Kronecker 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: <math display="block">\delta_{ij} = \begin{cases} 0 &\text{if } i \neq j, \\ 1 &\text{if } i=j. \end{cases}</math> or with use of Iverson brackets: <math display="block">\delta_{ij} = [i=j]\,</math> For example, <math>\delta_{12} = 0</math> because <math>1 \ne 2</math>, whereas <math>\delta_{33} = 1</math> because <math>3 = 3</math>.

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

In linear algebra, the <math>n\times n</math> identity matrix <math>\mathbf{I}</math> has entries equal to the Kronecker delta: <math display="block"> I_{ij} = \delta_{ij} </math> where <math>i</math> and <math>j</math> take the values <math>1,2,\cdots,n</math>, and the inner product of vectors can be written as <math display="block"> \mathbf{a}\cdot\mathbf{b} = \sum_{i,j=1}^n a_{i}\delta_{ij}b_{j} = \sum_{i=1}^n a_{i} b_{i}.</math> Here the Euclidean vectors are defined as Template:Mvar-tuples: <math> \mathbf{a} = (a_1, a_2, \dots, a_n)</math> and <math> \mathbf{b}= (b_1, b_2, ..., b_n) </math> and the last step is obtained by using the values of the Kronecker delta to reduce the summation over <math>j</math>.

It is common for Template:Mvar and Template:Mvar to be restricted to a set of the form Template:Math or Template:Math, but the Kronecker delta can be defined on an arbitrary set.

Properties

The following equations are satisfied: <math display="block">\begin{align} \sum_{j} \delta_{ij} a_j &= a_i,\\ \sum_{i} a_i \delta_{ij} &= a_j,\\ \sum_{k} \delta_{ik}\delta_{kj} &= \delta_{ij}. \end{align}</math> Therefore, the matrix Template:Math can be considered as an identity matrix.

Another useful representation is the following form: <math display="block">\delta_{nm} = \lim_{N\to\infty}\frac{1}{N} \sum_{k = 1}^N e^{2 \pi i \frac{k}{N}(n-m)}</math> This can be derived using the formula for the geometric series.

Alternative notation

Using the Iverson bracket: <math display="block">\delta_{ij} = [i=j ].</math>

Often, a single-argument notation <math>\delta_i</math> is used, which is equivalent to setting <math>j=0</math>: <math display="block">\delta_{i} = \delta_{i0} = \begin{cases} 0, & \text{if } i \neq 0 \\ 1, & \text{if } i = 0 \end{cases}</math>

In linear algebra, it can be thought of as a tensor, and is written <math>\delta_j^i</math>. Sometimes the Kronecker delta is called the substitution tensor.<ref name="Trowbridge">Template:Cite journal</ref>

Digital signal processing

File:Unit impulse.gif

Unit sample function

In the study of digital signal processing (DSP), the Kronecker delta function sometimes means the unit sample function <math>\delta[n]</math> , which represents a special case of the 2-dimensional Kronecker delta function <math>\delta_{ij}</math> where the Kronecker indices include the number zero, and where one of the indices is zero: <math display="block">\delta[n] \equiv \delta_{n0} \equiv \delta_{0n}~~~\text{where} -\infty<n<\infty</math>

Or more generally where: <math display="block">\delta[n-k] \equiv \delta[k-n] \equiv \delta_{nk} \equiv \delta_{kn}\text{where} -\infty<n<\infty, -\infty<k<\infty</math>

For discrete-time signals, it is conventional to place a single integer index in square braces; in contrast the Kronecker delta, <math>\delta_{ij}</math>, 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 imput. 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: <math display="block">\delta[n] = \begin{cases} 1 & n = 0 \\ 0 & n \text{ is another integer}\end{cases}</math>

In comparison, in continuous-time systems the Dirac delta function is often confused for both the Kronecker delta function and the unit sample function. The Dirac delta is defined as: <math display="block">\begin{cases} \int_{-\varepsilon}^{+\varepsilon}\delta(t)dt = 1 & \forall \varepsilon > 0 \\ \delta(t) = 0 & \forall t \neq 0\end{cases}</math>

Unlike the Kronecker delta function <math>\delta_{ij}</math> and the unit sample function <math>\delta[n]</math>, the Dirac delta function <math>\delta(t)</math> does not have an integer index, it has a single continuous non-integer value Template:Mvar.

