Simultaneous Diagonalization of Conics in $PG(2,q)$

TL;DR

This paper investigates conditions under which two conics in a finite projective plane can be simultaneously diagonalized via a collineation, linking algebraic matrix transformations to geometric properties of conics.

Contribution

It provides criteria for simultaneous diagonalization of conics in finite projective planes using combinatorial properties of pencils of conics.

Findings

Criteria for the existence of a nonsingular matrix S over GF(q) for simultaneous diagonalization.

Comparison of conic properties in PG(2,q) and the real projective plane.

Insights into collineations transforming conics into diagonal form.

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 $v^T Bv$ can be viewed as (possibly degenerate) conics in the finite projective coordinate 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 nonsingular 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 mapping two given conics into conics with matrices in diagonal form. For two proper conics, we will in particular compare the situation in $PG(2,q)$ to the real projective plane and point out some differences.