COURSE OF PROJECTIVE GEOMETRY

§ 16:

*O63* Let λ(x_{1},x_{2},x_{3}) be a fixed point (__x__ ≠ __0__).

We find μ=3 en x_{1} = x_{3} (axis) or μ=2 and (x_{1},x_{2},x_{3}) = λ(1/2,0,1) (centre).

The matrix is non-singular because 0 is not an eigenvalue.

*O64* Let *A* be a regular transformation of ℜ^{3} that induces φ. We construct the matrix of *A* on the natural base.
*A*(0,1,1) = σ(1,0,0), *A*(1,0,-1) = τ(1,1,0), *A*(0,1,0) = ρ(1,1,1), *A*(-2,2,3) = ω(1,0,1).

By sweeping we find
*A*(0,0,1) = (σ-ρ, -ρ, -ρ), *A*(1,0,0) = (τ+σ-ρ, τ-ρ, -ρ), *A*(0,1,0) = (ρ,ρ,ρ);
*A*(-2,2,3) = (ρ+σ-2τ, ρ-2τ, ρ) = (ω, 0, ω).

We find ρ = σ = 2τ and the matrix becomes