Write a projectivity $\tau: \mathbb{P^2} \to \mathbb{P^2}$ such that, on the affine plane, fixes the point $P=(-1,1)$ and has $r:x-y-1=0$ as invariant line.
Seen on the projective plane, defining $\infty:=x_0=0$, we have $P=[1:-1:1]$ and a point $Q \in r$, $Q=[1:2:1]$.
If the projectivity is induced by a matrix $T$, what I have to do is solve a linear system in 9 unknown given by the relations:
$T \begin{bmatrix} 1 \\ -1 \\ 1 \\ \end{bmatrix} $=$ \begin{bmatrix} \psi \\ -\psi \\ \psi \\ \end{bmatrix} $ and then, $T^{-T} \begin{bmatrix} 1 \\ 2 \\ 1 \\ \end{bmatrix} $ =$ \begin{bmatrix} \mu \\ 2\mu \\ \mu \\ \end{bmatrix} $
The first one is quite symple, but inverting a parametric matrix is really muddled.
Do you have any suggestion about? Thank you
The identity map satisfies the conditions that you’ve given in your question, but I suppose that whoever posed this problem meant for you to find some non-affine transformation.
The invariant conditions still leave quite a few of the original eight degrees of freedom, so trying to solve systems of linear equations for the elements of $T$ doesn’t seem like a fruitful way to go about this. Instead, you can construct the matrix directly: Choose two pairs of points $Q_1$, $Q_2$ and $Q_1'$, $Q_2'$ on the line, and a set of three non-zero weights $w_0$, $w_1$ and $w_2$. The matrix $$T = \begin{bmatrix}P&Q_1'&Q_2'\end{bmatrix} \operatorname{diag}(w_0,w_1,w_2)\,\begin{bmatrix}P&Q_1&Q_2\end{bmatrix}^{-1}$$ (using homogeneous coordinates of the points) maps $P$ to itself and maps $\lambda Q_1+\mu Q_2$ to $w_1\lambda Q_1'+w_2\mu Q_2'$, i.e., maps a point on $r$ to a point on $r$. For most choices, $T$ will not represent an affine map, but you should double-check, anyway.