Inverse Collineations of Projective Planes

# Inverse Collineations of Projective Planes

Recall from the Collineations of Projective Planes page that if $F$ is a field, $\mathbb{P}^2(F)$ is the projective plane over $F$, and $M$ is a $3 \times 3$ invertible matrix, then a collineation of $\mathbb{P}^2(F)$ is the bijective function $\phi_M : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ defined for all $\mathbf{x} \in \mathbb{P}^2(F)$ by $\phi_M(\mathbf{x}) = \mathbf{x}M$.

We already verified that $\phi_M$ is bijective, so we can define an inverse function $\phi_M^{-1} : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$.

 Definition: If $\phi_M : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ is the collineation $\phi_M(\mathbf{x}) = \mathbf{x}M$ then the corresponding Inverse Collineation is the bijective function $\phi_M^{-1} : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ defined by $\phi_M^{-1} (\mathbf{x}) = \mathbf{x}M^{-1}$.

Notice that for all $\mathbf{x} \in \mathbb{P}^2$ we have that:

(1)
\begin{align} \quad \phi_M^{-1} (\phi_M(\mathbf{x})) = \phi_M^{-1} (\mathbf{x}M) = \mathbf{x}MM^{-1} = \mathbf{x}I = \mathbf{x} \end{align}
(2)
\begin{align} \quad \phi_M(\phi_M^{-1}(\mathbf{x})) = \phi_M(\mathbf{x}M^{-1}) = \mathbf{x}M^{-1}M = \mathbf{x}I = \mathbf{x} \end{align}

Indeed we see that $\phi_M^{-1}$ is the inverse of $\phi_M$.

For example, consider the field $\mathbb{R}$ and the projective plane $\mathbb{P}^2(\mathbb{R})$. Also consider the following $3 \times 3$ matrix $M = \begin{bmatrix} 1 & 2 & 3\\ 2 & 0 & 2\\ 1 & 3 & 1 \end{bmatrix}$ that we saw earlier. We've already noted that $\det (M) = 12 \neq 0$ so the inverse of $M$ exists, and it's not hard to see that:

(3)
\begin{align} \quad M^{-1} = \begin{bmatrix} -\frac{1}{2} & \frac{7}{12} & \frac{1}{3}\\ 0 & -\frac{1}{6} & \frac{1}{3}\\ \frac{1}{2} & -\frac{1}{12} & -\frac{1}{3} \end{bmatrix} \end{align}

We saw earlier that for $\mathbf{x} = [2, 3, 1]$ that $\phi_M(\mathbf{x}) = [9, 7, 13]$ and so:

(4)
\begin{align} \quad \phi_M^{-1}(\phi_M(\mathbf{x})) = \phi_M(\mathbf{x})M^{-1} = \begin{bmatrix} 9 & 7 & 13 \end{bmatrix} \begin{bmatrix} -\frac{1}{2} & \frac{7}{12} & \frac{1}{3}\\ 0 & -\frac{1}{6} & \frac{1}{3}\\ \frac{1}{2} & -\frac{1}{12} & -\frac{1}{3} \end{bmatrix} = \begin{bmatrix} 2 & 3 & 1 \end{bmatrix} \end{align}