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