Collineations of Projective Planes on Lines

Recall from the Collineations of Projective Planes on Points page that if $F$ is any field, $\mathbb{P}^2(F)$ is the projective plane over $F$, and $M$ is any $3 \times 3$ invertible matrix whose entries are from $F$ 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 points $\mathbf{x} \in \mathbb{P}^2(F)$ by $\phi_M(\mathbf{x}) = \mathbf{x}M$.

From the duality of points and lines, we see that we can also apply the collineation $\phi_M$ to a line $\mathbf{a} = <a_1, a_2, a_3>$. Notice though that we define $\phi_M$ differently in the sense that lines are subsets of $\mathbb{P}^2(F)$, so $\phi_M : \mathcal P(\mathbb{P}^2(F)) \to \mathcal P(\mathbb{P}^2(F))$ is given for all lines $\mathbf{a} = <a_1, a_2, a_3>$ by:

(1)

\begin{align} \quad \phi_M(\mathbf{a}) = M^{-1} \mathbf{a} \end{align}

Hence if $\mathbf{x} = [x_1, x_2, x_3]$ in on the line $\mathbf{a} = <a_1, a_2, a_3>$ then $\mathbf{x} \cdot \mathbf{a} = 0$ and then $\phi_M(\mathbf{x})$ will be a point on the line $\phi_M(\mathbf{a})$ since:

(2)

\begin{align} \quad \phi_M(\mathbf{x}) \cdot \phi_M(\mathbf{a}) = \mathbf{x}M \cdot M^{-1} \mathbf{a} = \mathbf{x} \cdot \mathbf{a} = 0 \end{align}

Example 1

Consider the projective plane $\mathbb{P}^2(\mathbb{R})$, the matrix $M = \begin{bmatrix} 1 & 0 & 0\\ 1 & 0 & 2\\ 0 & 2 & 1 \end{bmatrix}$ and the line $\mathbf{a} = <1, 3, 4>$. Find $\phi_M(\mathbf{a})$.

$M$ is invertible (verify this) and is given by:

(3)

\begin{align} \quad M^{-1} = \begin{bmatrix} 1 & 0 & 0\\ \frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix} \end{align}

So then:

(4)

\begin{align} \quad \phi_M(\mathbf{a}) = \quad M^{-1} = \begin{bmatrix} 1 & 0 & 0\\ \frac{1}{4} & -\frac{1}{4} & \frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} & 0 \end{bmatrix} \begin{bmatrix} 1\\ 3\\ 4 \end{bmatrix} = \begin{bmatrix} 1\\ \frac{3}{2}\\ 1 \end{bmatrix} \end{align}

Hence $\phi_M(\mathbf{a}) = \left <1, \frac{3}{2}, 1 \right >$.