# Kronecker delta

The **Kronecker delta** is a mathematical symbol, written \( \delta_{ij} \), that equals 1 when its two indices are equal and 0 when they differ; it is defined for discrete (typically integer) indices and serves as the index-notation form of the identity matrix.<sup>[1](https://mathworld.wolfram.com/KroneckerDelta.html)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/wiki/Kronecker_symbol)</sup> It is named after the German mathematician Leopold Kronecker (Liegnitz, 7 December 1823 – Berlin, 29 December 1891), who first used the delta symbol in print in 1868, although the Encyclopedia of Mathematics dates his first use of the symbol to 1866.<sup>[3](https://planetmath.org/kroneckerdelta)</sup><sup> • </sup><sup>[4](https://proofwiki.org/wiki/Definition%3AKronecker_Delta)</sup><sup> • </sup><sup>[2](https://encyclopediaofmath.org/wiki/Kronecker_symbol)</sup><sup> • </sup><sup>[5](https://www.enciklopedija.hr/clanak/kronecker-leopold)</sup>

| Key fact | Detail |
|---|---|
| Definition | \( \delta_{ij} = 1 \) if \( i = j \), and \( \delta_{ij} = 0 \) if \( i \neq j \), for integer indices<sup>[1](https://mathworld.wolfram.com/KroneckerDelta.html)</sup> |
| Matrix form | For \( 1 \le i,j \le n \) the matrix \( (\delta^i_j) \) is the \( n \times n \) unit (identity) matrix<sup>[2](https://encyclopediaofmath.org/wiki/Kronecker_symbol)</sup> |
| Substitution property | \( \delta_{ij} a_j = a_i \) and \( \delta_{ik} A_{kj} = A_{ij} \); a repeated index on the delta replaces the summed index in the other factor<sup>[6](https://ceadpx.github.io/finite-element-methods/appendices/b-tensor-notation-index-calculus.html)</sup> |
| Trace | \( \delta_{ii} = d \), the dimension of the space; also \( \delta_{ij}\delta_{jk} = \delta_{ik} \)<sup>[6](https://ceadpx.github.io/finite-element-methods/appendices/b-tensor-notation-index-calculus.html)</sup> |
| Tensor status | \( \delta^i{}_j \) is a rank-(1,1) tensor, the identity map, with the same components in every coordinate system<sup>[7](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/SR-MT-2020/sr_mt_2020_tutorial_1_notes.pdf)</sup> |
| Discrete counterpart | The Kronecker delta takes discrete inputs and the value 1; the Dirac delta takes a continuous input and the value \( \infty \), picking terms out of sums versus integrals<sup>[8](https://books.physics.oregonstate.edu/GSF/discretecont.html)</sup> |

## Definition and basic properties

The symbol is defined by the piecewise relation

\[ \delta_{ij} = \begin{cases} 1, & i = j, \\ 0, & i \neq j, \end{cases} \]

with \( i \) and \( j \) integers (or, more generally, members of a discrete set).<sup>[1](https://mathworld.wolfram.com/KroneckerDelta.html)</sup><sup> • </sup><sup>[8](https://books.physics.oregonstate.edu/GSF/discretecont.html)</sup> It may also be written \( \delta^{ij} \) or \( \delta^i{}_j \), and it is the simplest case of the generalized Kronecker delta symbol.<sup>[3](https://planetmath.org/kroneckerdelta)</sup>

**Index substitution.** The property that makes the delta useful in computation is that contracting it with another indexed object replaces the summed index: \( \delta_{ij} a_j = a_i \) and \( \delta_{ik} A_{kj} = A_{ij} \).<sup>[6](https://ceadpx.github.io/finite-element-methods/appendices/b-tensor-notation-index-calculus.html)</sup> Repeated application simplifies chains of deltas, for example \( \delta_{ij}\delta_{jk}a_k = \delta_{ik}a_k = a_i \), and the contraction \( \delta_{ij}\delta_{ij} = \delta_{ii} \) equals 3 in three dimensions.<sup>[6](https://ceadpx.github.io/finite-element-methods/appendices/b-tensor-notation-index-calculus.html)</sup><sup> • </sup><sup>[9](https://ocw.mit.edu/courses/16-001-unified-engineering-materials-and-structures-fall-2021/mit16_001_f21_pset05_sol.pdf)</sup> The delta also picks out a single term of a sum: \( \sum_{i=1}^{\infty} a_i \, \delta_{iN} = a_N \).<sup>[8](https://books.physics.oregonstate.edu/GSF/discretecont.html)</sup>

## Relation to the identity matrix and the metric

When the indices run from 1 to \( n \), the \( n^2 \) components \( \delta^i_j \) form the unit matrix, and the delta plays an important role in matrix calculus.<sup>[2](https://encyclopediaofmath.org/wiki/Kronecker_symbol)</sup><sup> • </sup><sup>[5](https://www.enciklopedija.hr/clanak/kronecker-leopold)</sup> In tensor language, \( \delta^i{}_j \) represents the identity operator \( I: V \to V \), since \( \delta^i{}_j v^j = v^i \), and it therefore has the same numerical components in any basis.<sup>[10](https://danielmiranda.prof.ufabc.edu.br/livros/cvt/capitulo-10-tensores-em-coordenadas.html)</sup>

