# Collineation

projective

A projective transformation (projective isomorphism) of a projective space $\Pi _ {n}$ that is representable as the product of a finite number of perspectivities (cf. Perspective); if $\nu$ is a projective collineation, then for any subspace $S _ {q}$ there exists a product $\pi$ of not more than $q - 1$ perspectivities such that $\nu ( S _ {p} ) = \pi ( S _ {p} )$ for any $S _ {p} \subset S _ {q}$. For example, a projective transformation that leaves each point of some straight line fixed is a collineation, this is, a homology (in the narrow sense).

Let $\Pi _ {n}$ be interpreted as the collection of subspaces of the linear space $A _ {n + 1 } ^ {e} ( K)$ over a skew-field $K$. Then in order that a projective transformation be a projective collineation, it is necessary and sufficient that it be induced by a linear transformation of $A _ {n + 1 } ^ {e} ( K)$. The collection of all projective collineations forms a subgroup $G _ {0}$ of the group of projective transformations $G$ which is a normal subgroup of $G$.

The projective collineations exhaust all the projective transformations if and only if every automorphism of the skew-field $K$ is inner. A field possesses this property if and only if any of its automorphisms is the identity, such as, for example, the field of real numbers $\mathbf R$. The complex field $\mathbf C$ does not possess this property, whereas every automorphism of the skew-field of quaternions $\mathbf H$ is inner.

If $K$ is a non-commutative skew-field, then there exists a non-trivial projective collineation that leaves every point of a given simplex fixed. Each simplex can be mapped onto any other simplex by precisely one projective collineation, if and only if $K$ is a field (the second fundamental theorem of projective geometry).

There is a large amount of confusion in the literature on projective geometry about the terminology for the different kinds of transformations. The transformation defined above, a projective collineation, sometimes also called a projectivity, is usually defined as $any$ transformation that is the product of a finite number of perspectivities. Some authors use the term projectivity, however, only for mappings between lines that are the product of a finite number of perspectivities. A projective transformation, usually called a projective isomorphism, or in older literature simply a collineation, is a bijection between projective spaces that preserves incidence.