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$i,j = 1,2,\ldots$. 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).
 
$i,j = 1,2,\ldots$. 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 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
 
The Kronecker symbol is convenient in various problems of tensor calculus. For example, the determinant
 
+
$$
<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/k05590020.png" /></td> </tr></table>
+
\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
 
is equal to the sum
 
+
$$
<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/k05590021.png" /></td> </tr></table>
+
\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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590022.png" /> permutations of the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590023.png" />. The alternant of the tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590024.png" /> is given by
+
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>
 
<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>

Revision as of 19:26, 13 January 2016

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 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

How to Cite This Entry:
Kronecker symbol. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kronecker_symbol&oldid=37518
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article