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Mathematics for Physics

257 Kronecker Delta

A function of two integer indices that takes the value one when the indices match and zero otherwise that is used to pick matching components in sums and matrices.

definition [d] (Kronecker Delta = Kronecker Symbol) The discrete symbol \(\delta_{ij}\) defined by

  • \(\delta_{ij} = \begin{cases} 1 & \text{if } i = j \\ 0 & \text{if } i \neq j \end{cases}\) .

where

  • \(\delta_{ij}\) is the Kronecker delta.
  • \(i, j\) are discrete indices.
  • \(I\) is the identity matrix.

Note:

  • \(\delta_{ij}\) are the components of the identity matrix \(I\).
  • under contraction, \(\delta_{ij}\) acts as an index-replacement operator, for example \(b_{j}\delta_{ij} = b_{i}\).

definition [d] (Kronecker Delta = Kronecker Symbol) The symbol \(\delta_{ij}\) defined by

  • \(\delta_{ij} = 1\) if \(i = j\), and \(\delta_{ij} = 0\) if \(i \neq j\) .

where

  • \(\delta_{ij}\) is the Kronecker delta.
  • \(i, j\) are discrete indices.
  • \(I_{ik}\) are the components of the identity matrix.

Note:

  • the mixed form is written \(\delta^{i}_{\ j}\).
  • as a type \((1,1)\) tensor, \(\delta^{\mu}_{\ \nu}\) is the identity map on vectors and one-forms.
  • the components of the identity matrix satisfy \(I_{ik} = \delta_{ik}\).

257.1 Elementary Example

257.1.1 Simple

The Kronecker delta is \(1\) when the two indices match and \(0\) otherwise.

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

\[ \delta_{11} = 1,\quad \delta_{12} = 0,\quad \delta_{22} = 1 \]

where

  • \(i, j\) are integer indices.
  • \(\delta_{ij}\) is the Kronecker delta.

257.1.2 General

As a \(3 \times 3\) matrix of components, \(\delta^{i}_{\ j}\) is the identity.

\[ (\delta^{i}_{\ j}) = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \]

\[ \delta^{i}_{\ j}\, v^{j} = v^{i} \]

where

  • \(v^{j}\) are components of a vector \(v\).

257.2 References

  1. Arfken, G. B., Weber, H. J., & Harris, F. E. Mathematical Methods for Physicists, 7th ed. Elsevier / Academic Press, 2013. — Kronecker delta; identity matrix; index replacement.
  2. Carroll, S. Spacetime and Geometry. Cambridge University Press, 2021. — mixed Kronecker delta as identity map of type \((1,1)\).