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:

or with use of Iverson brackets:

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:

where and take the values , and the inner product of vectors can be written as

Here the Euclidean vectors are defined as n-tuples: and 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.

The following equations are satisfied:

Therefore, the matrix

Another useful representation is the following form:

This can be derived using the formula for the geometric series.

Using the Iverson bracket:

Often, a single-argument notation is used, which is equivalent to setting :

In linear algebra, it can be thought of as a tensor, and is written . Sometimes the Kronecker delta is called the substitution tensor.^{[1]}

In the study of digital signal processing (DSP), the unit sample function represents a special case of a 2-dimensional Kronecker delta function where the Kronecker indices include the number zero, and where one of the indices is zero. In this case:

Or more generally where:

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 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:

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:

Unlike the Kronecker delta function and the unit sample function , the Dirac delta function 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 , or the unit sample function .

The Kronecker delta has the so-called *sifting* property that for :

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 forms the multiplicative identity element of an incidence algebra.^{[2]}

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

Equivalently, the probability density function of the distribution can be written using the Dirac delta function as

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.

If it is considered as a type tensor, the Kronecker tensor can be written with a covariant index and contravariant index :

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 **generalized Kronecker delta** 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 *§ Properties of the generalized Kronecker delta* below disappearing.^{[3]}

In terms of the indices, the generalized Kronecker delta is defined as:^{[4]}^{[5]}

Let be the symmetric group of degree , then:

Using anti-symmetrization:

In terms of a determinant:^{[6]}

Using the Laplace expansion (Laplace's formula) of determinant, it may be defined recursively:^{[7]}

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:

More generally, for , using the Einstein summation convention:

Kronecker Delta contractions depend on the dimension of the space. For example,

where d is the dimension of the space. From this relation the full contracted delta is obtained as

The generalization of the preceding formulas is

The generalized Kronecker delta may be used for anti-symmetrization:

From the above equations and the properties of anti-symmetric tensors, we can derive the properties of the generalized Kronecker delta:

which are the generalized version of formulae written in

Reducing the order via summation of the indices may be expressed by the identity^{[8]}

Using both the summation rule for the case and the relation with the Levi-Civita symbol, the summation rule of the Levi-Civita symbol is derived:

The 4D version of the last relation appears in Penrose's spinor approach to general relativity

For any integer , 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.

The Kronecker comb function with period is defined (using DSP notation) as:

where and 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.

The Kronecker delta is also called degree of mapping of one surface into another.^{[12]} 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**:

while the normal has the direction of

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: