    Next: Projective Invariants & the Up: Linear Algebra and Homogeneous Previous: Lines in the Plane

## The Fixed Points of a Collineation

A point A is fixed by a collineation with matrix H exactly when , i.e. for some scalar . In other words, A must be a right eigenvector of H. Since an matrix typically has n+1 distinct eigenvalues, a collineation in typically has n+1 fixed points, although some of these may be complex.

H maps the line through any two fixed points A and B onto itself: . In addition, if A and B have the same eigenvalue ( ), is also an eigenvector and the entire line AB is pointwise invariant. In fact, the pointwise fixed subspaces of under H correspond exactly to the eigenspaces of H's repeated eigenvalues (if any).

Exercise 2.5   : Show that the matrix associated with a plane translation has a triple eigenvalue, but that the corresponding eigenspace is only two dimensional. Provide a geometric interpretation of this.

Exercise 2.6   : In 3D Euclidean space, consider a rotation by angle about the axis. Find the eigenvalues and eigenspaces, prove algebraically that the rotation axis is pointwise invariant, and show that in addition the circular points'' with complex coordinates and are fixed.    Next: Projective Invariants & the Up: Linear Algebra and Homogeneous Previous: Lines in the Plane
Bill Triggs
1998-11-13