Kronecker delta

Not to be confused with the Dirac delta function, nor with the Kronecker symbol.

In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: $${\delta }_{ij}=\{\begin{array}{ll}0& \text{if }i\ne j,\\ 1& \text{if }i=j.\end{array}$$ or with use of Iverson brackets: $${\delta }_{ij}=[i=j]$$ For example, ${\delta }_{12}=0$ because $1\ne 2$, whereas ${\delta }_{33}=1$ 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. In linear algebra, the $n\times n$ identity matrix $\mathbf{I}$ has entries equal to the Kronecker delta: $$I_{ij}={\delta }_{ij}$$ where $i$ and $j$ take the values $1,2,\cdots ,n$, and the inner product of vectors can be written as $$\mathbf{a}\cdot \mathbf{b}=\sum _{i,j=1}^n{a}_i{\delta }_{ij}{b}_j=\sum _{i=1}^n{a}_i{b}_i.$$ Here the Euclidean vectors are defined as n-tuples: $\mathbf{a}=(a_1,a_2,\dots ,a_n)$ and $\mathbf{b}=(b_1,b_2,...,b_n)$ and the last step is obtained by using the values of the Kronecker delta to reduce the summation over $j$. It is common for i and j to be restricted to a set of the form {1, 2, ..., n} or {0, 1, ..., n − 1}, but the Kronecker delta can be defined on an arbitrary set.

Properties

The following equations are satisfied: $$\begin{array}{rl}\sum _j{\delta }_{ij}{a}_j& =a_i,\\ \sum _i{a}_i{\delta }_{ij}& =a_j,\\ \sum _k{\delta }_{ik}{\delta }_{kj}& ={\delta }_{ij}.\end{array}$$ Therefore, the matrix δ can be considered as an identity matrix. Another useful representation is the following form: $${\delta }_{nm}=\underset{N\to \mathrm{\infty }}{\lim }\frac{1}{N}\sum _{k=1}^N{e}^{2\pi i\frac{k}{N}(n-m)}$$ This can be derived using the formula for the geometric series.

Alternative notation

Using the Iverson bracket: $${\delta }_{ij}=[i=j].$$ Often, a single-argument notation ${\delta }_i$ is used, which is equivalent to setting $j=0$: $${\delta }_i={\delta }_{i0}=\{\begin{array}{ll}0,& \text{if }i\ne 0\\ 1,& \text{if }i=0\end{array}$$ In linear algebra, it can be thought of as a tensor, and is written ${\delta }_j^i$. Sometimes the Kronecker delta is called the substitution tensor.

Digital signal processing

File:Unit impulse.gif

Unit sample function

In the study of digital signal processing (DSP), the unit sample function $\delta [n]$ represents a special case of a 2-dimensional Kronecker delta function ${\delta }_{ij}$ where the Kronecker indices include the number zero, and where one of the indices is zero. In this case: $$\delta [n]\equiv {\delta }_{n0}\equiv {\delta }_{0n}\text{where}-\mathrm{\infty }<n<\mathrm{\infty }$$ Or more generally where: $$\delta [n-k]\equiv \delta [k-n]\equiv {\delta }_{nk}\equiv {\delta }_{kn}\text{where}-\mathrm{\infty }<n<\mathrm{\infty },-\mathrm{\infty }<k<\mathrm{\infty }$$ However, this is only a 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 $\delta [n]\equiv {\delta }_{n0}\equiv {\delta }_{0n}$ does not exist, and in fact, the Kronecker delta function and the unit sample function are different functions that overlap in the 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. The discrete unit sample function is more simply defined as: $$\delta [n]=\{\begin{array}{ll}1& n=0\\ 0& n\text{ is another integer}\end{array}$$ In addition, the Dirac delta function is often confused for both the Kronecker delta function and the unit sample function. The Dirac delta is defined as: $$\{\begin{array}{ll}\underset{-\epsilon }{\overset{+\epsilon }{\int }}\delta (t)dt=1& \mathrm{\forall }\epsilon >0\\ \delta (t)=0& \mathrm{\forall }t\ne 0\end{array}$$ Unlike the Kronecker delta function ${\delta }_{ij}$ and the unit sample function $\delta [n]$, the Dirac delta function $\delta (t)$ does not have an 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 $\delta (t)$, or the unit sample function $\delta [n]$.