In continuous-time systems, the term "unit impulse function" is used to refer to the Dirac delta function <math>\delta(t)</math> or, in discrete-time systems, the Kronecker delta function <math>\delta[n]</math>.

Notable properties

The Kronecker delta has the so-called sifting property that for <math>j\in\mathbb{Z}</math>: <math display="block">\sum_{i=-\infty}^\infty a_i \delta_{ij} = a_j.</math> 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 <math display="block">\int_{-\infty}^\infty \delta(x-y)f(x)\, dx=f(y),</math> and in fact Dirac's delta was named after the Kronecker delta because of this analogous property.<ref>Template:Cite book</ref> In signal processing it is usually the context (discrete or continuous time) that distinguishes the Kronecker and Dirac "functions". And by convention, <math>\delta(t)</math> generally indicates continuous time (Dirac), whereas arguments like <math>i</math>, <math>j</math>, <math>k</math>, <math>l</math>, <math>m</math>, and <math>n</math> are usually reserved for discrete time (Kronecker). Another common practice is to represent discrete sequences with square brackets; thus: <math>\delta[n]</math>. 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.<ref>Template:Citation.</ref>

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 <math>\mathbf{x} = \{x_1,\cdots,x_n\}</math>, with corresponding probabilities <math>p_1,\cdots,p_n</math>, then the probability mass function <math>p(x)</math> of the distribution over <math>\mathbf{x}</math> can be written, using the Kronecker delta, as <math display="block">p(x) = \sum_{i=1}^n p_i \delta_{x x_i}.</math>

Equivalently, the probability density function <math>f(x)</math> of the distribution can be written using the Dirac delta function as <math display="block">f(x) = \sum_{i=1}^n p_i \delta(x-x_i).</math>

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 <math>(1,1)</math> tensor, the Kronecker tensor can be written <math>\delta^i_j</math> with a covariant index <math>j</math> and contravariant index <math>i</math>: <math display="block">\delta^{i}_{j} = \begin{cases} 0 & (i \ne j), \\ 1 & (i = j). \end{cases}</math>

This tensor represents:

• The identity mapping (or identity matrix), considered as a linear mapping <math>V\to V</math> or <math>V^*\to V^*</math>
• The trace or tensor contraction, considered as a mapping <math>V^* \otimes V\to K</math>
• The map <math>K\to V^*\otimes V</math>, representing scalar multiplication as a sum of outer products.

The Template:Visible anchor or multi-index Kronecker delta of order <math>2p</math> is a type <math>(p,p)</math> tensor that is completely antisymmetric in its <math>p</math> upper indices, and also in its <math>p</math> lower indices.

Two definitions that differ by a factor of <math>p!</math> are in use. Below, the version is presented has nonzero components scaled to be <math>\pm 1</math>. The second version has nonzero components that are <math>\pm 1/p!</math>, with consequent changes scaling factors in formulae, such as the scaling factors of <math>1/p!</math> in Template:Section link below disappearing.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Definitions of the generalized Kronecker delta

In terms of the indices, the generalized Kronecker delta is defined as:<ref>Template:Cite book</ref><ref>Template:Cite bookTemplate:ISBN missing</ref> <math display="block">\delta^{\mu_1 \dots \mu_p }_{\nu_1 \dots \nu_p} = \begin{cases} \phantom-1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are distinct integers and are an even permutation of } \mu_1 \dots \mu_p \\ -1 & \quad \text{if } \nu_1 \dots \nu_p \text{ are distinct integers and are an odd permutation of } \mu_1 \dots \mu_p \\ \phantom-0 & \quad \text{in all other cases}. \end{cases}</math>

