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''skew symmetry, anti-symmetry, alternance''
 
''skew symmetry, anti-symmetry, alternance''
  
One of the operations of tensor algebra, yielding a tensor that is skew-symmetric (over a group of indices) from a given tensor. Alternation is always effected over a few superscripts or over a few subscripts. A tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120601.png" /> with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120602.png" /> is the result of alternation of a tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120603.png" /> with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120604.png" />, for example, over superscripts, over a group of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120605.png" /> if
+
One of the operations of tensor algebra, yielding a tensor that is skew-symmetric (over a group of indices) from a given tensor. Alternation is always effected over a few superscripts or over a few subscripts. A tensor $  A $
 +
with components $  \{ a _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} },  1 \leq  i _  \nu  , j _  \mu  \leq  n \} $
 +
is the result of alternation of a tensor $  T $
 +
with components $  \{ t _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} },  1 \leq  i _  \nu  ,  j _  \mu  \leq  n \} $,  
 +
for example, over superscripts, over a group of indices $  I = (i _ {1} \dots i _ {m} ) $
 +
if
  
<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/a/a012/a012060/a0120606.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
a _ {j _ {1}  \dots j _ {q} } ^ {i _ {1} \dots i _ {p} }  = \
  
The summation is conducted over all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120607.png" /> rearrangements (permutations) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120608.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a0120609.png" />, the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206010.png" /> being <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206011.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206012.png" />, depending on whether the respective rearrangement is even or odd. Alternation over a group of subscripts is defined in a similar manner.
+
\frac{1}{m!}
 +
\sum _ {I \rightarrow \alpha }
 +
\sigma ( I , \alpha ) t _ {j _ {1}  \dots j _ {q} } ^ {\alpha _ {1} \dots \alpha _ {m} i _ {m+1} \dots i _ {p} } .
 +
$$
 +
 
 +
The summation is conducted over all $  m! $
 +
rearrangements (permutations) $  \alpha = ( \alpha _ {1} \dots \alpha _ {m} ) $
 +
of $  I $,  
 +
the number $  \sigma (I, \alpha ) $
 +
being $  +1 $
 +
or $  -1 $,  
 +
depending on whether the respective rearrangement is even or odd. Alternation over a group of subscripts is defined in a similar manner.
  
 
Alternation over a group of indices is denoted by enclosing the indices between square brackets. Secondary indices inside the square brackets are separated by vertical strokes. For instance:
 
Alternation over a group of indices is denoted by enclosing the indices between square brackets. Secondary indices inside the square brackets are separated by vertical strokes. For instance:
  
<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/a/a012/a012060/a01206013.png" /></td> </tr></table>
+
$$
 +
t _ {[ 4 | 23 | 1 ] }  = \
  
Successive alternation over groups of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206016.png" />, coincides with alternation over the group of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206017.png" />:
+
\frac{1}{2!}
 +
[ t _ {4 2 3 1 }  -
 +
t _ {1 2 3 4 }  ].
 +
$$
  
<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/a/a012/a012060/a01206018.png" /></td> </tr></table>
+
Successive alternation over groups of indices  $  I _ {1} $
 +
and  $  I _ {2} $,
 +
$  I _ {1} \subset  I _ {2} $,
 +
coincides with alternation over the group of indices  $  I _ {2} $:
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206019.png" /> is the dimension of the vector space on which the tensor is defined, alternation by a group of indices the number of which is larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206020.png" /> will always produce the zero tensor. Alternation over a given group of indices of a tensor which is symmetric with respect to this group (cf. [[Symmetrization (of tensors)|Symmetrization (of tensors)]]) also yields the zero tensor. A tensor that remains unchanged under alternation over a given group of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206021.png" /> is called skew-symmetric or alternating over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012060/a01206022.png" />. Interchanging any pair of such indices changes the sign of the component of the tensor.
+
$$
 +
t _ {[ i _ {1}  \dots [ i _ {k} \dots i _ {l} ]
 +
\dots i _ {q} ] }  = t _ {[ i _ {1}  \dots i _ {q} ] } .
 +
$$
 +
 
 +
If  $  n $
 +
is the dimension of the vector space on which the tensor is defined, alternation by a group of indices the number of which is larger than $  n $
 +
will always produce the zero tensor. Alternation over a given group of indices of a tensor which is symmetric with respect to this group (cf. [[Symmetrization (of tensors)|Symmetrization (of tensors)]]) also yields the zero tensor. A tensor that remains unchanged under alternation over a given group of indices $  I $
 +
is called skew-symmetric or alternating over $  I $.  
 +
Interchanging any pair of such indices changes the sign of the component of the tensor.
  
 
The operation of tensor alternation, together with the operation of symmetrization, is employed to decompose a tensor into simpler tensors.
 
