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''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 is ''isotropic'': has the same components relative to any other basis.

The Kronecker symbol is convenient in various problems of tensor calculus. For example, the determinant
$$
\left|{
\begin{array}{ccc} a^1_1 & \ldots & a^1_n \\ \vdots & \ddots & \vdots \\ a^n_1 & \ldots & a^n_n \end{array}
}\right|
$$
is equal to the sum
$$
\sum \delta^{i_1\ldots i_n}_{1\ldots n} a^1_{i_1} \cdots a^n_{i_n}
$$
where the summation is performed over all $n!$ permutations $( i_1\ldots i_n )$of the numbers $\{1,\ldots, n \}$. The ''alternant'' of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590024.png" /> is given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590025.png" /></td> </tr></table>

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Kronecker,   "Vorlesungen über die Theorie der Determinanten" , Leipzig  (1903)</TD></TR></table>

====Comment====
For the Kronecker symbol in number theory, see [[Legendre–Jacobi–Kronecker symbol‎]]

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