Simultaneous diagonalization of conics in $PG(2,q)$

Abstract

Consider two symmetric $3 \times 3$ matrices $A$ and $B$ with entries in $GF(q)$, for $q=p^n$, $p$ an odd prime. The zero sets of $v^T Av$ and $v^T Bv$, for $v \in GF(q)^3$ and $v\ne 0$, can be viewed as (possibly degenerate) conics in $PG(2,q)$, the Desarguesian plane of order $q$. Using combinatorial properties of pencils of conics in $PG(2,q)$, we are able to tell when it is possible to find a regular matrix $S$ with entries in $GF(q)$, such that $S^T A S$ and $S^T BS$ are both diagonal matrices. This is equivalent to the existence of a collineation, which maps two given conics into two conics in diagonal form. For two proper conics, we will in particular compare the situation in $PG(2,q)$ to the real projective plane and compare the geometrical properties of being diagonalizable with our combinatorial results.