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Kronecker delta

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	Kronecker delta
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	Topics Kronecker delta Topic Home In mathematicsMathematics Mathematics is the discipline that deals with concepts such as quantity, structure, space and change.... , the Kronecker delta or Kronecker's delta, named after Leopold KroneckerLeopold Kronecker Leopold Kronecker was a German mathematician and logician who argued that arithmetic and analysis must be founded on "whole ... , is a functionFunction (mathematics) In mathematics, a function relates each of its inputs to exactly one output.... of two variableVariable In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation.... s, usually integerInteger The integers consist of the positive natural numbers , their negatives and the number zero.... s, which is 1 if they are equal, and 0 otherwise. So, for example, , but . It is written as the symbol d_ij, and treated as a notational shorthand rather than as a function. Alternate notation Using the Iverson bracketIverson bracket In mathematics, the Iverson bracket, named after Kenneth E.... : Often, the notation is used. In linear algebraLinear algebra Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces , linear transformations, and... , it can be thought of as a tensorTensor In mathematics, a tensor is a generalized linear 'quantity' or 'geometrical entity' that can be expressed as a multi-dimen... , and is written . Digital signal processing Similarly, in digital signal processingDigital signal processing Digital signal processing is the study of signals in a digital representation and the processing methods of these signals.... , 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 responseImpulse response * Dirac delta function* Unit impulse function... 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 measureCounting measure In mathematics, the counting measure is an intuitive way to put a measure on any set: the "size" of a subset is taken to be ... , then this property coincides with the defining property of the Dirac delta functionDirac delta function The Dirac delta or Dirac's delta, often referred to as the unit impulse function and introduced by the British theoret... 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 sampling the Dirac delta function. The Kronecker delta is used in many areas of mathematics. Linear algebra In linear algebraFacts About Linear algebra Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces , linear transformations, and... , the identity matrixIdentity matrix In linear algebra, the identity matrix of size n is the n-by-n square matrix with ones on the main diagonal and ... can be written as . If it is considered as a tensorTensor Summary In mathematics, a tensor is a generalized linear 'quantity' or 'geometrical entity' that can be expressed as a multi-dimen... , the Kronecker tensor, it can be written with a covariant index j and contravariant index i. This (1,1) tensor represents: the identity matrix, considered as a linear mapping the traceTrace (linear algebra) In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main d... the inner product the map , representing scalar multiplication as a sum of outer productOuter product Outer product typically refers to the tensor product or to operations with similar cardinality such as exterior product.... s Extensions of the delta function In the same fashion, we may define an analogous, multi-dimensional function of many variables This function takes the value 1 if and only if all the upper indices match the corresponding lower ones, and the value zero otherwise. Integral representations For any integer n, using a standard residueResidue (complex analysis) In complex analysis, the residue is a complex number which describes the behavior of line integrals of a meromorphic functio... calculation we can write an integral representation for the Kronecker delta as where the contour of the integral goes counterclockwise around zero. This representation is also equivalent to by a rotation in the complex plane. See also Levi-Civita symbolLevi-Civita symbol The Levi-Civita symbol, also called the permutation symbol or antisymmetric symbol, is a mathematical symbol use... Dirac measureFacts About Dirac measure In mathematics, a Dirac measure is a measure on a set that gives the singleton set the measure 1, for a chosen element :... OR gateOR gate The OR gate is a digital logic gate that implements logical disjunction - it behaves according to the truth table to the rig...