Invertible symmetric 3 × 3 binary matrices and GQ(2, 4)

Andrea Blunck, Péter Lévay, Metod Saniga, Péter Vrana

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Abstract

We reveal an intriguing connection between the set of 27 (disregarding the identity) invertible symmetric 3 × 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 physics application of our findings is also mentioned.

Original languageEnglish
Pages (from-to)1143-1154
Number of pages12
JournalLinear and Multilinear Algebra
Volume60
Issue number10
DOIs
Publication statusPublished - Oct 1 2012

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Keywords

  • binary matrices of order 3 - GQ(2, 4) - PG(5, 2)
  • quadratic forms; symplectic polarity

ASJC Scopus subject areas

  • Algebra and Number Theory

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