Notable properties

The Kronecker delta has the so-called sifting property that for $j\in \mathbb{\mathbb{Z}}$: $$\sum _{i=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_i{\delta }_{ij}=a_j.$$ 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 $$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\delta (x-y)f(x)dx=f(y),$$ 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: $\delta [n]$. 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.

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 $\mathbf{x}=\{x_1,\cdots ,x_n\}$, with corresponding probabilities $p_1,\cdots ,p_n$, 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

If it is considered as a type $(1,1)$ tensor, the Kronecker tensor can be written ${\delta }_j^i$ with a covariant index $j$ and contravariant index $i$: $${\delta }_j^i=\{\begin{array}{ll}0& (i\ne j),\\ 1& (i=j).\end{array}$$ This tensor represents:

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

The generalized Kronecker delta 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 § Properties of the generalized Kronecker delta below disappearing.

Definitions of the generalized Kronecker delta

In terms of the indices, the generalized Kronecker delta is defined as: $${\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=\{\begin{array}{ll}\phantom{-}1& \text{if }{\nu }_1\dots {\nu }_p\text{ are distinct integers and are an even permutation of }{\mu }_1\dots {\mu }_p\\ -1& \text{if }{\nu }_1\dots {\nu }_p\text{ are distinct integers and are an odd permutation of }{\mu }_1\dots {\mu }_p\\ \phantom{-}0& \text{in all other cases}.\end{array}$$ Let ${\mathrm{S}}_p$ be the symmetric group of degree $p$, then: $${\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=\sum _{\sigma \in {\mathrm{S}}_p}\mathrm{sgn}(\sigma ){\delta }_{{\nu }_{\sigma (1)}}^{{\mu }_1}\cdots {\delta }_{{\nu }_{\sigma (p)}}^{{\mu }_p}=\sum _{\sigma \in {\mathrm{S}}_p}\mathrm{sgn}(\sigma ){\delta }_{{\nu }_1}^{{\mu }_{\sigma (1)}}\cdots {\delta }_{{\nu }_p}^{{\mu }_{\sigma (p)}}.$$ Using anti-symmetrization: $${\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=p!{\delta }_{[{\nu }_1}^{{\mu }_1}\dots {\delta }_{{\nu }_p]}^{{\mu }_p}=p!{\delta }_{{\nu }_1}^{[{\mu }_1}\dots {\delta }_{{\nu }_p}^{{\mu }_p]}.$$ In terms of a $p\times p$ determinant: $${\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=\left|\begin{array}{ccc}{\delta }_{{\nu }_1}^{{\mu }_1}& \cdots & {\delta }_{{\nu }_p}^{{\mu }_1}\\ ⋮& \ddots & ⋮\\ {\delta }_{{\nu }_1}^{{\mu }_p}& \cdots & {\delta }_{{\nu }_p}^{{\mu }_p}\end{array}\right|.$$ Using the Laplace expansion (Laplace's formula) of determinant, it may be defined recursively: $$\begin{array}{rl}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}& =\sum _{k=1}^p(-1{)}^{p+k}{\delta }_{{\nu }_k}^{{\mu }_p}{\delta }_{{\nu }_1\dots {\check{\nu }}_k\dots {\nu }_p}^{{\mu }_1\dots {\mu }_k\dots {\check{\mu }}_p}\\ & ={\delta }_{{\nu }_p}^{{\mu }_p}{\delta }_{{\nu }_1\dots {\nu }_{p-1}}^{{\mu }_1\dots {\mu }_{p-1}}-\sum _{k=1}^{p-1}{\delta }_{{\nu }_k}^{{\mu }_p}{\delta }_{{\nu }_1\dots {\nu }_{k-1}{\nu }_p{\nu }_{k+1}\dots {\nu }_{p-1}}^{{\mu }_1\dots {\mu }_{k-1}{\mu }_k{\mu }_{k+1}\dots {\mu }_{p-1}},\end{array}$$ where the caron, $\stackrel{}{ˇ}$, 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: $${\delta }_{{\nu }_1\dots {\nu }_n}^{{\mu }_1\dots {\mu }_n}={\epsilon }^{{\mu }_1\dots {\mu }_n}{\epsilon }_{{\nu }_1\dots {\nu }_n}.$$ More generally, for $m=n-p$, using the Einstein summation convention: $${\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=\frac{1}{m!}{\epsilon }^{{\kappa }_1\dots {\kappa }_m{\mu }_1\dots {\mu }_p}{\epsilon }_{{\kappa }_1\dots {\kappa }_m{\nu }_1\dots {\nu }_p}.$$

