Show that if v $\in$ V is an eigenvector of T, then [v] $\in$ P(V) is a fixed point of the projective transformation $\tau$ defined by T.

Question

Let T : V $\rightarrow$ V be an invertible transformation. Show that if v $\in$ V is an eigenvector of T, then [v] $\in$ P(V) is a fixed point of the projective transformation $\tau$ defined by T. Prove that any projective transformation of P$^{2}$(R) has a fixed point.

What is a good way to prove this problem? Thanks a lot.

Answer 1

If you can't see the answer to the first half of your question, then you really have not understood the terms being used. Consider:

1. The points of $P(V)$ are the lines of $V$.

2. You make a linear transformation of $V$ into a projective transformation$T$ on $P(V)$ by mapping the projective point $[v]$ to the projective point $[T(v)]$.

3. An eigenvector is just one that is scaled by the transformation, and therefore generates the same 1 dimensional space. Think about what that means in conjunction with the above two points.

Astor the second half of your question, represent your $P^2(\Bbb R)$ transformation as a 3 by 3 matrix over $\Bbb R$. Considered as a linear transformation, it has a real eigenvalue (since a polynomial over the reals of degree 3 must have a real root).