Kronecker symbol

Kronecker delta

The number $\delta^i_j$ defined by $$ \delta^i_j = \begin{cases} 1 & \text{if}\, i = j \\ 0 & \text{if}\, i \ne j \end{cases}\ \ , $$ $i,j = 1,2,\ldots$. When $1 \le i,j \le n$, the Kronecker symbol $\delta^i_j$ has $n^2$ components, and the matrix $(\delta^i_j)$ is the unit matrix. The Kronecker symbol was first used by L. Kronecker (1866).

The Kronecker symbol may be generalized, considering instead a set of quantities $\delta^{i_1\ldots i_p}_{j_1\ldots j_p}$ with $2p$ integer (upper and lower) indices, $i_\alpha\,,j_\beta = 1,\ldots,n$, equal to $+1$ (or $-1$) if the sequence $(i_1\ldots i_p)$ is an even (odd) permutation of the distinct indices $(j_1\ldots j_p)$ and zero otherwise. The numbers $\delta^{i_1\ldots i_p}_{j_1\ldots j_p}$ (when $p \ge 2$ often denoted by $\epsilon^{i_1\ldots i_p}_{j_1\ldots j_p}$) are called the components of the Kronecker symbol. An affine tensor of type $(p,p)$ whose components relative to some basis are equal to the components of the Kronecker symbol has the same components relative to any other basis.

The Kronecker symbol is convenient in various problems of tensor calculus. For example, the determinant

is equal to the sum

where the summation is performed over all permutations of the numbers . The alternant of the tensor is given by

References

[1]	L. Kronecker, "Vorlesungen über die Theorie der Determinanten" , Leipzig (1903)

Comment

For the Kronecker symbol in number theory, see Legendre–Jacobi–Kronecker symbol‎