Subspaces of matrices with special rank properties

Jean-Guillaume Dumas (1, 2) , Roderick Gow (1) , Gary Mcguire (1) , John Sheekey (1)

1 CSI - Claude Shannon Institute
2 CASYS - Calculs Algébriques et Systèmes Dynamiques

Let K be a field and let V be a vector space of finite dimension n over K. We investigate properties of a subspace M of EndK (V ) of dimension n(n − r + 1) in which each non-zero element of M has rank at least r and show that such subspaces exist if K has a cyclic Galois extension of degree n. We also investigate the maximum dimension of a constant rank r subspace of EndK (V ) when K is finite.

Mots clés: Constant rank subspace; Rank; Matrix; Skew-symmetric matrix; Character

Domaines: Calcul formel [cs.SC]; Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]

Jean-Guillaume Dumas, Roderick Gow, Gary Mcguire, John Sheekey. Subspaces of matrices with special rank properties. Linear Algebra and its Applications, 2010, 433 (1), pp.191-202. ⟨10.1016/j.laa.2010.02.015⟩. ⟨hal-00699674⟩