The operation of tensor alternation, together with the operation of symmetrization, is employed to decompose a tensor into simpler tensors.
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Alternation is also employed to produce sign-alternating sums of the form (*) with multi-indexed terms. For instance, a determinant with elements which commute under multiplication can be computed by the formulas
 
Alternation is also employed to produce sign-alternating sums of the form (*) with multi-indexed terms. For instance, a determinant with elements which commute under multiplication can be computed by the formulas
  
<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/a/a012/a012060/a01206023.png" /></td> </tr></table>
+
$$
 +
\left |
 +
 
 +
\begin{array}{ccc}
 +
a _ {1}  ^ {1}  &\dots  &a _ {n}  ^ {1}  \\
 +
.  &{}  & .  \\
 +
. &{}  & . \\
 +
a _ {1}  ^ {n}  &\dots  &a _ {n}  ^ {n}  \\
 +
\end{array}
 +
\
 +
\right |  =  n ! a _ {1} ^ {[1{} } \dots a _ {n} ^ { {}n] } =
 +
$$
  
<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/a/a012/a012060/a01206024.png" /></td> </tr></table>
+
$$
 +
= \
 +
n ! a _ {[1{} }  ^ {1} \dots a _ { {}n] }  ^ {n}  = \
 +
a _ {[1{} }  ^ {[1{} } \dots a _ { {}n] }  ^ { {}n] } .
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Shirokov,  "Tensor calculus. Tensor algebra" , Kazan'  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.V. Beklemishev,  "A course of analytical geometry and linear algebra" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Schouten,  "Tensor analysis for physicists" , Cambridge Univ. Press  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.V. Efimov,  E.R. Rozendorn,  "Linear algebra and multi-dimensional geometry" , Moscow  (1970)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P.A. Shirokov,  "Tensor calculus. Tensor algebra" , Kazan'  (1961)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.V. Beklemishev,  "A course of analytical geometry and linear algebra" , Moscow  (1971)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.A. Schouten,  "Tensor analysis for physicists" , Cambridge Univ. Press  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.V. Efimov,  E.R. Rozendorn,  "Linear algebra and multi-dimensional geometry" , Moscow  (1970)  (In Russian)</TD></TR></table>

Latest revision as of 16:10, 1 April 2020


skew symmetry, anti-symmetry, alternance

One of the operations of tensor algebra, yielding a tensor that is skew-symmetric (over a group of indices) from a given tensor. Alternation is always effected over a few superscripts or over a few subscripts. A tensor $ A $ with components $ \{ a _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} }, 1 \leq i _ \nu , j _ \mu \leq n \} $ is the result of alternation of a tensor $ T $ with components $ \{ t _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} }, 1 \leq i _ \nu , j _ \mu \leq n \} $, for example, over superscripts, over a group of indices $ I = (i _ {1} \dots i _ {m} ) $ if

$$ \tag{* } a _ {j _ {1} \dots j _ {q} } ^ {i _ {1} \dots i _ {p} } = \ \frac{1}{m!} \sum _ {I \rightarrow \alpha } \sigma ( I , \alpha ) t _ {j _ {1} \dots j _ {q} } ^ {\alpha _ {1} \dots \alpha _ {m} i _ {m+1} \dots i _ {p} } . $$

The summation is conducted over all $ m! $ rearrangements (permutations) $ \alpha = ( \alpha _ {1} \dots \alpha _ {m} ) $ of $ I $, the number $ \sigma (I, \alpha ) $ being $ +1 $ or $ -1 $, depending on whether the respective rearrangement is even or odd. Alternation over a group of subscripts is defined in a similar manner.

Alternation over a group of indices is denoted by enclosing the indices between square brackets. Secondary indices inside the square brackets are separated by vertical strokes. For instance:

$$ t _ {[ 4 | 23 | 1 ] } = \ \frac{1}{2!} [ t _ {4 2 3 1 } - t _ {1 2 3 4 } ]. $$

Successive alternation over groups of indices $ I _ {1} $ and $ I _ {2} $, $ I _ {1} \subset I _ {2} $, coincides with alternation over the group of indices $ I _ {2} $:

$$ t _ {[ i _ {1} \dots [ i _ {k} \dots i _ {l} ] \dots i _ {q} ] } = t _ {[ i _ {1} \dots i _ {q} ] } . $$

If $ n $ is the dimension of the vector space on which the tensor is defined, alternation by a group of indices the number of which is larger than $ n $ will always produce the zero tensor. Alternation over a given group of indices of a tensor which is symmetric with respect to this group (cf. Symmetrization (of tensors)) also yields the zero tensor. A tensor that remains unchanged under alternation over a given group of indices $ I $ is called skew-symmetric or alternating over $ I $. Interchanging any pair of such indices changes the sign of the component of the tensor.

The operation of tensor alternation, together with the operation of symmetrization, is employed to decompose a tensor into simpler tensors.

The product of two tensors with subsequent alternation over all indices is called an alternated product (exterior product).

Alternation is also employed to produce sign-alternating sums of the form (*) with multi-indexed terms. For instance, a determinant with elements which commute under multiplication can be computed by the formulas

$$ \left | \begin{array}{ccc} a _ {1} ^ {1} &\dots &a _ {n} ^ {1} \\ . &{} & . \\ . &{} & . \\ a _ {1} ^ {n} &\dots &a _ {n} ^ {n} \\ \end{array} \ \right | = n ! a _ {1} ^ {[1{} } \dots a _ {n} ^ { {}n] } = $$

$$ = \ n ! a _ {[1{} } ^ {1} \dots a _ { {}n] } ^ {n} = \ a _ {[1{} } ^ {[1{} } \dots a _ { {}n] } ^ { {}n] } . $$

References

[1] P.A. Shirokov, "Tensor calculus. Tensor algebra" , Kazan' (1961) (In Russian)
[2] D.V. Beklemishev, "A course of analytical geometry and linear algebra" , Moscow (1971) (In Russian)
[3] J.A. Schouten, "Tensor analysis for physicists" , Cambridge Univ. Press (1951)
[4] N.V. Efimov, E.R. Rozendorn, "Linear algebra and multi-dimensional geometry" , Moscow (1970) (In Russian)
How to Cite This Entry:
Alternation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alternation&oldid=45091
This article was adapted from an original article by L.P. Kuptsov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article