Let <math>\mathrm{S}_p</math> be the symmetric group of degree <math>p</math>, then: <math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = \sum_{\sigma \in \mathrm{S}_p} \sgn(\sigma)\, \delta^{\mu_1}_{\nu_{\sigma(1)}}\cdots\delta^{\mu_p}_{\nu_{\sigma(p)}} = \sum_{\sigma \in \mathrm{S}_p} \sgn(\sigma)\, \delta^{\mu_{\sigma(1)}}_{\nu_1}\cdots\delta^{\mu_{\sigma(p)}}_{\nu_p}. </math>

Using anti-symmetrization: <math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = p! \delta^{\mu_1}_{[ \nu_1} \dots \delta^{\mu_p}_{\nu_p ]} = p! \delta^{[ \mu_1}_{\nu_1} \dots \delta^{\mu_p ]}_{\nu_p}.</math>

In terms of a <math>p\times p</math> determinant:<ref>Template:Cite book</ref> <math display="block">\delta^{\mu_1 \dots \mu_p }_{\nu_1 \dots \nu_p} = \begin{vmatrix} \delta^{\mu_1}_{\nu_1} & \cdots & \delta^{\mu_1}_{\nu_p} \\ \vdots & \ddots & \vdots \\ \delta^{\mu_p}_{\nu_1} & \cdots & \delta^{\mu_p}_{\nu_p} \end{vmatrix}.</math>

Using the Laplace expansion (Laplace's formula) of determinant, it may be defined recursively:<ref>A recursive definition requires a first case, which may be taken as Template:Math for Template:Math, or alternatively Template:Math for Template:Math (generalized delta in terms of standard delta).</ref> <math display="block">\begin{align}

 \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p}
   &= \sum_{k=1}^p (-1)^{p+k} \delta^{\mu_p}_{\nu_k} \delta^{\mu_1 \dots \mu_{k} \dots \check\mu_p}_{\nu_1 \dots \check\nu_k \dots \nu_p} \\
   &= \delta^{\mu_p}_{\nu_p} \delta^{\mu_1 \dots \mu_{p - 1}}_{\nu_1 \dots \nu_{p-1}} - \sum_{k=1}^{p-1} \delta^{\mu_p}_{\nu_k} \delta^{\mu_1 \dots \mu_{k-1}\, \mu_k\, \mu_{k+1} \dots \mu_{p-1}}_{\nu_1 \dots \nu_{k-1}\, \nu_p\, \nu_{k+1} \dots \nu_{p-1}},

\end{align}</math> where the caron, <math>\check{}</math>, indicates an index that is omitted from the sequence.

When <math>p=n</math> (the dimension of the vector space), in terms of the Levi-Civita symbol: <math display="block">\delta^{\mu_1 \dots \mu_n}_{\nu_1 \dots \nu_n} = \varepsilon^{\mu_1 \dots \mu_n}\varepsilon_{\nu_1 \dots \nu_n}\,.</math> More generally, for <math>m=n-p</math>, using the Einstein summation convention: <math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = \tfrac{1}{m!} \varepsilon^{\kappa_1 \dots \kappa_m \mu_1 \dots \mu_p}\varepsilon_{\kappa_1 \dots \kappa_m \nu_1 \dots \nu_p}\,.</math>

Contractions of the generalized Kronecker delta

Kronecker Delta contractions depend on the dimension of the space. For example, <math display="block">\delta^{\nu_1}_{\mu_1} \delta^{\mu_1 \mu_2}_{\nu_1 \nu_2} = (d-1) \delta^{\mu_2}_{\nu_2} ,</math> where Template:Mvar is the dimension of the space. From this relation the full contracted delta is obtained as <math display="block">\delta^{\nu_1 \nu_2}_{\mu_1 \mu_2} \delta^{\mu_1 \mu_2}_{\nu_1 \nu_2} = 2d(d-1) .</math> The generalization of the preceding formulas isTemplate:Cn <math display="block">\delta^{\nu_1 \dots \nu_n}_{\mu_1 \dots \mu_n} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = n! \frac{(d-p+n)!}{(d-p)!} \delta^{\mu_{n+1} \dots \mu_p}_{\nu_{n+1} \dots \nu_p} .</math>

