Kronecker delta

 ( Eponymous Functions ) 

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative . The function is 1 if the variables are equal, and 0 otherwise: or with use of Iverson bracket: $$\backslash delta\_\{ij\}\; =\; i=j\backslash ,$$ For example, $\backslash delta\_\{12\}\; =\; 0$ because $1\; \backslash ne\; 2$, whereas $\backslash delta\_\{33\}\; =\; 1$ 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, the $n\backslash times\; n$ identity matrix $\backslash mathbf\{I\}$ has entries equal to the Kronecker delta: $$I\_\{ij\}\; =\; \backslash delta\_\{ij\}$$ where $i$ and $j$ take the values $1,2,\backslash cdots,n$, and the inner product of Euclidean vector can be written as $$\backslash mathbf\{a\}\backslash cdot\backslash mathbf\{b\}\; =\; \backslash sum\_\{i,j=1\}^n\; a\_\{i\}\backslash delta\_\{ij\}b\_\{j\}\; =\; \backslash sum\_\{i=1\}^n\; a\_\{i\}\; b\_\{i\}.$$ Here the Euclidean vector are defined as -tuples: $\backslash mathbf\{a\}\; =\; (a\_1,\; a\_2,\; \backslash dots,\; a\_n)$ and $\backslash mathbf\{b\}=\; (b\_1,\; b\_2,\; ...,\; b\_n)$ 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.

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Properties The following equations are satisfied: Therefore, the matrix can be considered as an identity matrix.

Another useful representation is the following form: $$\backslash delta\_\{nm\}\; =\; \backslash lim\_\{N\backslash to\backslash infty\}\backslash frac\{1\}\{N\}\; \backslash sum\_\{k\; =\; 1\}^N\; e^\{2\; \backslash pi\; i\; \backslash frac\{k\}\{N\}(n-m)\}$$ This can be derived using the formula for the geometric series.

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Alternative notation Using the Iverson bracket: $$\backslash delta\_\{ij\}\; =\; i=j.$$

Often, a single-argument notation $\backslash delta\_i$ is used, which is equivalent to setting $j=0$:

In linear algebra, it can be thought of as a tensor, and is written $\backslash delta\_j^i$. Sometimes the Kronecker delta is called the substitution tensor.

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Digital signal processing In the study of digital signal processing (DSP), the Kronecker delta function sometimes means the unit sample function $\backslash deltan$ , which represents a special case of the 2-dimensional Kronecker delta function $\backslash delta\_\{ij\}$ where the Kronecker indices include the number zero, and where one of the indices is zero:

Or more generally where:

For discrete-time signals, it is conventional to place a single integer index in square braces; in contrast the Kronecker delta, $\backslash delta\_\{ij\}$, can have any number of indexes. In LTI system theory, the discrete unit sample function is typically used as an input to a discrete-time system for determining the impulse response 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.

The discrete unit sample function is more simply defined as:

In comparison, in continuous-time systems the Dirac delta function is often confused for both the Kronecker delta function and the unit sample function. The Dirac delta is defined as:

Unlike the Kronecker delta function $\backslash delta\_\{ij\}$ and the unit sample function $\backslash deltan$, the Dirac delta function $\backslash delta(t)$ does not have an integer index, it has a single continuous non-integer value .

In continuous-time systems, the term "unit impulse function" is used to refer to the Dirac delta function $\backslash delta(t)$ or, in discrete-time systems, the Kronecker delta function $\backslash deltan$.

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Notable properties The Kronecker delta has the so-called sifting property that for $j\backslash in\backslash mathbb\{Z\}$: $$\backslash sum\_\{i=-\backslash infty\}^\backslash infty\; a\_i\; \backslash delta\_\{ij\}\; =\; a\_j.$$ 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 $$\backslash int\_\{-\backslash infty\}^\backslash infty\; \backslash delta(x-y)f(x)\backslash ,\; dx=f(y),$$ and in fact Dirac's delta was named after the Kronecker delta because of this analogous property.

(2025). 9780198520115, Oxford University Press. ISBN 9780198520115

In signal processing it is usually the context (discrete or continuous time) that distinguishes the Kronecker and Dirac "functions". And by convention, $\backslash delta(t)$ 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: $\backslash deltan$. 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..

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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 $\backslash mathbf\{x\}\; =\; \backslash \{x\_1,\backslash cdots,x\_n\backslash \}$, with corresponding probabilities $p\_1,\backslash cdots,p\_n$, then the probability mass function $p(x)$ of the distribution over $\backslash mathbf\{x\}$ can be written, using the Kronecker delta, as $$p(x)\; =\; \backslash sum\_\{i=1\}^n\; p\_i\; \backslash delta\_\{x\; x\_i\}.$$

Equivalently, the probability density function $f(x)$ of the distribution can be written using the Dirac delta function as $$f(x)\; =\; \backslash sum\_\{i=1\}^n\; p\_i\; \backslash delta(x-x\_i).$$

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.

