In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually non-negative integers or with use of Iverson brackets.

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 identity matrix, and the inner product of vectors Here the Euclidean vectors are defined as .

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.

Properties

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

Another useful representation is the following form: This can be derived using the formula for the geometric series.

Alternative notation

Using the Iverson bracket

In linear algebra, it can be thought of as a tensor. Sometimes the Kronecker delta is called the substitution tensor.

When juxtaposition creates ambiguity, often a comma is employed. For example, a comma separates two integers in .

Digital signal processing

In the study of digital signal processing

, 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 input. In contrast, the typical purpose of the Kronecker delta function is for filtering terms from an Einstein summation convention.

In comparison, in continuous-time systems the Dirac delta function

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.

Notable properties

The Kronecker delta has the so-called sifting 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 and in fact Dirac's delta was named after the Kronecker delta because of this analogous property.. 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.

The Kronecker delta is an elementary recursive function.

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, then the probability mass function

Equivalently, the probability density function

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

tensor with a covariant and contravariant

This tensor represents:

• The identity mapping (or identity matrix), considered as a linear mapping

• The trace or tensor contraction

• , representing scalar multiplication as a sum of outer products.

The or multi-index Kronecker delta tensor that is completely antisymmetric lower indices.

in below disappearing.

Definitions of the generalized Kronecker delta

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

be the symmetric group

Using anti-symmetrization

determinant:

Using the Laplace expansion (Laplace's formula) of determinant, it may be defined recursively:, indicates an index that is omitted from the sequence.

(the dimension of the vector space), in terms of the Levi-Civita symbol, using the Einstein summation convention

Contractions of the generalized Kronecker delta

where The generalization of the preceding formulas is

Properties of the generalized Kronecker delta

The generalized Kronecker delta may be used for anti-symmetrization

From the above equations and the properties of anti-symmetric tensors 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

and the relation with the Levi-Civita symbol, the summation rule of the Levi-Civita symbol 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

, 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

Kronecker comb

is defined (using DSP notation) as: 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.

See also

• Dirac measure

• Heaviside step function

• Indicator function

• Levi-Civita symbol

• Minkowski metric

• 't Hooft symbol

• Unit function

• XNOR gate

References