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''null system''
 
''null system''
  
An involutory correlation of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z0992501.png" />-dimensional projective space with an anti-symmetric operator. Suppose that the null system has the form
+
An involutory correlation of an $  n $-dimensional projective space with an anti-symmetric operator. Suppose that the null system has the form
  
<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/z/z099/z099250/z0992502.png" /></td> </tr></table>
+
$$
 +
{}  ^  \prime  u  = Ax .
 +
$$
  
Then the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z0992503.png" />, which is
+
Then the scalar product $  {}  ^  \prime  ux $,  
 +
which is
  
<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/z/z099/z099250/z0992504.png" /></td> </tr></table>
+
$$
 +
( x, Ax)  = -( x,Ax),
 +
$$
  
 
vanishes.
 
vanishes.
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Multi-dimensional spaces" , Moscow  (1966)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. Rozenfel'd,  "Multi-dimensional spaces" , Moscow  (1966)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
A null system is also called null polarity, a symplectic polarity or a symplectic correlation. As is clear from the above, it is a [[Polarity|polarity]] such that every point lies in its own polar hyperplane.
 
A null system is also called null polarity, a symplectic polarity or a symplectic correlation. As is clear from the above, it is a [[Polarity|polarity]] such that every point lies in its own polar hyperplane.
  
In projective <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z0992505.png" />-space, a correlation is a dualizing transformation (cf. [[Correlation|Correlation]]), taking points, lines and planes into planes, lines and points, while preserving incidence in accordance with the principle of duality. If every range of points on a line is transformed into a projectively related pencil of planes through the new line, the correlation is said to be projective. There is a unique projective correlation transforming five given points, no four in a plane, into five given planes, no four through a point.
+
In projective $  3 $-space, a correlation is a dualizing transformation (cf. [[Correlation|Correlation]]), taking points, lines and planes into planes, lines and points, while preserving incidence in accordance with the principle of duality. If every range of points on a line is transformed into a projectively related pencil of planes through the new line, the correlation is said to be projective. There is a unique projective correlation transforming five given points, no four in a plane, into five given planes, no four through a point.
  
A polarity is a projective correlation of period two (cf. [[Polarity|Polarity]]). In other words, it transforms each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z0992506.png" /> into a plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z0992507.png" /> and each point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z0992508.png" /> into a plane through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z0992509.png" />. One kind of polarity transforms each point on a quadric surface into the tangent plane at that point. Another kind, a null polarity, transforms every point of space into a plane through that point. It may be described as the unique projective correlation that transforms five points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925010.png" /> (no four collinear) into the respective planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925011.png" />. The line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925012.png" /> is self-polar, since it is the line of intersection of the polar planes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925016.png" />. In fact, all the lines through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925017.png" /> in its polar plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925018.png" /> are self-polar: there is a flat pencil of such lines in every plane, and the set of all self-polar lines is a linear complex.
+
A polarity is a projective correlation of period two (cf. [[Polarity|Polarity]]). In other words, it transforms each point $  A $
 +
into a plane $  \alpha $
 +
and each point of $  \alpha $
 +
into a plane through $  A $.  
 +
One kind of polarity transforms each point on a quadric surface into the tangent plane at that point. Another kind, a null polarity, transforms every point of space into a plane through that point. It may be described as the unique projective correlation that transforms five points $  A, B, C, D, E $ (no four collinear) into the respective planes $  EAB , ABC, BCD , CDE , DEA $.  
 +
The line $  AB $
 +
is self-polar, since it is the line of intersection of the polar planes $  EAB $
 +
and $  ABC $
 +
of $  A $
 +
and $  B $.  
 +
In fact, all the lines through $  A $
 +
in its polar plane $  EAB $
 +
are self-polar: there is a flat pencil of such lines in every plane, and the set of all self-polar lines is a linear complex.
  
In terms of projective coordinates, a null polarity takes each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925019.png" /> to the plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925020.png" />, where
+
In terms of projective coordinates, a null polarity takes each point $  ( x _ {0} , x _ {1} , x _ {2} , x _ {3} ) $
 +
to the plane $  [ X _ {0} , X _ {1} , X _ {2} , X _ {3} ] $,  
 +
where
  
<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/z/z099/z099250/z09925021.png" /></td> </tr></table>
+
$$
 +
X _  \mu  = \sum _ {\nu = 0 } ^ { 3 }  c _ {\mu \nu }  x _  \nu  $$
  
and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925023.png" />. In terms of the [[Plücker coordinates|Plücker coordinates]] of a line, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/z/z099/z099250/z09925024.png" />, where
+
and $  c _ {\mu \nu }  + c _ {\nu \mu }  = 0 $
 +
and $  c _ {01} c _ {23} + c _ {02} c _ {31} + c _ {03} c _ {12} \neq 0 $.  
 +
In terms of the [[Plücker coordinates|Plücker coordinates]] of a line, $  \{ p _ {01} , p _ {02} , p _ {03} , p _ {23} , p _ {31} , p _ {12} \} $,  
 +
where
  