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Generalizations If it is considered as a type $(1,1)$ tensor, the Kronecker tensor can be written $\backslash delta^i\_j$ with a covariant index $j$ and contravariant index $i$:

This tensor represents:

• The identity mapping (or identity matrix), considered as a linear mapping $V\backslash to\; V$ or $V^*\backslash to\; V^*$
• The trace or tensor contraction, considered as a mapping $V^*\; \backslash otimes\; V\backslash to\; K$
• The map $K\backslash to\; V^*\backslash otimes\; V$, representing scalar multiplication as a sum of .

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 $\backslash pm\; 1$. The second version has nonzero components that are $\backslash pm\; 1/p!$, with consequent changes scaling factors in formulae, such as the scaling factors of $1/p!$ in below disappearing.

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Definitions of the generalized Kronecker delta In terms of the indices, the generalized Kronecker delta is defined as:

Theodore Frankel (2025). 9781107602601, Cambridge University Press. ISBN 9781107602601

Let $\backslash mathrm\{S\}\_p$ be the symmetric group of degree $p$, then: $$\backslash delta^\{\backslash mu\_1\; \backslash dots\; \backslash mu\_p\}\_\{\backslash nu\_1\; \backslash dots\; \backslash nu\_p\}\; =\; \backslash sum\_\{\backslash sigma\; \backslash in\; \backslash mathrm\{S\}\_p\}\; \backslash sgn(\backslash sigma)\backslash ,\; \backslash delta^\{\backslash mu\_\{\backslash sigma(1)\}\}\_\{\backslash nu\_1\}\backslash cdots\backslash delta^\{\backslash mu\_\{\backslash sigma(p)\}\}\_\{\backslash nu\_p\}.$$

Using anti-symmetrization: $$\backslash delta^\{\backslash mu\_1\; \backslash dots\; \backslash mu\_p\}\_\{\backslash nu\_1\; \backslash dots\; \backslash nu\_p\}\; =\; p!\; \backslash delta^\{\backslash mu\_1\}\_\{\}\; =\; p!\; \backslash delta^\{\}\_\{\backslash nu\_p\}.$$

In terms of a $p\backslash times\; p$ determinant:

Using the Laplace expansion (Laplace's formula) of determinant, it may be defined Recursion:A recursive definition requires a first case, which may be taken as for , or alternatively for (generalized delta in terms of standard delta). $$\backslash begin\{align\}$$

 \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
   &= \sum_{k=1}^p (-1)^{p+k} \delta^{\mu_p}_{\nu_k} \delta^{\mu_1 \dots \mu_{k} \dots \check\mu_p}_{\nu_1 \dots \check\nu_k \dots \nu_p} \\
   &= \delta^{\mu_p}_{\nu_p} \delta^{\mu_1 \dots \mu_{p - 1}}_{\nu_1 \dots \nu_{p-1}} - \sum_{k=1}^{p-1} \delta^{\mu_p}_{\nu_k} \delta^{\mu_1 \dots \mu_{k-1}\, \mu_k\, \mu_{k+1} \dots \mu_{p-1}}_{\nu_1 \dots \nu_{k-1}\, \nu_p\, \nu_{k+1} \dots \nu_{p-1}},
     

\end{align} where the caron, $\backslash check\{\}$, 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: $$\backslash delta^\{\backslash mu\_1\; \backslash dots\; \backslash mu\_n\}\_\{\backslash nu\_1\; \backslash dots\; \backslash nu\_n\}\; =\; \backslash varepsilon^\{\backslash mu\_1\; \backslash dots\; \backslash mu\_n\}\backslash varepsilon\_\{\backslash nu\_1\; \backslash dots\; \backslash nu\_n\}\backslash ,.$$ More generally, for $m=n-p$, using the Einstein summation convention: $$\backslash delta^\{\backslash mu\_1\; \backslash dots\; \backslash mu\_p\}\_\{\backslash nu\_1\; \backslash dots\; \backslash nu\_p\}\; =\; \backslash tfrac\{1\}\{m!\}\; \backslash varepsilon^\{\backslash kappa\_1\; \backslash dots\; \backslash kappa\_m\; \backslash mu\_1\; \backslash dots\; \backslash mu\_p\}\backslash varepsilon\_\{\backslash kappa\_1\; \backslash dots\; \backslash kappa\_m\; \backslash nu\_1\; \backslash dots\; \backslash nu\_p\}\backslash ,.$$

