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).