<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/z/z099/z099250/z09925025.png" /></td> </tr></table>
+
$$
 +
p _ {\mu \nu }  + p _ {\nu \mu }  = 0 \ \
 +
\textrm{ and } \ \
 +
p _ {01} p _ {23} + p _ {02} p _ {31} +
 +
p _ {03} p _ {12}  = 0 ,
 +
$$
  
 
the linear complex of self-polar lines in the null polarity has the equation
 
the linear complex of self-polar lines in the null polarity has the equation
  
<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/z/z099/z099250/z09925026.png" /></td> </tr></table>
+
$$
 +
\sum \sum c _ {\mu \nu }  p _ {\mu \nu }  = 0.
 +
$$
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.G.C. von Staudt,  "Beiträge zur Geometrie der Lage" , Korn , Nürnberg  (1847)  pp. 60–69; 190–196</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1965)  pp. 65–70</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Pedoe,  "Geometry: a comprehensive course" , Dover, reprint  (1988)  pp. §85.5</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.G.C. von Staudt,  "Beiträge zur Geometrie der Lage" , Korn , Nürnberg  (1847)  pp. 60–69; 190–196</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  H.S.M. Coxeter,  "Non-Euclidean geometry" , Univ. Toronto Press  (1965)  pp. 65–70</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Pedoe,  "Geometry: a comprehensive course" , Dover, reprint  (1988)  pp. §85.5</TD></TR></table>

Latest revision as of 02:25, 16 June 2022


null system

An involutory correlation of an $ n $-dimensional projective space with an anti-symmetric operator. Suppose that the null system has the form

$$ {} ^ \prime u = Ax . $$

Then the scalar product $ {} ^ \prime ux $, which is

$$ ( x, Ax) = -( x,Ax), $$

vanishes.

References

[1] B.A. Rozenfel'd, "Multi-dimensional spaces" , Moscow (1966) (In Russian)

Comments

A null system is also called null polarity, a symplectic polarity or a symplectic correlation. As is clear from the above, it is a polarity such that every point lies in its own polar hyperplane.

In projective $ 3 $-space, a correlation is a dualizing transformation (cf. Correlation), taking points, lines and planes into planes, lines and points, while preserving incidence in accordance with the principle of duality. If every range of points on a line is transformed into a projectively related pencil of planes through the new line, the correlation is said to be projective. There is a unique projective correlation transforming five given points, no four in a plane, into five given planes, no four through a point.

A polarity is a projective correlation of period two (cf. Polarity). In other words, it transforms each point $ A $ into a plane $ \alpha $ and each point of $ \alpha $ into a plane through $ A $. One kind of polarity transforms each point on a quadric surface into the tangent plane at that point. Another kind, a null polarity, transforms every point of space into a plane through that point. It may be described as the unique projective correlation that transforms five points $ A, B, C, D, E $ (no four collinear) into the respective planes $ EAB , ABC, BCD , CDE , DEA $. The line $ AB $ is self-polar, since it is the line of intersection of the polar planes $ EAB $ and $ ABC $ of $ A $ and $ B $. In fact, all the lines through $ A $ in its polar plane $ EAB $ are self-polar: there is a flat pencil of such lines in every plane, and the set of all self-polar lines is a linear complex.

In terms of projective coordinates, a null polarity takes each point $ ( x _ {0} , x _ {1} , x _ {2} , x _ {3} ) $ to the plane $ [ X _ {0} , X _ {1} , X _ {2} , X _ {3} ] $, where

$$ X _ \mu = \sum _ {\nu = 0 } ^ { 3 } c _ {\mu \nu } x _ \nu $$

and $ c _ {\mu \nu } + c _ {\nu \mu } = 0 $ and $ c _ {01} c _ {23} + c _ {02} c _ {31} + c _ {03} c _ {12} \neq 0 $. In terms of the Plücker coordinates of a line, $ \{ p _ {01} , p _ {02} , p _ {03} , p _ {23} , p _ {31} , p _ {12} \} $, where

$$ p _ {\mu \nu } + p _ {\nu \mu } = 0 \ \ \textrm{ and } \ \ p _ {01} p _ {23} + p _ {02} p _ {31} + p _ {03} p _ {12} = 0 , $$

the linear complex of self-polar lines in the null polarity has the equation

$$ \sum \sum c _ {\mu \nu } p _ {\mu \nu } = 0. $$

References

[a1] K.G.C. von Staudt, "Beiträge zur Geometrie der Lage" , Korn , Nürnberg (1847) pp. 60–69; 190–196
[a2] H.S.M. Coxeter, "Non-Euclidean geometry" , Univ. Toronto Press (1965) pp. 65–70
[a3] D. Pedoe, "Geometry: a comprehensive course" , Dover, reprint (1988) pp. §85.5
How to Cite This Entry:
Zero system. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Zero_system&oldid=15550
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article