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Contractions of the generalized Kronecker delta Kronecker Delta contractions depend on the dimension of the space. For example, $$\backslash delta^\{\backslash nu\_1\}\_\{\backslash mu\_1\}\; \backslash delta^\{\backslash mu\_1\; \backslash mu\_2\}\_\{\backslash nu\_1\; \backslash nu\_2\}\; =\; (d-1)\; \backslash delta^\{\backslash mu\_2\}\_\{\backslash nu\_2\}\; ,$$ where is the dimension of the space. From this relation the full contracted delta is obtained as $$\backslash delta^\{\backslash nu\_1\; \backslash nu\_2\}\_\{\backslash mu\_1\; \backslash mu\_2\}\; \backslash delta^\{\backslash mu\_1\; \backslash mu\_2\}\_\{\backslash nu\_1\; \backslash nu\_2\}\; =\; 2d(d-1)\; .$$ The generalization of the preceding formulas is $$\backslash delta^\{\backslash nu\_1\; \backslash dots\; \backslash nu\_n\}\_\{\backslash mu\_1\; \backslash dots\; \backslash mu\_n\}\; \backslash delta^\{\backslash mu\_1\; \backslash dots\; \backslash mu\_p\}\_\{\backslash nu\_1\; \backslash dots\; \backslash nu\_p\}\; =\; n!\; \backslash frac\{(d-p+n)!\}\{(d-p)!\}\; \backslash delta^\{\backslash mu\_\{n+1\}\; \backslash dots\; \backslash mu\_p\}\_\{\backslash nu\_\{n+1\}\; \backslash dots\; \backslash nu\_p\}\; .$$

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Properties of the generalized Kronecker delta The generalized Kronecker delta may be used for anti-symmetrization: $$\backslash begin\{align\}$$

\frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a^{\nu_1 \dots \nu_p} &= a^{[ \mu_1 \dots \mu_p ]} , \\
\frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a_{\mu_1 \dots \mu_p} &= a_{[ \nu_1 \dots \nu_p ]} . \end{align}
     

From the above equations and the properties of anti-symmetric tensors, we can derive the properties of the generalized Kronecker delta: $$\backslash begin\{align\}$$

\frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a^{[ \nu_1 \dots \nu_p ]} &= a^{[ \mu_1 \dots \mu_p ]} , \\
\frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a_{[ \mu_1 \dots \mu_p ]} &= a_{[ \nu_1 \dots \nu_p ]} , \\
\frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} \delta^{\nu_1 \dots \nu_p}_{\kappa_1 \dots \kappa_p} &= \delta^{\mu_1 \dots \mu_p}_{\kappa_1 \dots \kappa_p} ,
     

\end{align} which are the generalized version of formulae written in . The last formula is equivalent to the Cauchy–Binet formula.

Reducing the order via summation of the indices may be expressed by the identity

$$\backslash delta^\{\backslash mu\_1\; \backslash dots\; \backslash mu\_s\; \backslash ,\; \backslash mu\_\{s+1\}\; \backslash dots\; \backslash mu\_p\}\_\{\backslash nu\_1\; \backslash dots\; \backslash nu\_s\; \backslash ,\; \backslash mu\_\{s+1\}\; \backslash dots\; \backslash mu\_p\}\; =\; \backslash frac\{(n-s)!\}\{(n-p)!\}\; \backslash delta^\{\backslash mu\_1\; \backslash dots\; \backslash mu\_s\}\_\{\backslash nu\_1\; \backslash dots\; \backslash nu\_s\}.$$

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: $$\backslash delta^\{\backslash mu\_1\; \backslash dots\; \backslash mu\_p\}\_\{\backslash nu\_1\; \backslash dots\; \backslash nu\_p\}\; =\; \backslash frac\{1\}\{(n-p)!\}\backslash varepsilon^\{\backslash mu\_1\; \backslash dots\; \backslash mu\_p\; \backslash ,\; \backslash kappa\_\{p+1\}\; \backslash dots\; \backslash kappa\_n\}\backslash varepsilon\_\{\backslash nu\_1\; \backslash dots\; \backslash nu\_p\; \backslash ,\; \backslash kappa\_\{p+1\}\; \backslash dots\; \backslash kappa\_n\}.$$ 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.Roger Penrose, "Applications of negative dimensional tensors," in Combinatorial Mathematics and its Applications, Academic Press (1971). Also, this relation is extensively used in S-duality theories, especially when written in the language of differential forms and Hodge duals.

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Integral representations For any integers $j$ and $k$, the Kronecker delta can be written as a complex contour integral using a standard residue calculation. The integral is taken over the unit circle in the complex plane, oriented counterclockwise. An equivalent representation of the integral arises by parameterizing the contour by an angle around the origin. $$\backslash delta\_\{jk\}\; =\; \backslash frac1\{2\backslash pi\; i\}\; \backslash oint\_\{|z|=1\}\; z^\{j-k-1\}\; \backslash ,dz=\backslash frac1\{2\backslash pi\}\; \backslash int\_0^\{2\backslash pi\}\; e^\{i(j-k)\backslash varphi\}\; \backslash ,d\backslash varphi$$

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Kronecker comb The Kronecker comb function with period $N$ is defined (using DSP notation) as: $$\backslash Delta\_Nn=\backslash sum\_\{k=-\backslash infty\}^\backslash infty\; \backslash deltan-kN,$$ where $N\backslash ne\; 0$, $k$ 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.

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See also

• Dirac measure
• Indicator function
• Heaviside step function
• Levi-Civita symbol
• Minkowski metric
• 't Hooft symbol
• Unit function
• XNOR gate

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