Contractions of the generalized Kronecker delta

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

Properties of the generalized Kronecker delta

The generalized Kronecker delta may be used for anti-symmetrization: $$\begin{array}{rl}\frac{1}{p!}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}{a}^{{\nu }_1\dots {\nu }_p}& =a^{[{\mu }_1\dots {\mu }_p]},\\ \frac{1}{p!}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}{a}_{{\mu }_1\dots {\mu }_p}& =a_{[{\nu }_1\dots {\nu }_p]}.\end{array}$$ From the above equations and the properties of anti-symmetric tensors, we can derive the properties of the generalized Kronecker delta: $$\begin{array}{rl}\frac{1}{p!}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}{a}^{[{\nu }_1\dots {\nu }_p]}& =a^{[{\mu }_1\dots {\mu }_p]},\\ \frac{1}{p!}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}{a}_{[{\mu }_1\dots {\mu }_p]}& =a_{[{\nu }_1\dots {\nu }_p]},\\ \frac{1}{p!}{\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}{\delta }_{{\kappa }_1\dots {\kappa }_p}^{{\nu }_1\dots {\nu }_p}& ={\delta }_{{\kappa }_1\dots {\kappa }_p}^{{\mu }_1\dots {\mu }_p},\end{array}$$ which are the generalized version of formulae written in § Properties. The last formula is equivalent to the Cauchy–Binet formula. Reducing the order via summation of the indices may be expressed by the identity $${\delta }_{{\nu }_1\dots {\nu }_s{\mu }_{s+1}\dots {\mu }_p}^{{\mu }_1\dots {\mu }_s{\mu }_{s+1}\dots {\mu }_p}=\frac{(n-s)!}{(n-p)!}{\delta }_{{\nu }_1\dots {\nu }_s}^{{\mu }_1\dots {\mu }_s}.$$ 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: $${\delta }_{{\nu }_1\dots {\nu }_p}^{{\mu }_1\dots {\mu }_p}=\frac{1}{(n-p)!}{\epsilon }^{{\mu }_1\dots {\mu }_p{\kappa }_{p+1}\dots {\kappa }_n}{\epsilon }_{{\nu }_1\dots {\nu }_p{\kappa }_{p+1}\dots {\kappa }_n}.$$ 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. Also, this relation is extensively used in S-duality theories, especially when written in the language of differential forms and Hodge duals.

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. $${\delta }_{x,n}=\frac{1}{2\pi i}{\oint }_{|z|=1}{z}^{x-n-1}dz=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}{e}^{i(x-n)\phi }d\phi$$

The Kronecker comb

The Kronecker comb function with period $N$ is defined (using DSP notation) as: $${\mathrm{\Delta }}_N[n]=\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\delta [n-kN],$$ 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

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_uvw 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 $$(u_s\mathbf{i}+v_s\mathbf{j}+w_s\mathbf{k})\times (u_t\mathbf{i}+v_t\mathbf{j}+w_t\mathbf{k}).$$ 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 onto S_xyz. Then the degree δ of mapping is ⁠1/4π⁠ 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, δ is given by the integral: $$\delta =\frac{1}{4\pi }\underset{R_{st}}{\iint }{(x^2+y^2+z^2)}^{-\frac{3}{2}}\left|\begin{array}{ccc}x& y& z\\ \frac{\partial x}{\partial s}& \frac{\partial y}{\partial s}& \frac{\partial z}{\partial s}\\ \frac{\partial x}{\partial t}& \frac{\partial y}{\partial t}& \frac{\partial z}{\partial t}\end{array}\right|dsdt.$$

See also

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