Properties of the generalized Kronecker delta

The generalized Kronecker delta may be used for anti-symmetrization: <math display="block">\begin{align}

\frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a^{\nu_1 \dots \nu_p} &= a^{[ \mu_1 \dots \mu_p ]} , \\
\frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a_{\mu_1 \dots \mu_p} &= a_{[ \nu_1 \dots \nu_p ]} . \end{align}</math>

From the above equations and the properties of anti-symmetric tensors, we can derive the properties of the generalized Kronecker delta: <math display="block">\begin{align}

\frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a^{[ \nu_1 \dots \nu_p ]} &= a^{[ \mu_1 \dots \mu_p ]} , \\
\frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} a_{[ \mu_1 \dots \mu_p ]} &= a_{[ \nu_1 \dots \nu_p ]} , \\
\frac{1}{p!} \delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} \delta^{\nu_1 \dots \nu_p}_{\kappa_1 \dots \kappa_p} &= \delta^{\mu_1 \dots \mu_p}_{\kappa_1 \dots \kappa_p} ,

\end{align}</math> which are the generalized version of formulae written in Template:Section link. The last formula is equivalent to the Cauchy–Binet formula.

Reducing the order via summation of the indices may be expressed by the identity<ref>Template:Cite book</ref> <math display="block"> \delta^{\mu_1 \dots \mu_s \, \mu_{s+1} \dots \mu_p}_{\nu_1 \dots \nu_s \, \mu_{s+1} \dots \mu_p} = \frac{(n-s)!}{(n-p)!} \delta^{\mu_1 \dots \mu_s}_{\nu_1 \dots \nu_s}.</math>

Using both the summation rule for the case <math>p=n</math> and the relation with the Levi-Civita symbol, the summation rule of the Levi-Civita symbol is derived: <math display="block">\delta^{\mu_1 \dots \mu_p}_{\nu_1 \dots \nu_p} = \frac{1}{(n-p)!}\varepsilon^{\mu_1 \dots \mu_p \, \kappa_{p+1} \dots \kappa_n}\varepsilon_{\nu_1 \dots \nu_p \, \kappa_{p+1} \dots \kappa_n}.</math> The 4D version of the last relation appears in Penrose's spinor approach to general relativity<ref>Template:Cite journal</ref> that he later generalized, while he was developing Aitken's diagrams,<ref>Template:Cite book</ref> to become part of the technique of Penrose graphical notation.<ref>Roger Penrose, "Applications of negative dimensional tensors," in Combinatorial Mathematics and its Applications, Academic Press (1971).</ref> Also, this relation is extensively used in S-duality theories, especially when written in the language of differential forms and Hodge duals.

Integral representations

For any integers <math>j</math> and <math>k</math>, 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, oriented counterclockwise. An equivalent representation of the integral arises by parameterizing the contour by an angle around the origin. <math display="block"> \delta_{jk} = \frac1{2\pi i} \oint_{|z|=1} z^{j-k-1} \,dz=\frac1{2\pi} \int_0^{2\pi} e^{i(j-k)\varphi} \,d\varphi</math>

The Kronecker comb

The Kronecker comb function with period <math>N</math> is defined (using DSP notation) as: <math display="block">\Delta_N[n]=\sum_{k=-\infty}^\infty \delta[n-kN],</math> where <math>N\ne 0</math> and <math>n</math> are integers. The Kronecker comb thus consists of an infinite series of unit impulses that are Template:Mvar 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
• Indicator function
• Heaviside step function
• Levi-Civita symbol
• Minkowski metric
• 't Hooft symbol
• Unit function
• XNOR gate

References

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