Kronecker Delta

The Kronecker delta is a mathematical symbol that is represented by a small delta with two indices (typically ) and is named after Leopold Kronecker . It is also sometimes referred to as the Kronecker symbol , although there is another Kronecker symbol . ${\displaystyle \delta _{ij}\,}$

The term delta function, which is also commonly used, is misleading because it is more often used to describe the delta distribution .

It is mainly used in sum formulas in connection with matrix or vector operations , or to avoid case distinctions in formulas.

definition

Given an arbitrary index set and a ring with zero element and one element . Be further . The Kronecker Delta is defined as: ${\displaystyle I}$ ${\displaystyle R}$ ${\displaystyle 0^{R}}$ ${\displaystyle 1^{R}}$ ${\displaystyle i,j\in I}$

${\displaystyle \delta _{ij}={\begin{cases}1^{R}&{\text{falls}}\quad i=j\\0^{R}&{\text{falls}}\quad i\neq j\end{cases}}}$

The index set is usually a finite subset of the natural numbers .

properties

The Kronecker Delta can take the form

${\displaystyle \delta =\mathrm {1} _{D}\colon I\times I\to \{0,1\}}$,

is the characteristic function of the diagonal set . Often, instead of an expanded image space, e.g. B. the real numbers considered. ${\displaystyle \mathrm {1} _{D}}$ ${\displaystyle D=\{(i,j)\in I\times I\mid i=j\}}$ ${\displaystyle \{0,1\}}$

This applies to products from Kronecker-Deltas with and for all with index quantities ${\displaystyle i,j,k\in I_{1}}$ ${\displaystyle b_{i}\in I_{2}}$ ${\displaystyle i}$ ${\displaystyle I_{1},I_{2}}$

${\displaystyle \prod _{i}\delta _{b_{i}b_{j}}=\prod _{i}\delta _{b_{i}b_{k}}\;\forall j,k}$

This expression compares almost everything with the fixed one and is only 1 if all expressions are equal, which is why instead of any one (expressed as ) can be used for it. ${\displaystyle b_{i}}$ ${\displaystyle b_{j}}$ ${\displaystyle b_{j}}$ ${\displaystyle b_{i}}$ ${\displaystyle b_{k}}$

For example with, this means (after deleting the same indices): ${\displaystyle I_{1}=\{1,2,3\}}$ ${\displaystyle b_{1}:=a,\;b_{2}:=b,\;b_{3}:=c}$

${\displaystyle \delta _{ba}\delta _{ca}=\delta _{ab}\delta _{cb}=\delta _{ac}\delta _{bc}}$

This expression is 1 if and only if holds. If the Kronecker delta is used together with the Einstein sums convention , this statement is not correct. The Kronecker delta together with Einstein's sum convention is discussed in the section “ As (r, s) -tensor ”. ${\displaystyle a=b=c}$

Trivially, the following also applies (for ): ${\displaystyle a,b\in I}$

${\displaystyle \prod \delta _{ab}=\delta _{ab}\,.}$

As an (r, s) -tensor

If one considers the Kronecker delta on a finite-dimensional vector space , one can understand it as a (0,2) -tensor . As a multilinear map ${\displaystyle V}$

${\displaystyle \delta \colon V\times V\to \mathbb {R} }$

the Kronecker delta is clearly determined by its effect on the basis vectors and it applies

${\displaystyle \delta (e_{i},e_{j})={\begin{cases}1,&{\mbox{falls }}\quad i=j,\\0,&{\mbox{falls }}\quad i\neq j.\end{cases}}}$

The Kronecker delta as a (0.2) -tensor is a special case of the general definitions from the beginning of the article. If the index set is finite in the general definition and if finite-dimensional vectors are indexed by it, then the general definition and the view as a (0,2) -tensor are the same. Another extension of the Kronecker delta, understood as a tensor, is the Levi-Civita symbol .

In connection with the tensor calculus, Einstein's summation convention is often used, in which double indices are used to sum up. That is, in an n-dimensional vector space

${\displaystyle \delta _{ab}\delta _{ab}=\sum _{a=1}^{n}\sum _{b=1}^{n}\delta _{ab}\delta _{ab}=\sum _{a=1}^{n}\delta _{aa}=\sum _{1}^{n}1=n\neq \delta _{ab}\,.}$

With this summation convention, attention is usually paid to which indices are at the top and which are at the bottom and it is only totaled if the same index is once above and once below. In the case of the Kronecker Delta it should then read. ${\displaystyle \delta _{b}^{a}\delta _{a}^{b}=n}$

Integral and sum display

If the set of whole numbers is selected as the index set , then the Kronecker delta can be represented using a curve integral . It is true ${\displaystyle \mathbb {Z} }$

${\displaystyle \delta _{xn}={\frac {1}{2\pi i}}\oint _{|z|=1}z^{x-n-1}dz={\frac {1}{2\pi }}\int _{0}^{2\pi }e^{i(x-n)\varphi }d\varphi \,,}$

where the curve that runs on the circle is directed counterclockwise. This representation can be proven using the residual theorem. ${\displaystyle |z|=1}$

Sometimes there is also a representation in the form

${\displaystyle \delta _{nm}={\frac {1}{N}}\sum _{k=1}^{N}e^{2\pi i{\frac {k}{N}}(n-m)}}$

helpful. This can be derived with the help of the partial sums sequence of the geometric series .

Examples

• In linear algebra , the - identity matrix can be written as. ${\displaystyle n\times n}$ ${\displaystyle (\delta _{ij})_{i,j\in \{1,\ldots ,n\}}}$
• With the Kronecker delta one can write the scalar product of orthonormal vectors as . ${\displaystyle e_{1},\dots ,e_{n}}$ ${\displaystyle \langle e_{i},e_{j}\rangle =\delta _{ij}}$

Alternative definition in digital signal processing

In the digital signal processing other similar definition of the Kronecker delta is used. The Kronecker delta is understood here as a function and is defined by ${\displaystyle \mathbb {Z} }$

${\displaystyle \delta [n]={\begin{cases}1,&n=0\\0,&n\neq 0\end{cases}}\,.}$

In this context, the function is referred to as “unit impulse ” and is used to determine the impulse response in discrete systems such as digital filters .

See also

• The delta distribution forms an analogue in distribution theory ; under integration it behaves like the Kronecker delta under summation over all possible values ​​for one of the two parameters.
• The Dirac measure, on the other hand, forms an analogue in measure theory ; under integration with regard to the measure , it behaves analogously to the Kronecker delta.

Web links

• Eric W. Weisstein : Kronecker Delta . In: MathWorld (English).

Individual evidence

1. ^ Alan V. Oppenheim, Ronald W. Schafer: Discrete-time signal processing . 3. Edition. Oldenbourg Verlag, 1999, ISBN 3-486-24145-1 .