On the Ranks of Skew Centrosymmetric Matrices over Finite Fields

Over a fieldFof characteristic ≠2, consider then×nmatrices that are skew and also symmetric around the secondary diagonal. For evenn, a suitable similarity transformation takes them precisely to the matrices of the form . With minor modification, this is true also for oddn. This result yields a formula for the number of them of any particular rank over finiteF.