The delta is a genuine tensor, not merely a table of numbers. The Oxford relativity tutorial notes that the Kronecker tensor \( \delta^i_j \) is a rank-(1,1) tensor, the identity map from vectors to vectors, which must have the same components regardless of coordinate system.<sup>[7](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/SR-MT-2020/sr_mt_2020_tutorial_1_notes.pdf)</sup> A tensor-calculus text verifies this directly from the transformation law and concludes that the Kronecker delta is indeed a tensor.<sup>[11](https://grinfeld.org/books/An-Introduction-To-Tensor-Calculus/Chapter14.html)</sup> Raising or lowering an index of \( \delta^i_j \) with the metric gives the metric tensor or its inverse, \( g_{ij}\delta^j{}_k = g_{ik} \) and \( g_{ij}g_{jk} = \delta^i{}_k \); the notations \( \delta_{ij} \) and \( \delta^{ij} \) make sense only in [Euclidean space](https://www.edgepedia.org/euclidean-space), where the metric itself is the unit matrix.<sup>[7](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/SR-MT-2020/sr_mt_2020_tutorial_1_notes.pdf)</sup> Consistently, in Euclidean space the metric tensor is \( g_{ij} = \delta_{ij} \), reproducing the usual Pythagorean form of distance.<sup>[12](https://mathworld.wolfram.com/MetricTensor.html)</sup>

## Identities with the permutation symbol

In three dimensions, with Einstein summation (repeated index means sum over it) assumed and \( \epsilon_{ijk} \) the permutation (Levi-Civita) symbol, the delta satisfies the identities \( \delta_{ii} = 3 \), \( \delta_{ij}\epsilon_{ijk} = 0 \), \( \epsilon_{ipq}\epsilon_{jpq} = 2\delta_{ij} \), and \( \epsilon_{ijk}\epsilon_{pqk} = \delta_{ip}\delta_{jq} - \delta_{iq}\delta_{jp} \).<sup>[1](https://mathworld.wolfram.com/KroneckerDelta.html)</sup> These contraction identities are the standard tool for proving vector identities in mechanics and electromagnetism.<sup>[13](https://arxiv.org/abs/1406.3060)</sup> The contraction \( \delta_{ij}\epsilon_{ijk} = 0 \) follows because the permutation symbol is nonzero only when all three indices are distinct, while the delta forces two of them to be equal.<sup>[9](https://ocw.mit.edu/courses/16-001-unified-engineering-materials-and-structures-fall-2021/mit16_001_f21_pset05_sol.pdf)</sup>

## Generalized Kronecker delta

For \( 1 \le p \le n \), the generalized Kronecker delta carries \( p \) upper and \( p \) lower indices and is defined as the determinant of a \( p \times p \) matrix of ordinary deltas,

\[ \delta^{i_1 \ldots i_p}_{j_1 \ldots j_p} = \det \begin{bmatrix} \delta^{i_1}_{j_1} & \cdots & \delta^{i_1}_{j_p} \\ \vdots & \ddots & \vdots \\ \delta^{i_p}_{j_1} & \cdots & \delta^{i_p}_{j_p} \end{bmatrix}. \]

It is totally antisymmetric in the upper indices and in the lower indices: it takes the value \( +1 \) or \( -1 \) when the two index lists contain the same \( p \) distinct indices, with the sign of the permutation relating the lists, and 0 otherwise.<sup>[10](https://danielmiranda.prof.ufabc.edu.br/livros/cvt/capitulo-10-tensores-em-coordenadas.html)</sup> The Encyclopedia of Mathematics describes the same generalization and notes that the corresponding affine tensor of type \( (p,p) \) is isotropic, having the same components in any basis; the determinant of a matrix \( a^i_j \) can be written as a sum over the \( n! \) permutations weighted by \( \delta^{i_1 \ldots i_n}_{1 \ldots n} \).<sup>[2](https://encyclopediaofmath.org/wiki/Kronecker_symbol)</sup>

## Kronecker delta versus Dirac delta

The two deltas look alike but live on different domains. For the Kronecker delta the inputs \( i \) and \( j \) are discrete variables, typically obtained by counting; for the Dirac delta the input \( x \) is a continuous variable, typically obtained by measuring.<sup>[8](https://books.physics.oregonstate.edu/GSF/discretecont.html)</sup> Their ranges differ for a practical reason: the Kronecker delta takes the value 1 and is used to pick out a term in a sum, while the Dirac delta takes the value \( \infty \) at its support and is used to pick out a value inside an integral, \( \int_{-\infty}^{\infty} f(x)\,\delta(x-a)\,dx = f(a) \).<sup>[8](https://books.physics.oregonstate.edu/GSF/discretecont.html)</sup> MathWorld describes the Kronecker delta as the discrete version of the delta function, and a quantum mechanics text calls the Dirac delta an analog of the Kronecker delta with continuous indices.<sup>[1](https://mathworld.wolfram.com/KroneckerDelta.html)</sup><sup> • </sup><sup>[14](https://library.uoh.edu.iq/admin/ebooks/50460-quantum-mechanics---concepts-and-applications.pdf)</sup>

