antisymmetric in its upper indices, and also in its lower indices. Two definitions that differ by a factor of are in use. Below, the version is presented has nonzero components scaled to be . The second version has nonzero components that are , with consequent changes scaling factors in formulae, such as the scaling factors of in ' below disappearing.


Definitions of the generalized Kronecker delta

In terms of the indices, the generalized Kronecker delta is defined as: :\delta^_ = \begin +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 \\ \;\;0 & \quad \text.\end Let be the
symmetric group In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...
of degree , 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
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
: :\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 . S ...
( Laplace's formula) of determinant, it may be defined
recursively Recursion (adjective: ''recursive'') occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics ...

recursively
: :\begin \delta^_ &= \sum_^p (-1)^ \delta^_ \delta^_ \\ &= \delta^_ \delta^_ - \sum_^ \delta^_ \delta^_, \end where the caron, , indicates an index that is omitted from the sequence. When (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 represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the natural numbers , for som ...
: :\delta^_ = \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) \delta^_ . The generalization of the preceding formulas is \left(x^2+y^2+z^2\right)^ \begin x & y & z \\ \frac & \frac & \frac \\ \frac & \frac & \frac \end \, ds \, dt.


See also

*
Dirac measure Image:Hasse diagram of powerset of 3.svg, 250px, A diagram showing all possible subsets of a 3-point set . The Dirac measure assigns a size of 1 to all sets in the upper-left half of the diagram and 0 to all sets in the lower-right half. In mathem ...
*
Indicator function In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
*
Levi-Civita symbol In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol represents a collection of numbers; defined from the parity of a permutation, sign of a permutation of the natural numbers , for som ...
* 't Hooft symbol *
Unit functionIn number theory, the unit function is a completely multiplicative function :''Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative f ...
* XNOR gate


References

{{Tensors Mathematical notation Elementary special functions