# central collineations

## Definitions and general properties

###### Definition 1.

A collineation^{} of a finite dimensional projective geometry^{} is a *central collineation*
if there is a hyperplane^{} of points fixed by the collineation.

Recall that collineations send any three collinear points to three collinear points. Thus
if a collineation fixes more than a hyperplane of points then it in fact fixes all the points
of the geometry^{} and so it is the identity map^{}. Therefore a central collineation can be
viewed the simplest of the non-identity collineations.

###### Theorem 2.

Every collineation of a finite dimensional projective geometry of dimension^{} $n\mathrm{>}\mathrm{1}$ is
a product^{} of at most $n$ central collineations. In particular, the automorphism group^{}
of a projective geometry of dimension $n\mathrm{>}\mathrm{1}$ is generated by central collineations.

Suppose that a central collineation is not the identity^{}. Then the hyperplane of fixed points
is unique and receives the title of *the axis of the central collineation*. There is
one further important result which justifies the name “central”.

###### Proposition 3.

Given a non-identity central collineation $f$, there is a
unique point $C$ such that for all other points $P$, it follows that $C$, $P$ and $P\mathit{}f$ are
collinear^{}.

The point $C$ determined by Proposition^{} 3 is called the *center*
of the non-identity central collineation. It is possible for the center to lie on the axis.

## Central collineations in coordinates

Suppose we have a projective geometry of dimension $n>2$, that is, we exclude now
the case of projective lines and planes. The the geometry can be coordinatized through
so that we may regard the projective geometry as the lattice^{}
of subspaces^{} of a vector space^{} $V$ of dimension $n+1$ over a division ring $\mathrm{\Delta}$.
Following the fundamental theorem of projective geometry^{} we further know that
every collineation is induced by a semi-linear transformation of $V$. So it is possible
to explore central collineations as semi-linear transformations.

Every hyperplane is a kernel of some linear functional^{}, so we let $\phi :V\to \mathrm{\Delta}$ be a linear functional of $V$ with $H\phi =0$. Furthermore, we fix $v\in V$ so that
$v\phi =1$ (which implieas also that $v\notin H$). Hence, for each $u\in V$,
$u=(u-(u\phi )v)+(u\phi )v$ where $u-(u\phi )v\in H$ and $(u\phi )v\in \u27e8v\u27e9$.

Let $f\in {\mathrm{GL}}_{\mathrm{\Delta}}(V)$ such that $f$ induces a central collineation $\stackrel{~}{f}$ on $PG(V)$ with axis $H\le V$. As every scalar multiple of $f$ induces the same collineation of $PG(V)$, we may assume that $f$ is the identity on $H$. Using the decomposition given by $\phi $ we have

$$uf=((u-(u\phi )v)+(u\phi )v)f=(u-(u\phi )v)+(u\phi )vf,u\in V.$$ |

Hence

$$uf=u+(u\phi )\widehat{v},\widehat{v}:=vf-v.$$ |

Suppose instead that $\phi $ is any linear functional of $V$. Then select some $\widehat{v}\in V$ such that $\widehat{v}\phi \ne -1$. Then

$$ug:=u+(u\phi )\widehat{v}$$ |

fixes all the points of $\mathrm{ker}\phi $ so $g$ induces a central collineation.

If we wish to do the same without appealing to linear functionals, we may select a basis $\{{v}_{1},\mathrm{\dots},{v}_{n+1}\}$ such that $H=\u27e8{v}_{1},\mathrm{\dots},{v}_{n}\u27e9$ and ${v}_{n+1}\phi =1$. As $f$ is selected to be the identity on $H$ we have so far specified $f$ by the matrix:

$$\left[\begin{array}{cccc}\hfill 1\hfill & \hfill \mathrm{\cdots}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \hfill & \hfill \hfill \\ \hfill 0\hfill & \hfill \hfill & \hfill 1\hfill & \hfill \hfill \\ \hfill {a}_{1}\hfill & \hfill {a}_{2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{n+1}\hfill \end{array}\right]$$ |

in the basis $\{{v}_{1},\mathrm{\dots},{v}_{n},{v}_{n+1}\}$.

Title | central collineations |

Canonical name | CentralCollineations |

Date of creation | 2013-03-22 16:03:13 |

Last modified on | 2013-03-22 16:03:13 |

Owner | Algeboy (12884) |

Last modified by | Algeboy (12884) |

Numerical id | 8 |

Author | Algeboy (12884) |

Entry type | Definition |

Classification | msc 51A10 |

Classification | msc 51A05 |

Related topic | Perspectivity^{} |

Defines | transvection |

Defines | center |

Defines | axis |

Defines | central collineation |