Invertible Symmetric 3 x 3 Binary Matrices and GQ(2,4)

Andrea Blunck (1) , Peter Levay (2) , Metod Saniga (3) , Peter Vrana (2)

1 Department of Mathematics
2 BME - Budapest University of Technology and Economics [Budapest]
3 Astronomical Institute of the Slovak Academy of Sciences

We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3 \times 3 matrices over GF(2) and the points of the generalized quadrangle GQ(2,4). The 15 matrices with eigenvalue one correspond to a copy of the subquadrangle GQ(2,2), whereas the 12 matrices without eigenvalues have their geometric counterpart in the associated double-six. The fine details of this correspondence, including the precise algebraic meaning/analogue of collinearity, are furnished by employing the representation of GQ(2,4) as a quadric in PG(5,2) of projective index one. An interesting physical application of our findings is also mentioned.

Mots clés: Binary Matrices of Order 3; GQ(2; 4); PG(5; 2): Quadratic Forms and Symplectic Polarity

Domaines: Géométrie algébrique [math.AG]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Andrea Blunck, Peter Levay, Metod Saniga, Peter Vrana. Invertible Symmetric 3 x 3 Binary Matrices and GQ(2,4). Linear and Multilinear Algebra, 2012, 60, pp.1143-1154. ⟨hal-00512669⟩