Kronecker's Delta

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:
$\delta_ = \begin 0 &\text i \neq j, \\ 1 &\text i=j. \end$
or with use of Iverson brackets:
\delta_ = =j,
where the Kronecker delta is a piecewise function of variables and . For example, , whereas .

The Kronecker delta appears naturally in many areas of mathematics, physics and engineering, as a means of compactly expressing its definition above.

In linear algebra, the identity matrix has entries equal to the Kronecker delta:
$I_ = \delta_$
where and take the values , and the inner product of vectors can be written as
$\mathbf\cdot\mathbf = \sum_^n a_\delta_b_ = \sum_^n a_ b_.$
Here the Euclidean vectors are defined as -tuples: $\mathbf = (a_1, a_2, \dots, a_n)$ and $\mathbf= (b_1, b_2, ..., b_n)$ and the last step is obtained by using the values of the Kronecker delta to reduce the summation over .

The restriction to positive or non-negative integers is common, but in fact, the Kronecker delta can be defined on an arbitrary set.

Properties

The following equations are satisfied:
$\begin \sum_ \delta_ a_j &= a_i,\\ \sum_ a_i \delta_ &= a_j,\\ \sum_ \delta_\delta_ &= \delta_. \end$
Therefore, the matrix can be considered as an identity matrix.

Another useful representation is the following form:
$\delta_ = \frac \sum_^N e^$
In the limit $N \to \infty$. This can be derived using the formula for the geometric series.

Alternative notation

Using the Iverson bracket: \delta_ = =j

Often, a single-argument notation is used, which is equivalent to setting :
$\delta_ = \begin 0, & \text i \neq 0 \\ 1, & \text i = 0 \end$

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

Digital signal processing

In the study of digital signal processing (DSP), the unit sample function \delta /math> represents a special case of a 2-dimensional Kronecker delta function $\delta_$ where the Kronecker indices include the number zero, and where one of the indices is zero. In this case:
\delta \equiv \delta_ \equiv \delta_~~~\text -\infty

Or more generally where:
\delta -k\equiv \delta -n\equiv \delta_ \equiv \delta_\text -\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 \equiv \delta_ \equiv \delta_ doesn't 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 = \begin 1 & n = 0 \\ 0 & n \text\end

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:
$\delta(t) = \begin \infty & t = 0 \\ 0 & t \text\end$

Unlike the Kronecker delta function $\delta_$ and the unit sample function \delta /math>, the Dirac delta function $\delta(t)$ doesn't have a integer index, it has a single continuous non-integer value .

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 /math>.

Properties of the delta function

The Kronecker delta has the so-called ''sifting'' property that for :
$\sum_^\infty a_i \delta_ = a_j.$
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
$\int_^\infty \delta(x-y)f(x)\, dx=f(y),$
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, generally indicates continuous time (Dirac), whereas arguments like , , , , , and are usually reserved for discrete time (Kronecker). Another common practice is to represent discrete sequences with square brackets; thus: . 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 , with corresponding probabilities , then the probability mass function of the distribution over can be written, using the Kronecker delta, as
$p(x) = \sum_^n p_i \delta_.$

Equivalently, the probability density function of the distribution can be written using the Dirac delta function as
$f(x) = \sum_^n p_i \delta(x-x_i).$

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 tensor, the Kronecker tensor can be written
with a covariant index and contravariant index :
$\delta^_ = \begin 0 & (i \ne j), \\ 1 & (i = j). \end$

This tensor represents:
* The identity mapping (or identity matrix), considered as a linear mapping or
* The trace or tensor contraction, considered as a mapping
* The map , representing scalar multiplication as a sum of outer products.

The or multi-index Kronecker delta of order is a type tensor that is completely antisymmetric in its upper indices, and also in its lower indices.

Two definitions that differ by a factor of are in use. Below, the version is presented has nonzero components scaled to be . The second version has nonzero components that are , with consequent changes scaling factors in formulae, such as the scaling factors of in ' below disappearing.

Definitions of the generalized Kronecker delta

In terms of the indices, the generalized Kronecker delta is defined as:
$\delta^_ = \begin +1 & \quad \text \nu_1 \dots \nu_p \text \mu_1 \dots \mu_p \\ -1 & \quad \text \nu_1 \dots \nu_p \text \mu_1 \dots \mu_p \\ \;\;0 & \quad \text. \end$

Let be the symmetric group of degree , then:
$\delta^_ = \sum_ \sgn(\sigma)\, \delta^_\cdots\delta^_ = \sum_ \sgn(\sigma)\, \delta^_\cdots\delta^_.$

Using anti-symmetrization:
$\delta^_ = p! \delta^_ \dots \delta^_ = p! \delta^_ \dots \delta^_.$

In terms of a determinant:
$\delta^_ = \begin \delta^_ & \cdots & \delta^_ \\ \vdots & \ddots & \vdots \\ \delta^_ & \cdots & \delta^_ \end.$

Using the Laplace expansion ( Laplace's formula) of determinant, it may be defined recursively:
$\begin \delta^_ &= \sum_^p (-1)^ \delta^_ \delta^_ \\ &= \delta^_ \delta^_ - \sum_^ \delta^_ \delta^_, \end$
where the caron, , indicates an index that is omitted from the sequence.

When (the dimension of the vector space), in terms of the Levi-Civita symbol:
$\delta^_ = \varepsilon^\varepsilon_.$

Contractions of the generalized Kronecker delta

Kronecker Delta contractions depend on the dimension of the space. For example,
$\delta^_ \delta^_ = (d-1) \delta^_ ,$
where is the dimension of the space. From this relation the full contracted delta is obtained as
$\delta^_ \delta^_ = 2d(d-1) \delta^_ .$
The generalization of the preceding formulas is
\left(x^2+y^2+z^2\right)^
\begin
x & y & z \\
\frac & \frac & \frac \\
\frac & \frac & \frac
\end \, ds \, dt.

See also

*Dirac measure
*Indicator function
*Levi-Civita symbol
* 't Hooft symbol
* Unit function
*XNOR gate

References

{{Tensors

Mathematical notation
Elementary special functions