Kronecker-Delta

The Kronecker delta is a mathematical character that is represented by a small delta with two indices (typically ) and is named after Leopold Kronecker. It is sometimes called the Kronecker symbol, even though there is still another Kronecker symbol.

The delta function also used term is misleading, because so often the Dirac delta is called.

It is mainly used in empirical formulas related to matrix or vector operations, or to avoid case distinctions in formulas.

Definition

The Kronecker delta is defined as:

It can be elements and an arbitrary index set, but usually a finite subset of the natural numbers.

Properties

The Kronecker delta can be in the form

Is written, that is, the characteristic function of the bias amount. Often this is in place of an extended image space, eg the real numbers is considered.

Sometimes, an alternative representation in the form

Helpful for large N.

For products of Kronecker deltas with and index sets

This expression compares virtually every with the fixed and is only 1 if all expressions are equal, why take any (expressed as ) can be used for it.

For example, the mean ( after deletion of the same indices):

This expression is exact then (and only then ) 1 if and only if. If the Kronecker delta used with the Einstein summation convention, so this statement is not correct. On the Kronecker delta, together with the Einstein summation convention tensor is the section as ( s r ) received.

Trivially applies ( for ):

As (R, S )-tensor

Considering the Kronecker delta on a finite dimensional vector space, one can understand it as a (0,2) tensor. As a multi- linear map

The Kronecker delta is uniquely determined by its effect on the basis vectors and it is

The Kronecker delta as a (0,2) tensor is a special case of the general definitions from the beginning of the article. Indeed, if in the general definition of the index set is finite and will be indexed by these finite vectors, then the general definition and the view as a (0,2) tensor are equal. Another extension of the tensor -conceived as a Kronecker delta is the Levi- Civita symbol.

In connection with the tensor calculus, Einstein's summation convention is often used in this summing over repeated indices. That is, applies in an n- dimensional vector space

In most cases, care is taken in this summation convention also which of the indices are above and which below and it will only summed if the same index is once above and once below. In the case of the Kronecker delta, it would then have to be so.

Integral representation

If you choose the index set of the set of integers, then the Kronecker delta can be represented using a contour integral. It is namely

Wherein the curve drawn on the circle, is directed anti-clockwise. This representation can be proved by using the residue theorem.

Examples

• In linear algebra, the identity matrix can be written as.

• With the Kronecker delta can be the scalar product of orthonormal vectors as write.

Alternative definition in the 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

The function will be referred to in this context as a " pulse unit ", and used for determining the impulse response in the discrete system, such as digital filters.