Identity Collineations of Projective Planes

# Identity 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 $\phi_M(\mathbf{x}) = \mathbf{x}M$.

Now recall that the projective plane $\mathbb{P}^2(F)$ is composed of homogeneous coordinates such that for any point $\mathbf{x} = [x_1, x_2, x_3] \in \mathbb{P}^2(F)$, $x_1, x_2, x_3$ not all zero, and for any $k \in F$, $k \neq 0$ we have that:

(1)
\begin{align} \quad \mathbf{x} = [x_1, x_2, x_3] = [kx_1, kx_2, kx_3] = k\mathbf{x} \end{align}

Now consider the $3 \times 3$ identity matrix, $I_3 = \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix}$. Then the map $\phi_{I_3} : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ is given for all $\mathbf{x} = [x_1, x_2, x_3] \in \mathbb{P}^2(F)$ by:

(2)
\begin{align} \quad \phi_{I_3} (\mathbf{x}) = \mathbf{x}I_3 = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1 \end{bmatrix} =\begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} = \mathbf{x} \end{align}

Furthermore, for any $k \in F$ and $k \neq 0$, we have that $kI_3 = \begin{bmatrix} k & 0 & 0\\ 0 & k & 0\\ 0 & 0 & k \end{bmatrix}$, and the map $\phi_{kI_3} : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ is given for all $\mathbf{x} = [x_1, x_2, x_3] \in \mathbb{P}^2(F)$ by:

(3)
\begin{align} \quad \phi_{kI_3} (\mathbf{x}) = \mathbf{x} kI_3 = \begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} \begin{bmatrix} k & 0 & 0\\ 0 & k & 0\\ 0 & 0 & k \end{bmatrix} =\begin{bmatrix} kx_1 & kx_2 & kx_3 \end{bmatrix} =\begin{bmatrix} x_1 & x_2 & x_3 \end{bmatrix} = \mathbf{x} \end{align}

Therefore $\mathbf{x}$ is mapped to itself once again.

 Definition: Let $F$ be a field and $\mathbb{P}^2(F)$ be the projective plane over $F$. The class of Identity Collineations is the set of bijective functions $\phi_{kI_3} : \mathbb{P}^2(F) \to \mathbb{P}^2(F)$ defined for all $\mathbf{x} \in \mathbb{P}^2(F)$ by $\phi_{kI_3}(\mathbf{x}) = \mathbf{x}kI_3$ for all $k \in F$ and $k \neq 0$.

Of course, these identity collineations aren't all that interesting, but it is important to note their existence.