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The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k0559001.png" /> defined by
+
''Kronecker delta''
  
<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/k0559002.png" /></td> </tr></table>
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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).
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k0559003.png" />. When <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k0559004.png" />, the Kronecker symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k0559005.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k0559006.png" /> components, and the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k0559007.png" /> 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 may be generalized, considering instead a set of quantities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k0559008.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k0559009.png" /> integer (upper and lower) indices, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590011.png" />, equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590012.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590013.png" />) if the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590014.png" /> is an even (odd) permutation of the distinct indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590015.png" /> and zero otherwise. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590016.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590017.png" /> often denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590018.png" />) are called the components of the Kronecker symbol. An [[Affine tensor|affine tensor]] of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/k/k055/k055900/k05590019.png" /> 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
 
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
 +
$$
 +
\sum \delta^{i_1\ldots i_n}_{1\ldots n} a^1_{i_1} \cdots a^n_{i_n}
 +
$$
 +
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 $\{ a^{\alpha_1\ldots\alpha_n} : 1 \le \alpha_i \le n \}$ is given by
 +
$$
 +
a^{[\alpha_1,\ldots,\alpha_p] } = \frac{1}{p!} \sum \delta^{\alpha_1\ldots \alpha_p}_{i_1\ldots i_p} a^{i_1\ldots i_p} \ .
 +
$$
  
<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>
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====References====
 
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Kronecker,  "Vorlesungen über die Theorie der Determinanten" , Leipzig  (1903)</TD></TR></table>
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
 
  
<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>
+
====Comment====
 +
For the Kronecker symbol in number theory, see [[Legendre–Jacobi–Kronecker symbol‎]]
  
====References====
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{{TEX|done}}
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L. Kronecker,  "Vorlesungen über die Theorie der Determinanten" , Leipzig  (1903)</TD></TR></table>
 

Latest revision as of 19:43, 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 $\{ a^{\alpha_1\ldots\alpha_n} : 1 \le \alpha_i \le n \}$ is given by $$ a^{[\alpha_1,\ldots,\alpha_p] } = \frac{1}{p!} \sum \delta^{\alpha_1\ldots \alpha_p}_{i_1\ldots i_p} a^{i_1\ldots i_p} \ . $$

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=18809
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article