Zero system

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