PolSys seminar
Spaces of matrices with rank bounded above
Vendredi 20 juin 2014Clément de Seguins Pazzis ( Lycéee Sainte-Genevièeve, Versailles, & Université de Versailles-Saint-Q
Let V be a linear subspace of the space M_{n,p}(K) of all n by p matrices with entries in the field K, with n geq p. The upper-rank of V is defined as the maximal rank among the matrices of V. In this talk, we shall review some contemporary advances to the theory of subspaces with given upper-rank r. One of the theory's basic results, proved by Flanders and Meshulam, states that a subspace with upper-rank r has dimension at most nr, and then goes on to classify the spaces with the critical dimension nr. In the 1980's, Atkinson and Lloyd further investigated the spaces with upper-rank r whose dimension is close to the critical one; we shall explain a recent generalization of their results to fields with small cardinality. We will include a short discussion of recent advances to a lesser known part of the theory, namely the one of primitive spaces with upper-rank r.
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