In signal processing the Kronecker delta serves as the discrete-time impulse signal, \( \delta[n] = 1 \) for \( n = 0 \) and 0 otherwise, in contrast to the continuous-time Dirac impulse.<sup>[15](https://my.ece.utah.edu/~ece3500/Fall2015/notes/class03.html)</sup> Unlike the Dirac delta, the Kronecker delta is an ordinary function with no mathematical subtlety: a Berkeley text notes that its sifting property holds trivially because no infinite value is required.<sup>[16](https://ptolemy.berkeley.edu/eecs20/week11/impulses.html)</sup> A finite Kronecker delta pulse has [Fourier series](https://www.edgepedia.org/fourier-series) coefficients all of the same magnitude, the basis for the interpretation that an impulse contains all frequencies in equal amounts.<sup>[16](https://ptolemy.berkeley.edu/eecs20/week11/impulses.html)</sup>

## Other representations and uses

MathWorld gives a contour integral representation for integers \( m \) and \( n \),

\[ \delta_{mn} = \frac{1}{2\pi i} \oint_{\gamma} z^{m-n-1}\,dz, \]

where \( \gamma \) is a contour corresponding to the unit circle.<sup>[1](https://mathworld.wolfram.com/KroneckerDelta.html)</sup> The Wolfram Language implements the symbol as `KroneckerDelta[i, j]`, with a generalized form `KroneckerDelta[i, j, ...]` that returns 1 if and only if all arguments are equal and 0 otherwise.<sup>[1](https://mathworld.wolfram.com/KroneckerDelta.html)</sup> In mechanics and engineering coursework, the delta appears throughout index notation, for example in the equilibrium equation \( \sigma_{ij,j} = 0 \) and in expressions such as \( \sigma_{kk}\delta_{ij} \), which expands to a diagonal matrix with \( \sigma_{kk} = \sigma_{11} + \sigma_{22} + \sigma_{33} \) on the diagonal.<sup>[17](https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Mechanics_of_Materials_%28Roylance%29/07%3A_Appendices/7.02%3A_Matrix_and_Index_Notation)</sup>

## References

1. [Kronecker Delta, Wolfram MathWorld](https://mathworld.wolfram.com/KroneckerDelta.html)
2. [Kronecker symbol, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Kronecker_symbol)
3. [Kronecker delta, PlanetMath](https://planetmath.org/kroneckerdelta)
4. [Definition: Kronecker Delta, ProofWiki](https://proofwiki.org/wiki/Definition%3AKronecker_Delta)
5. [Kronecker, Leopold, Hrvatska enciklopedija](https://www.enciklopedija.hr/clanak/kronecker-leopold)
6. [Appendix B: Tensor Notation and Index Calculus, Advanced Finite Element Methods](https://ceadpx.github.io/finite-element-methods/appendices/b-tensor-notation-index-calculus.html)
7. [Lecture notes on general relativity tutorials, Oxford Theoretical Physics](https://www-thphys.physics.ox.ac.uk/people/AndreiStarinets/TUTORIALS/SR-MT-2020/sr_mt_2020_tutorial_1_notes.pdf)
8. [Compare Kronecker and Dirac Deltas, Oregon State University](https://books.physics.oregonstate.edu/GSF/discretecont.html)
9. [Unified Engineering Materials and Structures, Problem Set 5 Solution, MIT OpenCourseWare](https://ocw.mit.edu/courses/16-001-unified-engineering-materials-and-structures-fall-2021/mit16_001_f21_pset05_sol.pdf)
10. [Tensores em coordenadas, Cálculo Vetorial e Análise Tensorial, UFABC](https://danielmiranda.prof.ufabc.edu.br/livros/cvt/capitulo-10-tensores-em-coordenadas.html)
11. [The Tensor Property, An Introduction to Tensor Calculus](https://grinfeld.org/books/An-Introduction-To-Tensor-Calculus/Chapter14.html)
12. [Metric Tensor, Wolfram MathWorld](https://mathworld.wolfram.com/MetricTensor.html)
13. [Proofs of Vector Identities Using Tensors, arXiv](https://arxiv.org/abs/1406.3060)
14. [Quantum Mechanics: Concepts and Applications, University of Hail library](https://library.uoh.edu.iq/admin/ebooks/50460-quantum-mechanics---concepts-and-applications.pdf)
15. [ECE 3500: Fundamentals of Signals and Systems, University of Utah](https://my.ece.utah.edu/~ece3500/Fall2015/notes/class03.html)
16. [Impulses, EECS 20, University of California, Berkeley](https://ptolemy.berkeley.edu/eecs20/week11/impulses.html)
17. [Matrix and Index Notation, Engineering LibreTexts](https://eng.libretexts.org/Bookshelves/Mechanical_Engineering/Mechanics_of_Materials_%28Roylance%29/07%3A_Appendices/7.02%3A_Matrix_and_Index_Notation)

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